variational convex analysis - virginia tech · duality for micro-magnetism, composites in...

Variational Convex Analysis

Fabio Silva Botelho

Dissertation submitted to the Faculty ofVirginia Polytechnic Institute and State University

in partial fulfillment of requirements for the degree of

Doctor of Philosophyin

Mathematics

Committee MembersRobert C. Rogers, Chair

Jeffrey T. BorggaardGeorge A. HagedornJames E. Thomson

July 15, 2009Blacksburg, Virginia

Keywords: calculus of variations, Banach spaces, duality, convex formulationsCopyright 2009, Fabio Silva Botelho

Variational Convex Analysis

Fabio Silva Botelho

(ABSTRACT)

This work develops theoretical and applied results for variational convex analysis. First wepresent the basic tools of analysis necessary to develop the core theory and applications.New results concerning duality principles for systems originally modeled by non-linear dif-ferential equations are shown in chapters 9 to 17. A key aspect of this work is that althoughthe original problems are non-linear with corresponding non-convex variational formulations,the dual formulations obtained are almost always concave and amenable to numerical com-putations. When the primal problem has no solution in the classical sense, the solution ofdual problem is a weak limit of minimizing sequences, and the evaluation of such averagebehavior is important in many practical applications. Among the results we highlight thedual formulations for micro-magnetism, phase transition models, composites in elasticityand conductivity and others. To summarize, in the present work we introduce convex anal-ysis as an interesting alternative approach for the understanding and computation of someimportant problems in the modern calculus of variations.

This work received partial support from Federal University of Pelotas, Pelotas-RS, Brasil.

Acknowledgments

I am especially grateful to Professor Robert C. Rogers by his excellent work as advisor.Also I would like to thank the Department of Mathematics by its constant support andthis opportunity of studying mathematics in advanced level. Finally, I am also gratefulto all Professors that have been teaching during the last years, by their valuable work.Among the Professors, I particularly thank Professors Martin Day (Calculus of Variations),James Thomson (Real Analysis) and George Hagedorn (Functional Analysis) by the excellentlectured courses.

iii

Contents

1 Introduction 1

1.1 Summary of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Duality Applied to a Plate Model . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Duality Applied to Finite Elasticity . . . . . . . . . . . . . . . . . . . 2

1.1.3 Duality Applied to a Shell Model . . . . . . . . . . . . . . . . . . . . 2

1.1.4 Duality Applied to Phase transitions . . . . . . . . . . . . . . . . . . 2

1.1.5 Duality Applied to Conductivity in Composites . . . . . . . . . . . . 3

1.1.6 Duality Applied to the Optimal Design in Elasticity . . . . . . . . . . 3

1.1.7 Duality Applied to Micro-Magnetism . . . . . . . . . . . . . . . . . . 3

1.1.8 Duality Applied to Fluid Mechanics . . . . . . . . . . . . . . . . . . . 3

1.1.9 Duality Applied to a Beam Model . . . . . . . . . . . . . . . . . . . . 4

2 Topological Vector Spaces 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Some Properties of Topological Vector Spaces . . . . . . . . . . . . . . . . . 9

2.4 Compactness in Topological Vector Spaces . . . . . . . . . . . . . . . . . . . 11

2.5 Normed and Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Linearity and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8 Some Classical Results in Banach Spaces . . . . . . . . . . . . . . . . . . . . 16

iv

CONTENTS v

3 The Hahn-Banach Theorems and Weak Topologies 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 The Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Weak Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 The Weak-star Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Weak-star Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 Separable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Measure and Integration 29

4.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Integration of Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Distributions 37

5.1 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Differentiation of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Lebesgue and Sobolev Spaces 42

6.1 Definition and Properties of Lp Spaces . . . . . . . . . . . . . . . . . . . . . 42

6.1.1 Spaces of Continuous Functions . . . . . . . . . . . . . . . . . . . . . 47

6.2 The Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3 The Sobolev Imbedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . 52

6.3.1 The Statement of Sobolev Imbedding Theorem . . . . . . . . . . . . . 53

6.4 The Rellich-Kondrachov Theorem . . . . . . . . . . . . . . . . . . . . . . . . 54

7 Basic Concepts on Convex Analysis 58

7.1 Convex Sets and Convex Functions . . . . . . . . . . . . . . . . . . . . . . . 58

7.2 Duality in Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.3 Relaxation for the Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . 69

CONTENTS vi

7.4 Duality Suitable for the Vectorial Case . . . . . . . . . . . . . . . . . . . . . 76

8 Constrained Variational Optimization 80

8.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.3 Lagrange Multiplier Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 85

9 Duality Applied to a Plate Model 86

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

9.2 The Primal Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . 90

9.3 The Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9.4 The Classical Dual Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 94

9.4.1 The Polar Functional Related to F : U → R . . . . . . . . . . . . . . 98

9.4.2 The First Duality Principle . . . . . . . . . . . . . . . . . . . . . . . 99

9.5 The Second Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9.6 The Third Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.7 A Convex Dual Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.8 A Final Result, Other Sufficient Conditions of Optimality . . . . . . . . . . . 109

9.9 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

10 Duality Applied to Elasticity 114

10.1 Introduction and Primal Formulation . . . . . . . . . . . . . . . . . . . . . . 114

10.2 The Duality Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

10.3 Final Results, Sufficient Conditions of Optimality . . . . . . . . . . . . . . . 119

11 Duality Applied to a Membrane Shell Model 121



11.3 The Polar Functional Related to F : U → R . . . . . . . . . . . . . . . . . . 124

11.4 The Final Format of First Duality Principle . . . . . . . . . . . . . . . . . . 124

CONTENTS vii

11.5 The Second Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 125

11.6 The Convex Primal Dual Formulation . . . . . . . . . . . . . . . . . . . . . . 128

11.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

12 Duality Applied to Phase Transition Problems 130

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

12.2 Existence of Solution for the Ginzburg-Landau Equation . . . . . . . . . . . 132

12.3 Convex Dual Formulations for the Ginzburg-Landau Equation . . . . . . . . 134

12.4 Applications to Phase Transition in Polymers . . . . . . . . . . . . . . . . . 140

12.4.1 Another Two Phase Model in Polymers . . . . . . . . . . . . . . . . 142

12.5 The Multi-Well Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

12.5.1 The Primal Variational Formulation . . . . . . . . . . . . . . . . . . . 144

12.5.2 A Scalar Multi-Well Formulation . . . . . . . . . . . . . . . . . . . . 146

12.5.3 An Example - A Two-dimensional Two-Well Problem . . . . . . . . 147

12.6 Duality Suitable for Vectorial Variational Problems . . . . . . . . . . . . . . 149

12.6.1 The Multi-Well Formulation Applied to Phase Transitions . . . . . . 149

12.6.2 A More Complex Phase Transition Problem . . . . . . . . . . . . . . 151

12.7 Another Multi-Well Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

12.8 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

12.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

13 Duality Applied to Conductivity in Composites 159

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

13.2 The Primal Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

13.3 The Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

13.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

14 Duality Applied to the Optimal Design in Elasticity 163

14.1 Optimal Design of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

14.1.1 The First Duality Principle . . . . . . . . . . . . . . . . . . . . . . . 164

CONTENTS viii

14.2 Optimal Design in Three-Dimensional Elasticity . . . . . . . . . . . . . . . . 166

14.3 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

14.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

15 Duality Applied to Micro-Magnetism 171

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

15.2 The Primal formulations and the Duality Principles . . . . . . . . . . . . . . 172

15.2.1 Summary of Results for the Hard Uniaxial Case . . . . . . . . . . . . 172

15.2.2 The Results for the Full Semi-linear Case . . . . . . . . . . . . . . . . 173

15.3 A Preliminary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

15.4 The Duality Principle for the Hard Case . . . . . . . . . . . . . . . . . . . . 174

15.5 The Full Semi-linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

15.6 Final Results, Convex Dual Formulations . . . . . . . . . . . . . . . . . . . . 182

15.7 The Cubic Case in Micro-magnetism . . . . . . . . . . . . . . . . . . . . . . 185

15.7.1 The Primal Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 186

15.7.2 The Duality Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 186

15.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

16 Duality Applied to Fluid Mechanics 189



16.3 The Dual Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . 191

16.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

17 Duality Applied to a Beam Model 196

17.1 Introduction and Statement of Primal Formulation . . . . . . . . . . . . . . 196

17.2 Existence and Regularity Results for Problem P1 . . . . . . . . . . . . . . . 197

17.3 A Convex Dual Formulation for the Beam Model . . . . . . . . . . . . . . . 199

17.4 A Necessary Condition for Problem P2 . . . . . . . . . . . . . . . . . . . . . 201

17.5 A Similar Two-dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . 202

CONTENTS ix

17.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

List of Figures

12.1 Vertical axis: u0(x)-weak limit of minimizing sequences for f(x)=0 . . . . . . . . 158

12.2 Vertical axis: u0(x)-weak limit of minimizing sequences for f(x) = 0.3 ∗ Sin(π ∗ x) 158

12.3 Vertical axis: u0(x)-weak limit of minimizing sequences for f(x) = 0.3 ∗ Cos(π ∗ x) 158

14.1 Vertical axis: solution t(x, y) for the dual problem with∫S tdS ≤ 0.68 . . . . . . . 168

14.2 Vertical axis: Field of displacements w0(x, y) (in m) for the dual problem, with∫S tdS ≤ 0.68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

14.3 Vertical axis: solution t(x, y) for the dual problem with∫S tdS ≤ 0.60 . . . . . . . 169

17.1 Vertical axis: solution u0(x, y) for the dual problem . . . . . . . . . . . . . . . . 203

17.2 Vertical axis: solution u0(x, y) for the dual problem . . . . . . . . . . . . . . . . 203

x

Chapter 1

Introduction

The main objective of this work is to present recent results of the author about applicationsof duality to non-convex problems in the calculus of variations. The text is divided intochapters described in the next page, and chapters 2 to 8 present the basic concepts onstandard analysis necessary to develop the applications.

Of course, the material presented in the first 8 chapters is not new, with exception of thesection on relaxation for the scalar case, where we show different proofs of some theoremspresented in Ekeland and Temam’s book Convex Analysis and Variational Problems (indeedsuch a book is the theoretical base of the present work), and the section about relaxationfor vectorial case. The applications, presented in chapters 9 to 17, correspond to the work ofthe present author along the last years, and almost all results including the applications ofduality for micro-magnetism, composites in elasticity and conductivity and phase transitions,were obtained during the PhD program at Virginia Tech.

The key feature of this work is that while all problems studied here are non-linear withcorresponding non-convex variational formulation, it has been almost always possible todevelop convex (in fact concave) dual variational formulations, which in general are moreamenable to numerical computations.

The section on relaxation for the vectorial case, as its title suggests, presents duality prin-ciples that are valid even for vectorial problems. It is worth noting that such results wereused within the text to develop concave dual variational formulations in situations such asfor conductivity in composites, vectorial examples in phase transitions, etc.

1.1 Summary of Main Results

The main results of this work are summarized as follows.

1

CHAPTER 1. INTRODUCTION 2

1.1.1 Duality Applied to a Plate Model

Chapter 9 develops dual variational formulations for the two dimensional equations of thenonlinear elastic Kirchhoff-Love plate model. The first duality principle presented is theclassical one and may be found in similar format in Telega [33], Gao [18]. It is worthnoting that such results are valid only for positive definite membrane forces. However, weobtain new dual variational formulations which relax or even remove such constraints. Inparticular we exhibit a convex dual variational formulation which allows non positive definitemembrane forces. In the last section, similar to the triality criterion introduced in Gao [20],we obtain sufficient conditions of optimality for the present case. The results are basedon fundamental tools of Convex Analysis and the Legendre Transform, which can easily beanalytically expressed for the model in question.

1.1.2 Duality Applied to Finite Elasticity

Chapter 10 develops duality for a model in finite elasticity. The dual formulations obtainedallow the matrix of stresses to be non positive definite. This is in some sense, an extensionof earlier results (which establish the complementary energy as a perfect global optimizationduality principle only if the stress tensor is positive definite at the equilibrium point). Also, itis important to emphasize that one of the formulations obtained is convex. Again, the resultsare based on standard tools of convex analysis and on the concept of Legendre Transform.

1.1.3 Duality Applied to a Shell Model

The main focus of Chapter 11 is the development of dual variational formulations for a non-linear elastic membrane shell model. In the present literature, the concept of complementaryenergy can be established only if the external load produces a critical point with positivedefinite membrane forces matrix. Our idea is to obtain dual variational formulations forwhich the mentioned constraint is relaxed or even eliminated. It is important to emphasizethat one of the dual (in fact, primal dual) formulations here presented is convex. Again,the results are obtained through basic tools of convex analysis and the concept of LegendreTransform, which can be analytically established for the concerned shell model.

1.1.4 Duality Applied to Phase transitions

The first part of Chapter 12 is concerned with the development of dual variational formula-tions for Ginzburg-Landau type equations. Since the primal formulations are non-convex, weuse specific results for distance between two convex functions to obtain the dual approaches.Note that we obtain two different convex dual formulations (in fact one of them is a kind


of primal-dual formulation). As a second objective we present duality as an alternativeperspective to address the multi-well and related phase transition problems. In these lattercases the dual formulations are concave and the solution of the dual problems reflect theaverage behavior of minimizing sequences, as weak cluster points, considering that we maynot have minimizers in the classical sense through the primal approaches.

1.1.5 Duality Applied to Conductivity in Composites

The main focus of Chapter 13 is the development of a dual variational formulation for atwo-phase optimization problem in conductivity. The primal formulation may not haveminimizers in the classical sense. In this case, the solution through the dual formulationmay be a weak limit of minimizing sequences for the original problem.

1.1.6 Duality Applied to the Optimal Design in Elasticity

The first part of Chapter 14 develops a dual variational formulation for the optimal designof a plate of variable thickness. The design variable, namely the plate thickness, is supposedto minimize the plate deformation work due to a given external load. The second part isconcerned with the optimal design for a two-phase problem in elasticity. In this case, we arelooking for the mixture of two constituents that minimizes the structural internal work. Inboth applications the dual formulations were obtained through basic tools of convex analysis.

1.1.7 Duality Applied to Micro-Magnetism

The main focus of Chapter 15 is the development of dual variational formulations for func-tionals related to ferromagnetism models. We develop duality principles for the so-calledhard and full (semi-linear) uniaxial and cubic cases. It is important to emphasize that thenew dual formulations here presented are convex and are useful to compute the averagebehavior of minimizing sequences, specially as the primal formulation has no minimizers inthe classical sense. The results are obtained through standard tools of convex analysis.

1.1.8 Duality Applied to Fluid Mechanics

In Chapter 16 we use the concept of Legendre Transform to obtain dual variational formu-lations for the Navier-Stokes system.


1.1.9 Duality Applied to a Beam Model

Chapter 17 develops existence, duality and numerical results for a non-linear beam model.Our final result is a convex variational formulation for the concerned beam model.

Chapter 2

Topological Vector Spaces

2.1 Introduction

The main objective of this chapter is to present an outline of the basic tools of analysisnecessary to develop the subsequent chapters. We assume the reader has a background inlinear algebra and elementary real analysis at an undergraduate level. Some short proofsare given but many are left to other sources. More details on this subject may be found inChapter 1 of ”Functional Analysis” by W. Rudin (reference [31]).

2.2 Vector Spaces

We denote by F a scalar field. In practice this is either R or C, the set of real or complexnumbers.

Definition 2.2.1 (Vector Spaces). A vector space over F is a set denoted by U , whoseelements are called vectors, for which are defined two operations, namely, addition denotedby (+) : U × U → U , and, scalar multiplication, denoted by (.) : F × U → U , so that thefollowing relations are valid

1. u + v = v + u,∀u, v ∈ U,

2. u + (v + w) = (u + v) + w, ∀u, v, w ∈ U,

3. there exists a vector denoted by θ such that u + θ = u,∀u ∈ U,

4. for each u ∈ U, there exists a unique vector denoted by − u such that u + (−u) = θ,

5. α.(β.u) = (α.β).u, ∀α, β ∈ F, u ∈ U,

5

CHAPTER 2. TOPOLOGICAL VECTOR SPACES 6

6. α.(u + v) = α.u + β.u, ∀α ∈ F, u, v ∈ U,

7. (α + β).u = α.u + β.u, ∀α, β ∈ F, u ∈ U,

8. 1.u = u,∀u ∈ U.

Remark 2.2.2. Now and on we will drop the dot (.) in scalar multiplication and denoteα.u as αu.

Definition 2.2.3 (Vector Subspace). Let U be a vector space. A set V ⊂ U is said to be avector subspace of U if V is also a vector space with the same operations as those of U . IfV 6= U we say that V is a proper subspace of U .

Definition 2.2.4 (Finite dimensional Space). A vector space is said to be of finite dimensionif there exists fixed u1, u2, ..., un ∈ U such that for each u ∈ U there are correspondingα1, ...., αn ∈ F for which

u =n∑

i=1

αiui. (2.1)

Definition 2.2.5 (Topological Spaces). A set U is said to be a topological space if it ispossible to define a collection σ of subsets of U called a topology in U , for which are validthe following properties:

1. U ∈ σ,

2. ∅ ∈ σ,

3. if A ∈ σ and B ∈ σ then A ∩B ∈ σ, and

4. arbitrary unions of elements in σ also belong to σ.

Any A ∈ σ is said to be an open set.

Remark 2.2.6. When necessary, to clarify the notation, we shall denote the vector space Uendowed with the topology σ by (U, σ).

Definition 2.2.7 (Closed Sets). Let U be a topological space. A set A ⊂ U is said to beclosed if U − A is open. We also denote U − A = Ac.

Proposition 2.2.8. For closed sets we have the following properties:

1. U and ∅ are closed,

2. If A and B are closed sets then A ∪B is closed,

3. Arbitrary intersections of closed sets are closed.


Proof:

1. Since ∅ is open and U = ∅c, by Definition 2.2.7 U is closed. Similarly, since U is openand ∅ = U − U = U c, ∅ is closed.

2. A,B closed implies that Ac and Bc are open, and by Definition 2.2.5, Ac ∪Bc is open,so that A ∩B = (Ac ∪Bc)c is closed.

3. Consider A = ∩λ∈LAλ, where L is a collection of indices and Aλ is open, ∀λ ∈ L. Wemay write A = (∪λ∈LAc

λ)c and since Ac

λ is open ∀λ ∈ L we have, by Definition 2.2.5,that A is closed. ¤

Definition 2.2.9 (Closure). Given A ⊂ U we define the closure of A, denoted by A, as theintersection of all closed sets that contain A.

Remark 2.2.10. From Proposition 2.2.8 Item 3 we have that A is the smallest closed setthat contains A, in the sense that, if C is closed and A ⊂ C then A ⊂ C.

Definition 2.2.11 (Interior). Given A ⊂ U we define its interior, denoted by A, as theunion of all open sets contained in A.

Remark 2.2.12. It is not difficult to prove that if A is open then A = A.

Definition 2.2.13 (Neighborhood). Given u0 ∈ U we say that V is a neighborhood of u0 ifsuch a set is open and contains u0. We denote such neighborhoods by Vu0.

Proposition 2.2.14. If A ⊂ U is a set such that for each u ∈ A there exists a neighborhoodVu 3 u such that Vu ⊂ A, then A is open.

Proof: This follows from the fact that A = ∪u∈UVu and arbitrary union of open sets areopen. ¤

Definition 2.2.15 (Function). Let U and V be two topological spaces. We say that f : U →V is a function if f is a collection of pairs (u, v) ∈ U × V such that for each u ∈ U thereexists only one v ∈ V such that (u, v) ∈ f .

Definition 2.2.16 (Continuity at a Point). A function f : U → V is continuous at u ∈ Uif for each neighborhood V2(f(u)) there exists a neighborhood V1(u) such that f(V1(u)) ⊂V2(f(u)).

Definition 2.2.17 (Continuous Function). A function f : U → V is continuous if it iscontinuous at each u ∈ U .

Proposition 2.2.18. A function f : U → V is continuous if and only if f−1(V) is open foreach open V ⊂ V , where

f−1(V) = u ∈ U | f(u) ∈ V. (2.2)


Proof: Suppose f−1(V) is open whenever V ⊂ V is open. Pick u ∈ U and any V suchthat f(u) ∈ V . Since u ∈ f−1(V) and f(f−1(V)) = V , we have that f is continuous atu ∈ U . Since u ∈ U is arbitrary we have that f is continuous. Conversely, suppose f iscontinuous and pick V ⊂ V open. If f−1(V) = ∅ we are done, since ∅ is open,. Thus, supposeu ∈ f−1(V), since f is continuous, there exists Vu a neighborhood of u such that f(Vu) ⊂ V .This means Vu ⊂ f−1(V) and therefore, from Proposition 2.2.14, f−1(V) is open. ¤Definition 2.2.19. We say that (U, σ) is a Hausdorff topological space if, given u1, u2 ∈ U ,u1 6= u2, there exists V1, V2 ∈ σ such that

u1 ∈ V1 , u2 ∈ V2 and V1 ∩ V2 = ∅. (2.3)

Definition 2.2.20 (Base). A collection σ′ ⊂ σ is said to be a base for σ if every element ofσ may be represented as a union of elements of σ′.

Definition 2.2.21 (Local Base). A collection σ of neighborhoods of a point u ∈ U is saidto be a local base at u if each neighborhood of u contains a member of σ.

Definition 2.2.22 (Topological Vector Space). A vector space endowed with a topology,denoted by (U, σ), is said to be a topological vector space if and only if

1. Every single point of U is a closed set,

2. The vector space operations (addition and scalar multiplication) are continuous withrespect to σ.

More specifically, addition is continuous if, given u, v ∈ U and V ∈ σ such that u + v ∈ Vthen there exists Vu 3 u and Vv 3 v such that Vu + Vv ⊂ V. On the other hand, scalarmultiplication is continuous if given α ∈ F, u ∈ U and V 3 α.u, there exists δ > 0 andVu 3 u such that, ∀β ∈ F satisfying |β − α| < δ we have βVu ⊂ V.

Given (U, σ), let us associate with each u0 ∈ U and α0 ∈ F (α0 6= 0) the functions Tu0 : U →U and Mα0 : U → U defined by

Tu0(u) = u0 + u (2.4)

andMα0(u) = α0.u. (2.5)

The continuity of such functions is a straightforward consequence of the continuity of vectorspace operations (addition and scalar multiplication). It is clear that the respective inversemaps, namely T−u0 and M1/α0 are also continuous. So if V is open then u0 + V , that is(T−u0)

−1(V) = Tu0(V) = u0 + V is open. By analogy α0V is open. Thus σ is completelydetermined by a local base, so that the term local base will be understood henceforth as alocal base at 0. So to summarize, a local base of a topological vector space is a collection Ωof neighborhoods of 0, such that each neighborhood of 0 contains a member of Ω.

Now we present some simple results, namely:


Proposition 2.2.23. If A ⊂ U is open, then ∀u ∈ A there exists a neighborhood V of θ suchthat u + V ⊂ A

Proof : Just take V = A− u. ¤Proposition 2.2.24. Given a topological vector space (U, σ), any element of σ may beexpressed as a union of translates of members of Ω, so that the local base Ω generates thetopology σ.

Proof: Let A ⊂ U open and u ∈ U . V = A − u is a neighborhood of θ and by definition oflocal base, there exists a set VΩu ⊂ V such that VΩu ∈ Ω. Thus, we may write

A = ∪u∈A(u + VΩu). ¤ (2.6)

2.3 Some Properties of Topological Vector Spaces

In this section we study some fundamental properties of topological vector spaces. We startwith the following proposition:

Proposition 2.3.1. Any topological vector space U is a Hausdorff space.

Proof: Pick u0, u1 ∈ U such that u0 6= u1. Thus V = U −u1−u0 is an open neighborhoodof zero. As θ + θ = θ, by the continuity of addition, there exist V1 and V2 neighborhoods ofθ such that

V1 + V2 ⊂ V (2.7)

define U = V1 ∩ V2 ∩ (−V1) ∩ (−V2), thus U = −U (symmetric) and U + U ⊂ V and hence

u0 + U + U ⊂ u0 + V ⊂ U − u1 (2.8)

so thatu0 + v1 + v2 6= u1, ∀v1, v2 ∈ U , (2.9)

oru0 + v1 6= u1 − v2, ∀v1, v2 ∈ U , (2.10)

and since U = −U(u0 + U) ∩ (u1 + U) = ∅. ¤ (2.11)

Definition 2.3.2 (Bounded Sets). A set A ⊂ U is said to be bounded if to each neighborhoodof zero V there corresponds a number s > 0 such that A ⊂ tV for each t > s.

Definition 2.3.3 (Convex Sets). A set A ⊂ U such that

if u, v ∈ A then λu + (1− λ)v ∈ A, ∀λ ∈ [0, 1], (2.12)

is said to be convex.


Definition 2.3.4 (Locally Convex Spaces). A topological vector space U is said to be locallyconvex if there is a local base Ω whose elements are convex.

Definition 2.3.5 (Balanced sets). A set A ⊂ U is said to be balanced if αA ⊂ A, ∀α suchthat 0 < |α| < 1.

Theorem 2.3.6. In a topological vector space U we have:

1. Every neighborhood of zero contains a balanced neighborhood of zero,

2. Every convex neighborhood of zero contains a balanced convex neighborhood of zero.

Proof:

1. Suppose U is a neighborhood of zero. From the continuity of scalar multiplication,there exist V (neighborhood of zero) and δ > 0, such that αV ⊂ U whenever |α| < δ.Define W = ∪|α|<δαV , thus W ⊂ U is a balanced neighborhood of zero.

2. Suppose U is a convex neighborhood of zero in U . Define

A = ∩αU | α ∈ C, |α| = 1. (2.13)

As 0.θ = θ (where θ ∈ U denotes the zero vector) from the continuity of scalarmultiplication there exists δ > 0 and there is a neighborhood of zero V such that if|β| < δ then βV ⊂ U . Define W as the union of all such βV . Thus W is balanced andα−1W = W as |α| = 1, so that W = αW ⊂ αU , and hence W ⊂ A, which impliesthat the interior A is a neighborhood of zero. Also A ⊂ U . Since A is intersection ofconvex sets, it is convex and so is A. Now will show that A is balanced and completethe proof. For this, it suffices to prove that A is balanced. Choose r and β such that0 ≤ r ≤ 1 and |β| = 1. Then

rβA = ∩|α|=1rβαU = ∩|α|=1rαU . (2.14)

Since αU is a convex set that contains zero, we obtain rαU ⊂ αU , so that rβA ⊂ A,which completes the proof. ¤

Proposition 2.3.7. Let U be a topological vector space and V a neighborhood of zero in U .Given u ∈ U , there exists r ∈ R+ such that βu ∈ V, ∀β such that |β| < r.

Proof: Observe that u + V is a neighborhood of 1.u, then by the continuity of scalar multi-plication, there exists W neighborhood of u and r > 0 such that

βW ⊂ u + V ,∀β such that |β − 1| < r, (2.15)

so thatβu ∈ u + V , (2.16)


or(β − 1)u ∈ V , where |β − 1| < r, (2.17)

and thusβu ∈ V ,∀β such that |β| < r, (2.18)

which completes the proof. ¤

Corollary 2.3.8. Let V be a neighborhood of zero in U , if rn is a sequence such thatrn > 0, ∀n ∈ N and limn→∞ rn = ∞, then U ⊂ ∪∞n=1rnV.

Proof: Let u ∈ U , then αu ∈ V for α sufficiently small, from the last proposition u ∈ 1αV .

As rn →∞ we have that rn > 1α

for n sufficiently big, so that u ∈ rnV , which completes theproof. ¤

Proposition 2.3.9. Suppose δn is sequence such that δn → 0, δn < δn−1, ∀n ∈ N and Va bounded neighborhood of zero in U , then δnV is a local base for U .

Proof: Let U be a neighborhood of zero, as V is bounded, there exists t0 ∈ R+ such thatV ⊂ tU for any t > t0. As limn→∞ δn = 0, there exists n0 ∈ N such that if n ≥ n0 thenδn < 1

t0, so that δnV ⊂ U ,∀n such that n ≥ n0. ¤

Definition 2.3.10 (Convergence in topological vector spaces). Let U be a topological vectorspace. We say un converges to u0 ∈ U , if for each neighborhood V of u0 then, there existsN ∈ N such that

un ∈ V ,∀n ≥ N.

2.4 Compactness in Topological Vector Spaces

We start this section with the definition of open covering.

Definition 2.4.1 (Open Covering). Given B ⊂ U we say that Oα, α ∈ A is a coveringof B if B ⊂ ∪α∈AOα. If Oα is open ∀α ∈ A then Oα is said to be an open covering of B.

Definition 2.4.2 (Compact Sets). A set B ⊂ U is said to be compact if each open coveringof B has a finite sub-covering. More explicitly, if B ⊂ ∪α∈AOα, where Oα is open ∀α ∈ A,then, there exist α1, ..., αn ∈ A such that B ⊂ Oα1 ∪ ... ∪ Oαn, for some n, a finite positiveinteger.

Proposition 2.4.3. A compact subset of a Hausdorff space is closed.

Proof : Let U be a Hausdorff space and consider A ⊂ U , A compact. Given x ∈ A andy ∈ Ac, there exist open sets Ox and Ox

y such that x ∈ Ox, y ∈ Oxy and Ox ∩ Ox

y = ∅. Itis clear that A ⊂ ∪x∈AOx and since A is compact, we may find x1, x2, ..., xn such that


A ⊂ ∪ni=1Oxi

. For the selected y ∈ Ac we have y ∈ ∩ni=1Oxi

y and (∩ni=1Oxi

y ) ∩ (∪ni=1Oxi

) = ∅.Since ∩n

i=1Oxiy is open, and y is an arbitrary point of Ac we have that Ac is open, so that A

is closed, which completes the proof. ¤

Proposition 2.4.4. A closed subset of a compact U space is compact.

Proof: Consider Oα an open cover of A. Thus Ac, Oα, α ∈ A is a cover of U . As U iscompact, there exist α1, α2, ..., αn such that Ac∪(∪n

i=1Oαi) ⊃ U , so that Oαi

, i ∈ 1, ..., ncovers A, so that A is compact. The proof is complete. ¤

Definition 2.4.5 (Countably Compact Sets). A set A is said to be countably compact ifevery infinite subset of A has a limit point in A.

Proposition 2.4.6. Every compact subset of a topological space U is countably compact.

Proof: Let B an infinite subset of A compact and suppose B has no limit point. Choosex1, x2, .... ⊂ B and define F = x1, x2, x3, .... It is clear that F has no limit point. Thusfor each n ∈ N, there exist On open such that On ∩ F = xn. Also, for each x ∈ A − F ,there exist Ox such that x ∈ Ox and Ox ∩ F = ∅. Thus Ox, x ∈ A− F, O1,O2, ... is anopen cover of A without a finite subcover, which contradicts the fact that A is compact. ¤

2.5 Normed and Metric Spaces

We start with the definition of norm.

Definition 2.5.1 (Norm). A vector space U is said to be a normed space, if it is possibleto define a function ‖.‖U : U → R+ = [0, +∞), called a norm, which satisfies the followingproperties:

1. ‖u‖U > 0, if u 6= θ and ‖u‖U = 0 ⇔ u = θ

2. ‖u + v‖U ≤ ‖u‖U + ‖v‖U , ∀ u, v ∈ U ,

3. ‖αu‖U = |α|‖u‖U ,∀u ∈ U, α ∈ F.

Now we present the definition of metric.

Definition 2.5.2 (Metric Space). A vector space U is said to be a metric space if it ispossible to define a function d : U × U → R+, called a metric on U , such that

1. 0 ≤ d(u, v) < ∞, ∀u, v ∈ U ,

2. d(u, v) = 0 ⇔ u = v,


3. d(u, v) = d(v, u), ∀u, v ∈ U ,

4. d(u,w) ≤ d(u, v) + d(v, w), ∀u, v, w ∈ U .

A metric can be defined through a norm, that is

d(u, v) = ‖u− v‖U . (2.19)

In this case we say that the metric is induced by the norm.

The set Br(u) = v ∈ U | d(u, v) < r is called the open ball with center at u and radiusr. A metric d : U × U → R+ is said to be invariant if

d(u + w, v + w) = d(u, v),∀u, v, w ∈ U. (2.20)

The following are some basic definitions concerning metric and normed spaces:

Definition 2.5.3 (Convergent Sequences). Given a metric space U , we say that un ⊂ Uconverges to u0 ∈ U as n →∞, if given any ε > 0, there exists n0 ∈ N, such that if n ≥ n0

then d(un, u0) < ε. In this case we write un → u0 as n → +∞.

Definition 2.5.4 (Cauchy Sequence). un ⊂ U is said to be a Cauchy sequence if givenε > 0 there exist n0 ∈ N such that d(un, um) < ε, ∀m,n ≥ n0

Definition 2.5.5 (Completeness). A metric space U is said to be complete if each Cauchysequence related to d : U × U → R+ converges to an element of U .

Definition 2.5.6 (Banach Spaces). A normed vector space U is said to be a Banach Spaceif each Cauchy sequence related to the metric induced by the norm converges to an elementof U .

Remark 2.5.7. We say that a topology σ is compatible with a metric d if any A ⊂ σ isrepresented by unions and/or finite intersections of open balls. In this case we say thatd : U × U → R+ induces the topology σ.

Definition 2.5.8 (Metrizable Spaces). A topological vector space (U, σ) is said to be metriz-able if σ is compatible with some metric d.

Definition 2.5.9 (Normable Spaces). A topological vector space (U, σ) is said to be normableif the induced metric (by this norm) is compatible with σ.

2.6 Linear Mappings

Given U, V topological vector spaces, a function (mapping) f : U → V , A ⊂ U and B ⊂ V ,we define:

f(A) = f(u) | u ∈ A, (2.21)


and the inverse image of B, denoted f−1(B) as

f−1(B) = u ∈ U | f(u) ∈ B. (2.22)

Definition 2.6.1 (Linear Functions). A function f : U → V is said to be linear if

f(αu + βv) = αf(u) + βf(v),∀u, v ∈ U, α, β ∈ F. (2.23)

Definition 2.6.2 (Null Space and Range). Given f : U → V , we define the null space andthe range of f, denoted by N(f) and R(f) respectively, as

N(f) = u ∈ U | f(u) = θ (2.24)

andR(f) = v ∈ V | ∃u ∈ U such that f(u) = v. (2.25)

Note that N(f) and R(f) are subspaces of U .

Proposition 2.6.3. Let U, V be topological vector spaces. If f : U → V is linear andcontinuous at θ, then it is continuous everywhere.

Proof: Since f is linear we have f(θ) = θ. Since f is continuous at θ, given V ⊂ V aneighborhood of zero, there exists U ⊂ U neighborhood of zero, such that

f(U) ⊂ V . (2.26)

Thusv − u ∈ U ⇒ f(v − u) = f(v)− f(u) ∈ V , (2.27)

orv ∈ u + U ⇒ f(v) ∈ f(u) + V , (2.28)

which means that f is continuous at u. Since u is arbitrary, f is continuous everywhere. ¤

2.7 Linearity and Continuity

Definition 2.7.1 (Bounded Functions). A function f : U → V is said to be bounded if itmaps bounded sets into bounded sets.

Proposition 2.7.2. A set E is bounded if and only if the following condition is satisfied:whenever un ⊂ E and αn ⊂ F are such that αn → 0 as n → ∞ we have αnun → θ asn →∞.


Proof: Suppose E is bounded. Let U be a balanced neighborhood of θ in U , then E ⊂ tUfor some t. For un ⊂ E, as αn → 0, there exists N such that if n > N then t < 1

|αn| .Since t−1E ⊂ U and U is balanced, we have that αnun ∈ U , ∀n > N , and thus αnun → θ.Conversely, if E is not bounded, there is a neighborhood V of θ and rn such that rn →∞and E * rnV , that is, we can choose un such that r−1

n un is not in V , ∀n ∈ N, so that r−1n un

does not converge to θ. ¤

Proposition 2.7.3. Let f : U → V be a linear function. Consider the following statements

1. f is continuous,

2. f is bounded,

3. If un → θ then f(un) is bounded,

4. If un → θ then f(un) → θ.

Then,

• 1 implies 2,

• 2 implies 3,

• if U is metrizable then 3 implies 4, which implies 1.

Proof:

1. 1 implies 2: Suppose f is continuous, for W ⊂ V neighborhood of zero, there exists aneighborhood of zero in U , denoted by V , such that

f(V) ⊂ W . (2.29)

If E is bounded, there exists t0 ∈ R+ such that E ⊂ tV , ∀t ≥ t0, so that

f(E) ⊂ f(tV) = tf(V) ⊂ tW , ∀t ≥ t0, (2.30)

and thus f is bounded.

2. 2 implies 3: Suppose un → θ and let W be a neighborhood of zero. Then there existsN ∈ N such that if n ≥ N then un ∈ V ⊂ W where V is a balanced neighborhoodof zero. On the other hand, for n < N , there exists Kn such that un ∈ KnV . DefineK = max1, K1, ..., Kn. Then un ∈ KV ,∀n ∈ N and hence un is bounded, a finallyfrom 2, we have that f(un) is bounded.


3. 3 implies 4: Suppose U is metrizable and let un → θ. Given K ∈ N, there existsnK ∈ N such that if n > nK then d(un, θ) < 1

K2 . Define γn = 1 if n < n1 and γn = K,if nK ≤ n < nK+1 so that

d(γnun, θ) = d(Kun, θ) ≤ Kd(un, θ) < K−1. (2.31)

Thus since 2 implies 3 we have f(γnun) is bounded so that, by Proposition 2.7.2f(un) = γ−1

n f(γnun) → θ as n →∞.

4. 4 implies 1: suppose 1 fails. Thus there exists a neighborhood of zero W ⊂ V suchthat f−1(W) contains no neighborhood of zero in U . Particularly, we can select unsuch that un ∈ B1/n(θ) and f(un) not in W so that f(un) does not converge to zero.Thus 4 fails. ¤

2.8 Some Classical Results in Banach Spaces

Now we state some very important theorems in Banach spaces, which the proofs may befound in [27]. We start with the definition of nowhere dense set.

Definition 2.8.1 (Nowhere Dense Sets). A set S in a metric space M is called nowheredense if S has an empty interior.

Theorem 2.8.2 (Baire Category Theorem). A complete metric space is never the union ofa countable number of nowhere dense sets.

Theorem 2.8.3 (Principle of Uniform Boundedness). Let U be a Banach space. Let F bea family of bounded linear transformations from U to a normed linear space V . Suppose foreach u ∈ U , ‖Tu‖V | T ∈ F is bounded. Then ‖T‖ | T ∈ F is bounded.

Theorem 2.8.4 (Open Mapping Theorem). Let T : U → V be a bounded linear transfor-mation of the Banach space U onto the Banach space V . Then if M is open in U then T (M)is open in V .

Theorem 2.8.5 (Inverse Mapping Theorem). A continuous bijection of one Banach spaceonto another has a continuous inverse.

Here we introduce the definition of graph and finish this chapter by stating the Closed GraphTheorem.

Definition 2.8.6 (Graph of a Mapping). Let T be a mapping of a normed linear space Uinto a normed linear space V . The graph of T , denoted by Γ(T ), is defined as

Γ(T ) = (u, v) ∈ U × V | v = T (u).Theorem 2.8.7 (The closed Graph Theorem). Let U and V be Banach spaces and T alinear map of U into V . Then T is bounded if and only if its graph is closed.

Chapter 3

The Hahn-Banach Theorems andWeak Topologies

3.1 Introduction

The notion of weak topologies and weak convergence is fundamental in the modern varia-tional analysis. Many important problems are non-convex and have no minimizers in theclassical sense. However the minimizing sequences in reflexive spaces may be weakly con-vergent, and it is important to evaluate the average behavior of such sequences in manypractical applications.

3.2 The Hahn-Banach Theorem

In this chapter U denotes a Banach space, unless otherwise indicated. We start this sectionby stating and proving the Hahn-Banach theorem for real vector spaces, which is sufficientfor our purposes.

Theorem 3.2.1 (The Hahn-Banach Theorem). Consider a functional p : U → R satisfying

p(λu) = λp(u),∀u ∈ U, λ > 0, (3.1)

p(u + v) ≤ p(u) + p(v), ∀u, v ∈ U. (3.2)

Let V ⊂ U a vector subspace and let g : V → R be a linear functional such that

g(u) ≤ p(u),∀u ∈ V. (3.3)

Then there exists a linear functional f : U → R such that

g(u) = f(u),∀u ∈ V, (3.4)

17

CHAPTER 3. THE HAHN-BANACH THEOREMS AND WEAK TOPOLOGIES 18

andf(u) ≤ p(u),∀u ∈ U. (3.5)

Proof: Pick z ∈ U − V . Denote by V the space spanned by V and z, that is

V = v + αz | v ∈ V and α ∈ R. (3.6)

We may define an extension of g to V , denoted by g, as

g(αz + v) = αg(z) + g(v), (3.7)

where g(z) will be appropriately defined. Suppose given v1, v2 ∈ V , α > 0, β > 0. Then

βg(v1) + αg(v2) = g(βv1 + αv2)

= (α + β)g(β

α + βv1 +

α

α + βv2)

≤ (α + β)p(β

α + β(v1 − αz) +

α

α + β(v2 + αz))

≤ βp(v1 − αz) + αp(v2 + βz) (3.8)

and therefore

1

α[−p(v1 − αz) + g(v1)] ≤ 1

β[p(v2 + βz)− g(v2)],∀v1, v2 ∈ V, α, β > 0. (3.9)

Thus, there exists a ∈ R such that

supv∈V,α>0

[1

α(−p(v − αz) + g(v))] ≤ a ≤ inf

v∈V,α>0[1

α(p(v + αz)− g(v))]. (3.10)

If we define g(z) = a we obtain g(u) ≤ p(u),∀u ∈ V . Define by E the set of extensions e of g,which satisfy e(u) ≤ p(u) on the subspace where e is defined. We define a partial order in Eby setting e1 ≺ e2 if e2 is defined in a larger set than e1 and e1 = e2 where both are defined.Let eαα∈A be a linearly ordered subset of E . Let Vα be the subspace on which eα is defined.Define e on ∪α∈AVα by setting e(u) = eα on Vα. Clearly eα ≺ e so each linearly ordered setof E has an upper bound. By the Zorn’s lemma, E has a maximal element f defined on someset U such that f(u) ≤ p(u),∀u ∈ U . We can conclude that U = U , otherwise if there wasan z1 ∈ U − U , as above we could have a new extension f1 to the subspace spanned by z1

and U , contradicting the maximality of f . ¤

Definition 3.2.2 (Topological Dual Space). For a Banach space U , we define its TopologicalDual Space, as the set of all linear continuous functionals defined on U . We suppose thatsuch dual space of U , may be identified with a space denoted by U∗ through a bilinear form


〈., .〉U : U × U∗ → R. That is, given f : U → R linear continuous functional, there existsu∗ ∈ U∗ such that

f(u) = 〈u, u∗〉U , ∀u ∈ U. (3.11)

The norm of f , denoted by ‖f‖U∗, is defined as

‖f‖U∗ = supu∈U

|〈u, u∗〉U | | ‖u‖U ≤ 1. (3.12)

Corollary 3.2.3. Let V ⊂ U a vector subspace of U and let g : V → R a linear continuousfunctional of norm

‖g‖V ∗ = supu∈U

|g(u)| | ‖u‖V ≤ 1. (3.13)

Then, there exists an u∗ in U∗ such that

〈u, u∗〉U = g(u),∀u ∈ V, (3.14)

and‖u∗‖U∗ = ‖g‖V ∗ . (3.15)

Proof: Apply Theorem 3.2.1 with p(x) = ‖g‖V ∗‖u‖V . ¤

Corollary 3.2.4. Given u0 ∈ U there exists u∗0 ∈ U∗ such that

‖u∗0‖U∗ = ‖u0‖U and 〈u0, u∗0〉U = ‖u0‖2

U . (3.16)

Proof: Apply Corollary 3.2.3 with V = αu0 | α ∈ R and g(tu0) = t‖u0‖2U so that

‖g‖V ∗ = ‖u0‖U . ¤

Corollary 3.2.5. Given u ∈ U we have

‖u‖U = supu∗∈U∗

|〈u, u∗〉U | | ‖u∗‖U∗ ≤ 1. (3.17)

Proof: Suppose u 6= θ. Since

|〈u, u∗〉U | ≤ ‖u‖U‖u∗‖U∗ ,∀u ∈ U, u∗ ∈ U∗

we havesup

u∗∈U∗|〈u, u∗〉U | | ‖u∗‖U∗ ≤ 1 ≤ ‖u‖U . (3.18)

However, from last corollary we have that there exists u∗0 ∈ U∗ such that ‖u∗0‖U∗ = ‖u‖U and〈u, u∗0〉U = ‖u‖2

U . Define u∗1 = ‖u‖−1U u∗0. Then ‖u∗1‖U = 1 and 〈u, u∗1〉U = ‖u‖U . ¤

Definition 3.2.6 (Affine Hyper-Plane). Let U be a Banach space. An affine hyper-plane His a set of the form

H = u ∈ U | 〈u, u∗〉U = α (3.19)

for some u∗ ∈ U∗ and α ∈ R.


Proposition 3.2.7. A hyper-plane H defined as above is closed.

Proof: The result follows from the continuity of 〈u, u∗〉U as a functional defined in U . ¤

Definition 3.2.8 (Separation). Given A,B ⊂ U we say that a hyper-plane H, defined asabove separates A and B if

〈u, u∗〉U ≤ α, ∀u ∈ A, and 〈u, u∗〉U ≥ α, ∀u ∈ B. (3.20)

We say that H separates A and B strictly if there exists ε > 0 such that

〈u, u∗〉U ≤ α− ε, ∀u ∈ A, and 〈u, u∗〉U ≥ α + ε, ∀u ∈ B, (3.21)

Theorem 3.2.9 (Hahn-Banach theorem, geometric form). Consider A,B ⊂ U two con-vex disjoint non-empty sets, where A is open. Then there exists a closed hyper-plane thatseparates A and B.

We need the following Lemma.

Lemma 3.2.10. Consider C ⊂ U a convex open set such that θ ∈ C. Given u ∈ U , define

p(u) = infα > 0, α−1u ∈ C. (3.22)

Thus, p is such that there exists M ∈ R+ satisfying

0 ≤ p(u) ≤ M‖u‖U , ∀u ∈ U, (3.23)

andC = u ∈ U | p(u) < 1. (3.24)

Alsop(u + v) ≤ p(u) + p(v), ∀u, v ∈ U.

Proof: Let r > 0 be such that B(θ, r) ⊂ C, thus

p(u) ≤ ‖u‖U

r,∀u ∈ U (3.25)

which proves (3.23). Now suppose u ∈ C. Since C is open (1 + ε)u ∈ C for ε sufficientlysmall. Therefore p(u) ≤ 1

1+ε< 1. Conversely, if p(u) < 1 there exists 0 < α < 1 such that

α−1u ∈ C and therefore, since C is convex, u = α(α−1u) + (1− α)θ ∈ C.

Also, let u, v ∈ C and ε > 0. Thus up(u)+ε

∈ C and vp(v)+ε

∈ C so that tup(u)+ε

+ (1−t)vp(v)+ε

∈C, ∀t ∈ [0, 1]. Particularly for t = p(u)+ε

p(u)+p(v)+2εwe obtain u+v

p(u)+p(v)+2ε∈ C, which means

p(u + v) ≤ p(u) + p(v) + 2ε, ∀ε > 0 ¤


Lemma 3.2.11. Consider C ⊂ U a convex open set and let u0 ∈ U be a vector not in C.Then there exists u∗ ∈ U∗ such that 〈u, u∗〉U < 〈u0, u

∗〉U ,∀u ∈ C

Proof: By a translation, we may assume θ ∈ C. Consider the functional p as in the lastlemma. Define V = αu0 | α ∈ R. Define g on V , by

g(tu0) = t, t ∈ R. (3.26)

We have that g(u) ≤ p(u), ∀u ∈ V . From the Hahn-Banach theorem, there exist a linearfunctional f on U which extends g such that

f(u) ≤ p(u) ≤ M‖u‖U . (3.27)

Here we have used lemma 3.2.10. In Particular, f(u0) = 1, and (also from the last lemma)f(u) < 1,∀u ∈ C. The existence of u∗ satisfying the theorem follows from the continuity off indicated in (3.27). ¤Proof of Theorem 3.2.9 Define C = A + (−B) so that C is convex and θ is not in C. FromLemma 3.2.11, there exists u∗ ∈ U∗ such that 〈w, u∗〉U < 0, ∀w ∈ C, which means

〈u, u∗〉U < 〈v, u∗〉U ,∀u ∈ A, v ∈ B. (3.28)

Thus, there exists α ∈ R such that

supu∈A

〈u, u∗〉U ≤ α ≤ infv∈B

〈v, u∗〉U , (3.29)

which completes the proof. ¤Theorem 3.2.12 (Hahn-Banach theorem, second geometric form). Consider A,B ⊂ U twoconvex disjoint non-empty sets. Suppose A is closed and B is compact. Then there exists anhyper-plane which separates A and B strictly.

Proof: There exists ε > 0 sufficiently small such that Aε = A+B(0, ε) and Bε = B +B(0, ε)are convex disjoint sets. From Theorem 3.2.9, there exists u∗ ∈ U∗ such that u∗ 6= θ and

〈u + εw1, u∗〉U ≤ 〈u + εw2, u

∗〉U ,∀u ∈ A, v ∈ B, w1, w2 ∈ B(0, 1). (3.30)

Thus, there exists α ∈ R such that

〈u, u∗〉U + ε‖u∗‖U∗ ≤ α ≤ 〈v, u∗〉U − ε‖u∗‖U∗ ,∀u ∈ A, v ∈ B. ¤ (3.31)

Corollary 3.2.13. Suppose V ⊂ U is a vector subspace such that V 6= U . Then there existsu∗ ∈ U∗ such that u∗ 6= θ and

〈u, u∗〉U = 0,∀u ∈ V. (3.32)

Proof: Consider u0 ∈ U such that u0 does not belong to V . Applying Theorem 3.2.9 toA = V and B = u0 we obtain u∗ ∈ U∗ and α ∈ R such that u∗ 6= θ and

〈u, u∗〉U < α < 〈u0, u∗〉U ,∀u ∈ V. (3.33)

Since V is a subspace we must have 〈u, u∗〉U = 0,∀u ∈ V . ¤.


3.3 Weak Topologies

Definition 3.3.1 (Weak Neighborhoods and Weak Topologies). For the topological space Uand u0 ∈ U , we define a weak neighborhood of u0, denoted by Vw as

Vw = u ∈ U | |〈u− u0, u∗i 〉U | < ε, ∀i ∈ 1, ..., m, (3.34)

for some m ∈ N, ε > 0, and u∗i ∈ U∗, ∀i ∈ 1, ..., m. Also, we define the weak topologyfor U , denoted by σ(U,U∗) as the set of arbitrary unions and finite intersections of weakneighborhoods in U .

Proposition 3.3.2. Consider Z a topological vector space and ψ a function of Z into U .Then ψ is continuous on U endowed with the weak topology, if and only if u∗ψ is continuous,for all u∗ ∈ U∗.

Proof: It is clear that if ψ is continuous with U endowed with the weak topology, then u∗ ψis continuous for all u∗ ∈ U∗. Conversely, consider U a weakly open set in U . We have toshow that ψ−1(U) is open in Z. But observe that U = ∪λ∈LVλ, where each Vλ is a weakneighborhood. Thus ψ−1(U) = ∪λ∈Lψ−1(Vλ). The result follows considering that u∗ ψ iscontinuous for all u∗ ∈ U∗, so that ψ−1(Vλ) is open, for all λ ∈ L. ¤

Proposition 3.3.3. A Banach space U is Hausdorff as endowed with the weak topologyσ(U,U∗).

Proof: Pick u1, u2 ∈ U such that u1 6= u2. From the Hahn-Banach theorem, second geometricform, there exists a hyper-plane separating u1 and u2. That is, there exist u∗ ∈ U∗ andα ∈ R such that

〈u1, u∗〉U < α < 〈u2, u

∗〉U . (3.35)

DefiningVw1 = u ∈ U | |〈u− u1, u

∗〉| < α− 〈u1, u∗〉U, (3.36)

andVw2 = u ∈ U | |〈u− u2, u

∗〉U | < 〈u2, u∗〉U − α, (3.37)

we obtain u1 ∈ Vw1, u2 ∈ Vw2 and Vw1 ∩ Vw2 = ∅. ¤

Remark 3.3.4. if un ∈ U is such that un converges to u in σ(U,U∗) then we write un u.

Proposition 3.3.5. Let U be a Banach space. Considering un ⊂ U we have

1. un u, for σ(U,U∗) ⇔ 〈un, u∗〉U → 〈u, u∗〉U ,∀u∗ ∈ U∗,

2. If un → u strongly (in norm) then un u weakly,

3. If un u weakly, then ‖un‖U is bounded and ‖u‖U ≤ lim infn→∞ ‖un‖U ,


4. If un u weakly and u∗n → u∗ strongly in U∗ then 〈un, u∗n〉U → 〈u, u∗〉U .

Proof:

1. The result follows directly from the definition of topology σ(U,U∗).

2. This follows from the inequality

|〈un, u∗〉U − 〈u, u∗〉U | ≤ ‖u∗‖U∗‖un − u‖U . (3.38)

3. Since for every u∗ ∈ U∗ the sequence 〈un, u∗〉U is bounded, from the uniform bound-edness principle we have that there exists M > 0 such that ‖un‖U ≤ M, ∀n ∈ N.Furthermore, for u∗ ∈ U∗ we have

|〈un, u∗〉U | ≤ ‖u∗‖U∗‖un‖U , (3.39)

and taking the limit, we obtain

|〈u, u∗〉U | ≤ lim infn→∞

‖u∗‖U∗‖un‖U . (3.40)

Thus‖u‖U = sup

‖u‖U∗≤1

|〈u, u∗〉U | ≤ lim infn→∞

‖u∗‖U∗‖un‖U . (3.41)

4. Just observe that

|〈un, u∗n〉U − 〈u, u∗〉U | ≤ |〈un, u

∗n − u∗〉U |+ |〈u− un, u∗〉U | ≤

‖u∗n − u∗‖U∗‖un‖U + |〈un − u, u∗〉U |. ¤

Theorem 3.3.6. Consider A ⊂ U a convex set. Thus A is weakly closed if and only if it isstrongly closed.

Proof: Suppose A is strongly closed. Consider u0 not in A. By the Hahn-Banach theoremthere exists a closed hyper-plane which separates u0 and A strictly. Therefore there existsα ∈ R and u∗ ∈ U∗ such that

〈u0, u∗〉U < α < 〈v, u∗〉U ,∀v ∈ A. (3.42)

DefineV = u ∈ U | 〈u, u∗〉U < α, (3.43)

so that u0 ∈ V , V ⊂ U −A. Since V is open for σ(U,U∗) we have that U −A is weakly open,hence A is weakly closed. The converse is obvious. ¤


3.4 The Weak-star Topology

Definition 3.4.1 (Reflexive Spaces). Let U be a Banach space. We say that U is reflexiveif the canonical injection J : U → U∗∗ defined by

〈u, u∗〉U = 〈u∗, J(u)〉U∗ ,∀u ∈ U, u∗ ∈ U∗, (3.44)

is onto.

The weak topology for U∗ is denoted by σ(U∗, U∗∗). By analogy, we can define the topologyσ(U∗, U), which is called the weak-star topology. A standard neighborhood of u∗0 ∈ U∗ forthe weak-star topology, which we denoted by Vw∗ , is given by

Vw∗ = u∗ ∈ U∗ | |〈ui, u∗ − u∗0〉U | < ε, ∀i ∈ 1, ..., m (3.45)

for some ε > 0, m ∈ N, ui ∈ U,∀i ∈ 1, ..., m. It is clear that the weak topology for U∗ andthe weak-star topology coincide if U is reflexive.

Proposition 3.4.2. Let U be a Banach space. U∗ as endowed with the weak-star topologyis a Hausdorff space.

Proof: The proof similar to that of Proposition 3.3.3. ¤

3.5 Weak-star Compactness

We start with an important theorem about weak-* compactness.

Theorem 3.5.1 (Banach Alaoglu Theorem). The set BU∗ = f ∈ U∗ | ‖f‖U∗ ≤ 1 iscompact for the topology σ(U∗, U) (the weak-star topology).

Proof: For each u ∈ U , we will associate a real number ωu and denote ω =∏

u∈U ωu. Wehave that ω ∈ RU and let us consider the projections Pu : RU → R, where Pu(ω) = ωu.Consider the weakest topology σ for which the functions Pu (u ∈ U) are continuous. For U∗,with the topology σ(U∗, U) define φ : U∗ → RU , by

φ(u∗) =∏u∈U

〈u, u∗〉U ,∀u∗ ∈ U∗. (3.46)

Since for each fixed u the mapping u∗ → 〈u, u∗〉U is weakly-star continuous, we see that, φis σ continuous, since weak-star convergence and convergence in σ are equivalent in U∗. Toprove that φ−1 is continuous, from Proposition 3.3.2, it suffices to show that the function


ω → 〈u, φ−1(ω)〉U is continuous on φ(U∗). This is true because 〈u, φ−1(ω)〉U = ωu on φ(U∗).On the other hand, it is also clear that φ(BU∗) = K where

K = ω ∈ RU | |ωu| ≤ ‖u‖U , ωu+v = ωu + ωv, ωλu = λωu,∀u, v ∈ U, λ ∈ R. (3.47)

To finish the proof, it is sufficient, from the continuity of φ−1, to show that K is compact inRU , concerning the topology σ. Observe that K = K1 ∩K2 where

K1 = ω ∈ RU | |ωu| ≤ ‖u‖U ,∀u ∈ U, (3.48)

andK2 = ω ∈ RU | ωu+v = ωu + ωv, ωλu = λωu,∀u, v ∈ U, λ ∈ R. (3.49)

The set∏

u∈U [−‖u‖U , ‖u‖U ] is compact as a Cartesian product of compact intervals. SinceK1 ⊂ K and K1 is closed, we have that K1 is compact (for the topology in question) . Onthe other hand, K2 is closed, because defining the closed sets Au,v and Bλ,u as

Au,v = ω ∈ RU | ωu+v − ωu − ωv = 0, (3.50)

andBλ,u = ω ∈ RU ωλu − λωu = 0 (3.51)

we may writeK2 = (∩u,v∈UAu,v) ∩ (∩(λ,u)∈R×UBλ,u). (3.52)

We recall that the K2 is closed because arbitrary intersections of closed sets are closed.Finally, we have that K1 ∩K2 is compact, which completes the proof. ¤

Theorem 3.5.2 (Kakutani). Let U be a Banach space. Then U is reflexive if and only if

BU = u ∈ U | ‖u‖U ≤ 1 (3.53)

is compact for the weak topology σ(U,U∗).

Proof: Suppose U is reflexive, then J(BU) = BU∗∗ . From the last theorem BU∗∗ is compactfor the topology σ(U∗∗, U∗). Therefore it suffices to verify that J−1 : U∗∗ → U is continuousfrom U∗∗ with the topology σ(U∗∗, U∗) to U , with the topology σ(U,U∗).

From Proposition 3.3.2 it is sufficient to show that the function u 7→ 〈f, J−1u〉U is continuousfor the topology σ(U∗∗, U∗), for each f ∈ U∗. Since 〈f, J−1u〉U = 〈u, f〉U∗ we have completedthe first part of the proof. For the second we need two lemmas.

Lemma 3.5.3 (Helly). Let U be a Banach space, f1, ..., fn ∈ U∗ and α1, ..., αn ∈ R, then 1and 2 are equivalent, where:

1.Given ε > 0, there exists uε ∈ U such that ‖uε‖U ≤ 1 and

|〈uε, fi〉U − αi| < ε, ∀i ∈ 1, ..., n.


2. ∣∣∣∣∣n∑

i=1

βiαi

∣∣∣∣∣ ≤∥∥∥∥∥

n∑i=1

βifi

∥∥∥∥∥U∗

,∀β1, ..., βn ∈ R. (3.54)

Proof: 1 ⇒ 2: Fix β1, ..., βn ∈ R, ε > 0 and define S =∑n

i=1 |βi|. From 1, we have

∣∣∣∣∣n∑

i=1

βi〈uε, fi〉U −n∑

i=1

βiαi

∣∣∣∣∣ < εS (3.55)

and therefore ∣∣∣∣∣n∑

i=1

βiαi

∣∣∣∣∣−∣∣∣∣∣

n∑i=1

βi〈uε, fi〉U∣∣∣∣∣ < εS (3.56)

or ∣∣∣∣∣n∑

i=1

βiαi

∣∣∣∣∣ <

∥∥∥∥∥n∑

i=1

βifi

∥∥∥∥∥U∗

‖uε‖U + εS ≤∥∥∥∥∥

n∑i=1

βifi

∥∥∥∥∥U∗

+ εS (3.57)

so that ∣∣∣∣∣n∑

i=1

βiαi

∣∣∣∣∣ ≤∥∥∥∥∥

n∑i=1

βifi

∥∥∥∥∥U∗

(3.58)

since ε is arbitrary.

Now let us show that 2 ⇒ 1. Define ~α = (α1, ..., αn) ∈ Rn and consider the functionϕ(u) = (〈f1, u〉U , ..., 〈fn, u〉U). Item 1 implies that ~α belongs to the closure of ϕ(BU). Let ussuppose that ~α does not belong to the closure of ϕ(Bu) and obtain a contradiction. Thus

we can separate ~α and the closure of ϕ(Bu) strictly, that is there exists ~β = (β1, ..., βn) ∈ Rn

and γ ∈ R such thatϕ(u).~β < γ < ~α.~β, ∀u ∈ BU (3.59)

Taking the supremum in u we contradict 2.

Also we need the lemma.

Lemma 3.5.4. Let U be a Banach space. Then J(BU) is dense in BU∗∗ for the topologyσ(U∗∗, U∗).

Proof: Let u∗∗ ∈ BU∗∗ and consider Vu∗∗ a neighborhood of u∗∗ for the topology σ(U∗∗, U∗).It suffices to show that J(BU) ∩ Vu∗∗ 6= ∅. As Vu∗∗ is a weak neighborhood, there existsf1, ..., fn ∈ U∗ and ε > 0 such that

Vu∗∗ = η ∈ U∗∗ | 〈fi, η − u∗∗〉U∗| < ε, ∀i ∈ 1, ..., n. (3.60)


Define αi = 〈fi, u∗∗〉U∗ and thus for any given β1, ..., βn ∈ R we have

∣∣∣∣∣n∑

i=1

βiαi

∣∣∣∣∣ =

∣∣∣∣∣〈n∑

i=1

βifi, u∗∗〉U∗

∣∣∣∣∣ ≤∥∥∥∥∥

n∑i=1

βifi

∥∥∥∥∥U∗

, (3.61)

so that from Helly lemma, there exists uε ∈ U such that ‖uε‖U ≤ 1 and

|〈uε, fi〉U − αi| < ε, ∀i ∈ 1, ..., n (3.62)

or,|〈fi, J(uε)− u∗∗〉U∗ | < ε, ∀i ∈ 1, ..., n (3.63)

and henceJ(uε) ∈ Vu∗∗ . ¤ (3.64)

Now we will complete the proof of Kakutani Theorem. Suppose BU is weakly compact (thatis, compact for the topology σ(U,U∗)). Observe that J : U → U∗∗ is weakly continuous,that is, it is continuous with U endowed with the topology σ(U,U∗) and U∗∗ endowed withthe topology σ(U∗∗, U∗). Thus as BU is weakly compact, we have that J(BU) is compactfor the topology σ(U∗∗, U∗). From the last lemma, J(BU) is dense BU∗∗ for the topologyσ(U∗∗, U∗). Hence J(BU) = BU∗∗ , or J(U) = U∗∗, which completes the proof. ¤

Proposition 3.5.5. Let U be a reflexive Banach space. Let K ⊂ U be a convex closedbounded set. Then K is weakly compact.

Proof: From Theorem 3.3.6, K is weakly closed (closed for the topology σ(U,U∗)). SinceK is bounded, there exists α ∈ R+ such that K ⊂ αBU . Since K is weakly closed andK = K ∩ αBU , we have that it is weakly compact. ¤

Proposition 3.5.6. Let U be a reflexive Banach space and M ⊂ U a closed subspace. ThenM with the norm induced by U is reflexive.

Proof: We can identify two weak topologies in M , namely:

σ(M, M∗) and the trace of σ(U,U∗). (3.65)

It can be easily verified that these two topologies coincide (through restrictions and exten-sions of linear forms). From theorem 2.4.2, it suffices to show that BM is compact for thetopology σ(M,M∗). But BU is compact for σ(U,U∗) and M ⊂ U is closed (strongly) andconvex so that it is weakly closed, thus from last proposition, BM is compact for the topologyσ(U,U∗), and therefore it is compact for σ(M, M∗). ¤


3.6 Separable Sets

Definition 3.6.1 (Separable Spaces). A metric space U is said to be separable if there exista set K ⊂ U such that K is countable and dense in U .

The next Proposition is proved in Brezis [6].

Proposition 3.6.2. Let U be a separable metric space. If V ⊂ U then V is separable.

Theorem 3.6.3. Let U be a Banach space such that U∗ is separable. Then U is separable.

Proof: Consider u∗n a countable dense set in U∗. Observe that

‖u∗n‖U∗ = sup|〈u∗n, u〉U | | u ∈ U and ‖u‖U = 1 (3.66)

so that for each n ∈ N, there exists un ∈ U such that ‖un‖U = 1 and 〈u∗n, un〉U ≥ 12‖u∗n‖U∗ .

Define U0 as the vector space on Q spanned by un, and U1 as the vector space on Rspanned by un. It is clear that U0 is dense in U1 and we will show that U1 is dense in U ,so that U0 is a dense set in U . For, suppose u∗ is such that 〈u∗, u〉U = 0,∀u ∈ U1. Sinceu∗n is dense in U∗, given ε > 0, there exists n ∈ N such that ‖u∗n − u∗‖U∗ < ε, so that

1

2‖u∗n‖U∗ ≤ 〈un, u

∗n〉U = 〈un, u∗n − u∗〉U + 〈un, u

∗〉U ≤ ‖u∗n − u∗‖U∗‖un‖U + 0 < ε (3.67)

or‖u∗‖U∗ ≤ ‖u∗n − u∗‖U∗ + ‖u∗n‖U∗ < ε + 2ε = 3ε. (3.68)

Therefore, since ε is arbitrary, ‖u∗‖U∗ = θ, that is u∗ = 0. By Corollary 3.2.13 this completesthe proof. ¤

Proposition 3.6.4. U is reflexive if and only if U∗ is reflexive.

Proof: Suppose U is reflexive, as BU∗ is compact for σ(U∗, U) and σ(U∗, U) = σ(U∗, U∗∗)we have that BU∗ is compact for σ(U∗, U∗∗), which means that U∗ is reflexive.

Suppose U∗ is reflexive, from above U∗∗ is reflexive. Since J(U) is a closed subspace of U∗∗,from Proposition 3.5.6, J(U) is reflexive. Thus, U is reflexive, since J is a isometry.

Proposition 3.6.5. Let U be a Banach space. Then U is reflexive and separable if and onlyif U∗ is reflexive and separable.

Chapter 4

Measure and Integration

4.1 Basic Concepts

In this chapter U denotes a topological space.

Definition 4.1.1 (σ-Algebra ). A collection M of subsets of U is said to be a σ-Algebra ifM has the following properties:

1. U ∈M,

2. if A ∈M then U − A ∈M,

3. if An ∈M,∀n ∈ N, then ∪∞n=0An ∈M.

Definition 4.1.2 (Measurable Spaces). If M is a σ-algebra in U we say that U is a mea-surable space. The elements of M are called the measurable sets of U .

Definition 4.1.3 (Measurable Function). If U is a measurable space and V is a topologicalspace, we say that f : U → V is a measurable function if f−1(V) is measurable wheneverV ⊂ V is an open set.

Remark 4.1.4. 1. Observe that ∅ = U − U so that from 1 and 2 in Definition 4.1.1, wehave that ∅ ∈ M.

2. From 1 and 3 from Definition 4.1.1, it is clear that ∪ni=1Ai ∈M whenever Ai ∈M,∀i ∈

1, ..., n.3. Since ∩∞i=1Ai = (∪∞i=1A

ci)

c also from Definition 4.1.1, it is clear that M is closed undercountable intersections.

4. Since A−B = Bc ∩ A we obtain: if A,B ∈M then A−B ∈M.

29

CHAPTER 4. MEASURE AND INTEGRATION 30

Theorem 4.1.5. Let F be any collection of subsets of U . Then there exists a smallestσ-algebra M0 in U such that F ⊂ M0.

Proof: Let Ω be the family of all σ-Algebras that contain F . Since the set of all subsets inU is a σ-algebra, Ω is non-empty.

Let M0 = ∩Mλ⊂ΩMλ, it is clear that M0 ⊃ F , it remains to prove that in fact M0 is aσ-algebra. Observe that:

1. U ∈Mλ, ∀Mλ ∈ Ω, so that, U ∈M0,

2. A ∈ M0 implies A ∈ Mλ, ∀Mλ ∈ Ω, so that Ac ∈ Mλ, ∀Mλ ∈ Ω, which meansAc ∈M0,

3. An ⊂ M0 implies An ⊂ Mλ, ∀Mλ ∈ Ω, so that ∪∞n=1An ∈ Mλ, ∀Mλ ∈ Ω, whichmeans ∪∞n=1An ∈M0.

From Definition 4.1.1 the proof is complete. ¤

Definition 4.1.6 (Borel Sets). Let U be a topological space, considering the last theoremthere exists a smallest σ-algebra in U , denoted by B, which contains the open sets of U . Theelements of B are called the Borel sets.

Theorem 4.1.7. Suppose M is a σ-algebra in U and V is a topological space. For f : U →V , we have:

1. If Ω = E ⊂ V | f−1(E) ∈M, then Ω is a σ-algebra.

2. If V = [−∞,∞], and f−1((α,∞]) ∈M, for each α ∈ R, then f is measurable.

Proof:

1. (a) V ∈ Ω since f−1(V ) = U and U ∈M.

(b) E ∈ Ω ⇒ f−1(E) ∈M⇒ U − f−1(E) ∈M⇒ f−1(V − E) ∈M⇒ V − E ∈ Ω.

(c) Ei ⊂ Ω ⇒ f−1(Ei) ∈ M, ∀i ∈ N ⇒ ∪∞i=1f−1(Ei) ∈ M ⇒ f−1(∪∞i=1Ei) ∈ M ⇒

∪∞i=1Ei ∈ Ω.Thus Ω is a σ-algebra.

2. Define Ω = E ⊂ [−∞,∞] | f−1(E) ∈ M from above Ω is a σ- algebra. Givenα ∈ R, let αn be a real sequence such that αn → α as n → ∞, αn < α, ∀n ∈ N .Since (αn,∞] ∈ Ω for each n and

[−∞, α) = ∪∞n=1[−∞, αn] = ∪∞n=1(αn,∞]C , (4.1)


we obtain, [−∞, α) ∈ Ω. Furthermore, we have (α, β) = [−∞, β) ∩ (α,∞] ∈ Ω. Sinceevery open set in [−∞,∞] may be expressed as a countable union of intervals (α, β)we have that Ω contains all the open sets. Thus, f−1(E) ∈M whenever E is open, sothat f is measurable. ¤

Proposition 4.1.8. If fn : U → [−∞,∞] is a sequence of measurable functions andg = supn≥1 fn and h = lim supn→∞ fn then g and h are measurable.

Proof: Observe that g−1((α,∞]) = ∪∞n=1f−1n ((α,∞]). From last theorem g is measurable.

By analogy h = infk≥1supi≥k fi is measurable. ¤

4.2 Simple Functions

Definition 4.2.1 (Simple Functions). A function f : U → C is said to be a simple functionif its range (R(f)) has only finitely many points. If α1, ..., αn = R(f) and we set Ai =u ∈ U | f(u) = αi, clearly we have: f =

∑ni=1 αiχAi

, where

χAi(u) =

1, if u ∈ Ai,0, otherwise.

(4.2)

Theorem 4.2.2. Let f : U → [0,∞] be a measurable function. Thus there exists a sequenceof simple functions sn : U → [0,∞] such that

1. 0 ≤ s1 ≤ s2 ≤ ... ≤ f ,

2. sn(u) → f(u) as n →∞,∀u ∈ U.

Proof: Define δn = 2−n. To each n ∈ N and each t ∈ R+, there corresponds a unique integerK = Kn(t) such that

Kδn ≤ t ≤ (K + 1)δn. (4.3)

Defining

ϕn(t) =

Kn(t)δn, if 0 ≤ t < n,n, if t ≥ n,

(4.4)

we have that each ϕn is a Borel function on [0,∞], such that

1. t− δn < ϕn(t) ≤ t if 0 ≤ t ≤ n,

2. 0 ≤ ϕ1 ≤ ... ≤ t,

3. ϕn(t) → t as n →∞,∀t ∈ [0,∞].

It follows that the sequence sn = ϕn f corresponds to the results indicated above. ¤


4.3 Measures

Definition 4.3.1 (Measure). Let M be a σ-algebra on a topological space U . A functionµ : M→ [0,∞] is said to be a measure if µ(∅) = 0 and µ is countably additive, that is, givenAi ⊂ U , a sequence of pairwise disjoint sets then

µ(∪∞i=1Ai) =∞∑i=1

µ(Ai). (4.5)

In this case (U,M, µ) is called a measure space.

Proposition 4.3.2. Let µ : M→ [0,∞], where M is a σ-algebra of U . Then we have thefollowing.

1. µ(A1∪ ...∪An) = µ(A1)+ ...+µ(An) for any given Ai of pairwise disjoint measurablesets of M.

2. If A,B ∈M and A ⊂ B then µ(A) ≤ µ(B).

3. If An ⊂ M, A = ∪∞n=1An and

A1 ⊂ A2 ⊂ A3 ⊂ ... (4.6)

then, limn→∞ µ(An) = µ(A).

4. If An ⊂ M, A = ∩∞n=1An, A1 ⊃ A2 ⊃ A3 ⊃ .... and µ(A1) is finite then,

limn→∞

µ(An) = µ(A). (4.7)

Proof:

1. Take An+1 = An+2 = .... = ∅ in Definition 4.1.1 item 1,

2. Observe that B = A∪ (B−A) and A∩ (B−A) = ∅ so that by above µ(A∪ (B−A)) =µ(A) + µ(B − A) ≥ µ(A),

3. Let B1 = A1 and let Bn = An − An−1 then Bn ∈ M, Bi ∩ Bj = ∅ if i 6= j, An =B1 ∪ ... ∪Bn and A = ∪∞i=1Bi. Thus

µ(A) = µ(∪∞i=1Bi) =∞∑

n=1

µ(Bi) = limn→∞

n∑i=1

Bi = limn→∞

µ(An) (4.8)

4. Let Cn = A1 − An. Then C1 ⊂ C2 ⊂ ..., µ(Cn) = µ(A1) − µ(An), A1 − A = ∪∞n=1Cn.Thus by 3 we have

µ(A1)− µ(A) = µ(A1 − A) = limn→∞

µ(Cn) = µ(A1)− limn→∞

µ(An). ¤ (4.9)


4.4 Integration of Simple Functions

Definition 4.4.1 (Integral for Simple Functions). For s : U → [0,∞], a measurable simplefunction, that is,

s =n∑

i=1

αiχAi, (4.10)

where

χAi(u) =

1, if u ∈ Ai,0, otherwise,

(4.11)

we define the integral of s over E ⊂M, denoted by∫

Es dµ as

∫

E

s dµ =n∑

i=1

αiµ(Ai ∩ E). (4.12)

The convention 0.∞ = 0 is used here.

Definition 4.4.2 (Integral for Non-negative Measurable Functions). If f : U → [0,∞] ismeasurable, for E ∈M, we define the integral of f on E, denoted by

∫E

fdµ, as

∫

E

fdµ = sups∈A∫

E

sdµ, (4.13)

whereA = s simple and measurable | 0 ≤ s ≤ f. (4.14)

Definition 4.4.3 (Integral for Measurable Functions). For a measurable f : U → [−∞,∞]and E ∈ M, we define f+ = maxf, 0, f− = max−f, 0 and the integral of f on E,denoted by

∫E

f dµ, as ∫

E

f dµ =

∫

E

f+ dµ−∫

E

f− dµ.

Theorem 4.4.4 (Lebesgue’s Monotone Convergence Theorem). Let fn be a sequence ofreal measurable functions on U and suppose that

1. 0 ≤ f1(u) ≤ f2(u) ≤ ... ≤ ∞,∀u ∈ U,

2. fn(u) → f(u) as n →∞, ∀u ∈ U .

Then,

(a) f is measurable,

(b)∫

Ufndµ → ∫

Ufdµ as n →∞.


Proof: Since∫

Ufndµ ≤ ∫

Ufn+1dµ,∀n ∈ N, there exists α ∈ [0,∞] such that

∫

U

fndµ → α, as n →∞, (4.15)

By Proposition 4.1.8, f is measurable, and since fn ≤ f we have∫

U

fndµ ≤∫

U

fdµ. (4.16)

From (4.15) and (4.16), we obtain

α ≤∫

U

fdµ. (4.17)

Let s be any simple function such that 0 ≤ s ≤ f , and let c ∈ R such that 0 < c < 1. Foreach n ∈ N we define

En = u ∈ U | fn(u) ≥ cs(u). (4.18)

Clearly En is measurable and E1 ⊂ E2 ⊂ ... and U = ∪n∈NEn. Observe that∫

U

fndµ ≥∫

En

fndµ ≥ c

∫

En

sdµ. (4.19)

Letting n →∞ and applying Proposition 4.3.2, we obtain

α = limn→∞

∫

U

fndµ ≥ c

∫

U

sdµ, (4.20)

so that

α ≥∫

U

sdµ, ∀s simple such that 0 ≤ s ≤ f. (4.21)

This implies

α ≥∫

U

fdµ. (4.22)

From (4.17) and (4.22) the proof is complete. ¤.

Theorem 4.4.5 (Fatou’s Lemma). If fn : U → [0,∞] is a sequence of measurable func-tions, then ∫

U

lim infn→∞

fndµ ≤ lim infn→∞

∫

U

fndµ. (4.23)

Proof: For each k ∈ N define gk : U → [0,∞] by

gk(u) = infi≥kfi(u). (4.24)

Thengk ≤ fk (4.25)


so that ∫

U

gkdµ ≤∫

U

fkdµ, ∀k ∈ N. (4.26)

Also 0 ≤ g1 ≤ g2 ≤ ..., each gk is measurable, and

limk→∞

gk(u) = lim infn→∞

fn(u), ∀u ∈ U. (4.27)

From the monotone convergence theorem

lim infk→∞

∫

U

gkdµ = limk→∞

∫

U

gkdµ =

∫

U

lim infn→∞

fndµ. (4.28)

From (4.26) we have

lim infk→∞

∫

U

gkdµ ≤ lim infk→∞

∫

U

fkdµ. (4.29)

Thus, from (4.28) and (4.29) we obtain∫

U

lim infn→∞

fndµ ≤ lim infn→∞

∫

U

fndµ. ¤ (4.30)

Theorem 4.4.6 (Lebesgue’s Dominated Convergence Theorem). Suppose fn is sequenceof complex measurable functions on U such that

limn→∞

fn(u) = f(u), ∀u ∈ U. (4.31)

If there exists a measurable function g : U → R+ such that∫

Ugdµ < ∞ and |fn(u)| ≤

g(u),∀u ∈ U, n ∈ N, then

1.∫

U|f |dµ < ∞,

2. limn→∞∫

U|fn − f |dµ = 0.

Proof:

1. This inequality holds since f is measurable and |f | ≤ g.

2. Since 2g − |fn − f | ≥ 0 we may apply the Fatou’s Lemma and obtain:∫

U

2gdµ ≤ lim infn→∞

∫

U

(2g − |fn − f |)dµ, (4.32)

so that

lim supn→∞

∫

U

|fn − f |dµ ≤ 0. (4.33)

Hence

limn→∞

∫

U

|fn − f |dµ = 0. (4.34)

This completes the proof. ¤


We finish this chapter with an important remark:

Remark 4.4.7. In a measurable space U we say that a property holds almost everywhere(a.e.) if it holds on U except for a set of measure zero.

Chapter 5

Distributions

5.1 Basic Definitions and Results

Definition 5.1.1 (Test Functions, the Space D(Ω)). Let Ω ∈ Rn be a nonempty open set.For each K ⊂ Ω compact, consider the space DK, the set of all C∞(Ω) functions with supportin K. We define the space of test functions, denoted by D(Ω) as

D(Ω) = ∪K⊂ΩDK , K compact. (5.1)

Thus φ ∈ D(Ω) if and only if φ ∈ C∞(Ω) and the support of φ is a compact subset of Ω.

Definition 5.1.2 (Topology for D(Ω)). Let Ω ⊂ Rn be an open set.

1. For every K ⊂ Ω compact, σK denotes the topology which a local base is defined byVN,n, where N,n ∈ N,

VN,n = φ ∈ DK | ‖φ‖N < 1/n (5.2)

and‖φ‖N = max|Dαφ(x)| | x ∈ Ω, |α| ≤ N. (5.3)

2. σ denotes the collection of all convex balanced sets W ∈ D(Ω) such that W∩DK ⊂ σK

for every compact K ⊂ Ω.

3. We define σ in D(Ω) as the collection of all unions of sets of the form φ + W, forφ ∈ D(Ω) and W ∈ σ.

Theorem 5.1.3. Concerning the last definition we have the following.

1. σ is a topology in D(Ω).

37

CHAPTER 5. DISTRIBUTIONS 38

2. Through σ, D(Ω) is made into a locally convex topological vector space.

Proof:

1. From item 3 of Definition 5.1.2, it is clear that arbitrary unions of elements of σ areelements of σ. Let us now show that finite intersections of elements of σ also belongs toσ. Suppose V1 ∈ σ and V2 ∈ σ, if V1 ∩V2 = ∅ we are done. Thus, suppose φ ∈ V1 ∩V2.By the definition of σ there exist two sets of indices L1 and L2, such that

Vi = ∪λ∈Li(φiλ +Wiλ), for i = 1, 2, (5.4)

and as φ ∈ V1 ∩ V2 there exist φi ∈ D(Ω) and Wi ∈ σ such that

φ ∈ φi +Wi, for i = 1, 2. (5.5)

Thus there exists K ∈ Ω such that φi ⊂ DK for i ∈ 1, 2. Since DK ∩ Wi ∈ σK ,DK ∩Wi is open in DK so that from (5.5) there exists 0 < δi < 1 such that

φ− φi ∈ (1− δi)Wi, for i ∈ 1, 2. (5.6)

From (5.6) and from the convexity of Wi we have

φ− φi + δiWi ⊂ (1− δi)Wi + δiWi = Wi (5.7)

so thatφ + δiWi ⊂ φi +Wi ⊂ Vi, for i ∈ 1, 2. (5.8)

Define Wφ = (δ1W1) ∩ (δ2W2) so that

φ +Wφ ⊂ Vi, (5.9)

and therefore we may write

V1 ∩ V2 = ∪φ∈V1∩V2(φ +Wφ) ∈ σ. (5.10)

This completes the proof.

2. It suffices to show that single points are closed sets in D(Ω) and the vector spaceoperations are continuous.

(a) Pick φ1, φ2 ∈ D(Ω) such that φ1 6= φ2 and define

V = φ ∈ D(Ω) | ‖φ‖0 < ‖φ1 − φ2‖0. (5.11)

Thus V ∈ σ and φ1 * φ2 + V . As φ2 + V is open and belongs to D(Ω)− φ1and φ2 6= φ1 is arbitrary, it follows that D(Ω) − φ1 is open, so that φ1 isclosed.


(b) The proof that addition is σ-continuous follows from the convexity of any elementof σ. Thus given φ1, φ2 ∈ D(Ω) and V ∈ σ we have

φ1 +1

2V + φ2 +

1

2V = φ1 + φ2 + V . (5.12)

(c) To prove the continuity of scalar multiplication, first consider φ0 ∈ D(Ω) andα0 ∈ R. Then:

αφ− α0φ0 = α(φ− φ0) + (α− α0)φ0. (5.13)

For V ∈ σ there exists δ > 0 such that δφ0 ∈ 12V . Let us define c = 1

2(|α0| + δ).

Thus if |α− α0| < δ then (α− α0)φ0 ∈ 12V . Let φ ∈ D(Ω) such that

φ− φ0 ∈ cV =1

2(|α0|+ δ)V , (5.14)

so that

(|α0|+ δ)(φ− φ0) ∈ 1

2V . (5.15)

This means

α(φ− φ0) + (α− α0)φ0 ∈ 1

2V +

1

2V = V . (5.16)

Therefore αφ− α0φ0 ∈ V whenever |α− α0| < δ and φ− φ0 ∈ cV . ¤

For the next result the proof may be found in Rudin [31].

Proposition 5.1.4. A convex balanced set V ⊂ Ω is open if and only if V ∈ σ.

Proposition 5.1.5. The topology σK of DK ⊂ D(Ω) coincides with the topology that DK

inherits from D(Ω).

Proof: From Proposition 5.1.4 we have

V ∈ σ implies DK ∩ V ∈ σK . (5.17)

Now suppose V ∈ σK , we must show that there exists A ∈ σ such that V = A ∩ DK . Thedefinition of σK implies that for every φ ∈ V , there exist N ∈ N and δφ > 0 such that

ϕ ∈ DK | ‖ϕ− φ‖N < δφ ⊂ V . (5.18)

DefineUφ = ϕ ∈ D(Ω) | ‖ϕ‖N < δφ. (5.19)

Then Uφ ∈ σ andDK ∩ (φ + Uφ) = φ + (DK ∩ Uφ) ⊂ V . (5.20)

Defining A = ∪φ∈V(φ + Uφ), we have completed the proof. ¤The proof for the next result may also be found in Rudin [31].


Proposition 5.1.6. If A is a bounded set of D(Ω) then A ⊂ DK for some K ⊂ Ω, and thereare MN < ∞ such that ‖φ‖N ≤ MN ,∀φ ∈ A, N ∈ N.

Proposition 5.1.7. If φn is a Cauchy sequence in D(Ω), then φn ⊂ DK for someK ⊂ Ω compact, and

limi,j→∞

‖φi − φj‖N = 0,∀N ∈ N. (5.21)

Proof: Since Cauchy sequences are bounded, we have that φn ⊂ DK for some K ⊂ Ωcompact. The result indicated in (5.21) follows from the fact that φn is also a Cauchysequence in σK . ¤

Proposition 5.1.8. If φn → 0 in D(Ω), then there exists a compact K ⊂ Ω which containsthe support of φn,∀n ∈ N and Dαφn → 0 uniformly, for each multi-index α.

The proof follows directly from last proposition.

Theorem 5.1.9. Suppose T : D(Ω) → V is linear, where V is a locally convex space. Thenthe following statements are equivalent.

1. T is continuous.

2. T is bounded.

3. If φn → 0 in D(Ω) then T (φn) → 0 as n →∞.

4. The restrictions of T to each DK are continuous.

Proof:1 ⇒ 2. This follows from Proposition 2.7.3 .

2 ⇒ 3. Suppose T is bounded and φn → 0 in D(Ω), by last proposition φn → 0 insome DK so that φn is bounded and T (φn) is also bounded. Hence by Proposition 2.7.3,T (φn) → 0 in V .

3 ⇒ 4. Assume 3 holds and consider φn ⊂ DK . If φn → 0 then, by Proposition 5.1.5,φn → 0 in D(Ω), so that, by above T (φn) → 0 in V . Since DK is metrizable, also by propo-sition 2.7.3 we have that 4 follows.

4 ⇒ 1. Assume 4 holds and let V be a convex balanced neighborhood of zero in V . DefineU = T−1(V). Thus U is balanced and convex. By Proposition 5.1.5, U is open in D(Ω) ifand only if DK ∩ U is open in DK for each compact K ⊂ Ω, thus if the restrictions of T toeach DK are continuous at 0, then T is continuous at 0, hence 4 implies 1. ¤

Definition 5.1.10 (Distribution). A linear functional in D(Ω) which is continuous withrespect to σ is said to be a Distribution.


Proposition 5.1.11. Every differential operator is a continuous mapping from D(Ω) intoD(Ω).

Proof: Since ‖Dαφ‖N ≤ ‖φ‖|α|+N ,∀N ∈ N, Dα is continuous on eachDK , so that by Theorem5.1.9, Dα is continuous on D(Ω). ¤

Theorem 5.1.12. Denoting by D′(Ω) the dual space of D(Ω) we have that T : D(Ω) → R ∈D′(Ω) if and only if for each compact set K ⊂ Ω there exists an N ∈ N and c ∈ R+ suchthat

|T (φ)| ≤ c‖φ‖N ,∀φ ∈ DK . (5.22)

The proof follows from the equivalence of 1 and 4 in Theorem 5.1.9. ¤

5.2 Differentiation of Distributions

Definition 5.2.1 (Derivatives for Distributions). Given T ∈ D′(Ω) and a multi-index α, wedefine the Dα derivative of T as

DαT (φ) = (−1)|α|T (Dαφ), ∀φ ∈ D(Ω). (5.23)

Remark 5.2.2. Observe that if |T (φ)| ≤ c‖φ‖N ,∀φ ∈ D(Ω) for some c ∈ R+, then

|DαT (φ)| ≤ c‖Dαφ‖N ≤ c‖φ‖N+|α|,∀φ ∈ D(Ω), (5.24)

thus DαT ∈ D′(Ω). Therefore, derivatives of distributions are also distributions.

Theorem 5.2.3. Suppose Tn ⊂ D′(Ω). Let T : D(Ω) → R be defined by

T (φ) = limn→∞

Tn(φ),∀φ ∈ D(Ω). (5.25)

Then T ∈ D′(Ω), andDαTn → DαT in D′(Ω). (5.26)

Proof: Let K be an arbitrary compact subset of Ω. Since (5.25) holds for every φ ∈ DK ,and since DK is a Frechet space, the Banach-Steinhaus theorem implies that the restrictionof T to DK is continuous. It follows from Theorem 5.1.9 that T is continuous in D(Ω), thatis, T ∈ D′(Ω). On the other hand

(DαT )(φ) = (−1)|α|T (Dαφ) = (−1)|α| limn→∞

Tn(Dαφ) = limn→∞

(DαTn(φ)), ∀φ ∈ D(Ω). ¤(5.27)

Chapter 6

Lebesgue and Sobolev Spaces

We start with the definition of Lebesgue spaces, denoted by Lp(Ω), where 1 ≤ p ≤ ∞ andΩ ⊂ Rn is an open set.

6.1 Definition and Properties of Lp Spaces

Definition 6.1.1 (Lp Spaces). For 1 ≤ p < ∞, we say that u ∈ Lp(Ω) if u : Ω → R ismeasurable and ∫

Ω

|u|pdx < ∞. (6.1)

We also denote ‖u‖p = [∫Ω|u|pdx]1/p and will show that ‖.‖p is a norm.

Definition 6.1.2 (L∞ Spaces). We say that u ∈ L∞(Ω) if u is measurable and there existsM ∈ R+, such that |u(x)| ≤ M, a.e. in Ω. We define

‖u‖∞ = infM > 0 | |u(x)| ≤ M, a.e. in Ω. (6.2)

We will show that ‖.‖∞ is a norm. For 1 ≤ p ≤ ∞, we define q by the relations

q =

+∞, if p = 1,p

p−1, if 1 < p < +∞,

1, if p = +∞,

so that symbolically we have1

p+

1

q= 1.

The next result is fundamental in the proof of the Sobolev Imbedding Theorem.

42

CHAPTER 6. LEBESGUE AND SOBOLEV SPACES 43

Theorem 6.1.3 (Holder Inequality). Consider u ∈ Lp(Ω) and v ∈ Lq(Ω), with 1 ≤ p ≤ ∞.Then uv ∈ L1(Ω) and ∫

Ω

|uv|dx ≤ ‖u‖p‖v‖q. (6.3)

Proof: The result is clear if p = 1 or p = ∞. You may assume ‖u‖p, ‖v‖q > 0, otherwisethe result is also obvious. Thus suppose 1 < p < ∞. From the concavity of log function on(0,∞) we obtain

log

(1

pap +

1

qbq

)≥ 1

plog ap +

1

qlog bq = log(ab). (6.4)

Thus,

ab ≤ 1

p(ap) +

1

q(bq), ∀a ≥ 0, b ≥ 0. (6.5)

Therefore

|u(x)||v(x)| ≤ 1

p|u(x)|p +

1

q|v(x)|q, a.e. in Ω. (6.6)

Hence |uv| ∈ L1(Ω) and ∫

Ω

|uv|dx ≤ 1

p‖u‖p

p +1

q‖v‖q

q. (6.7)

Replacing u by λu in (6.7) λ > 0, we obtain

∫

Ω

|uv|dx ≤ λp−1

p‖u‖p

p +1

λq‖v‖q

q. (6.8)

For λ = ‖u‖−1p ‖v‖q/p

q we obtain the Holder inequality. ¤The next step is to prove that ‖.‖p is a norm.

Theorem 6.1.4. Lp(Ω) is a vector space and ‖.‖p is norm ∀p such that 1 ≤ p ≤ ∞.

Proof: If p = 1 or p = ∞ the result is clear. Thus, suppose 1 < p < ∞. For u, v ∈ Lp(Ω) wehave

|u(x) + v(x)|p ≤ (|u(x)|+ |v(x)|)p ≤ 2p(|u(x)|p + |v(x)|p), (6.9)

so that u + v ∈ Lp(Ω). On the other hand

‖u + v‖pp =

∫

Ω

|u + v|p−1|u + v|dx ≤∫

Ω

|u + v|p−1|u|dx +

∫

Ω

|u + v|p−1|v|dx, (6.10)

and hence, from Holder’s inequality

‖u + v‖pp ≤ ‖u + v‖p−1

p ‖u‖p + ‖u + v‖p−1p ‖v‖p, (6.11)

that is,‖u + v‖p ≤ ‖u‖p + ‖v‖p,∀u, v ∈ Lp(Ω). ¤ (6.12)


Theorem 6.1.5. Lp(Ω) is a Banach space for any p such that 1 ≤ p ≤ ∞.

Proof: Suppose p = ∞. Suppose un is Cauchy sequence in L∞(Ω). Thus, given k ∈ Nthere exists Nk ∈ N such that, if m,n ≥ Nk then

‖um − un‖∞ <1

k. (6.13)

Therefore, for each k, there exist a set Ek such that m(Ek) = 0, and

|um(x)− un(x)| < 1

k, ∀x ∈ Ω− Ek, ∀m,n ≥ Nk. (6.14)

Observe that E = ∪∞k=1Ek is such that m(E) = 0. Thus un(x) is a real Cauchy sequenceat each x ∈ Ω − E. Define u(x) = limn→∞ un(x) on Ω − E. Letting m → ∞ in (6.14) weobtain

|u(x)− un(x)| < 1

k, ∀x ∈ Ω− E, ∀n ≥ Nk. (6.15)

Thus u ∈ L∞(Ω) and ‖un − u‖∞ → 0 as n →∞.

Now suppose 1 ≤ p < ∞. Let un a Cauchy sequence in Lp(Ω). We can extract asubsequence unk

such that

‖unk+1− unk

‖p ≤ 1

2k,∀k ∈ N. (6.16)

To simplify the notation we write uk in place of unk, so that

‖uk+1 − uk‖p ≤ 1

2k,∀k ∈ N. (6.17)

Defining

gn(x) =n∑

k=1

|uk+1(x)− uk(x)|, (6.18)

we obtain‖gn‖p ≤ 1, ∀n ∈ N. (6.19)

From the monotone convergence theorem and (6.19), gn(x) converges to a limit g(x) withg ∈ Lp(Ω). On the the other hand, for m ≥ n ≥ 2 we have

|um(x)− un(x)| ≤ |um(x)− um−1(x)|+ ... + |un+1(x)− un(x)| ≤ g(x)− gn−1(x), a.e. in Ω.(6.20)

Hence un(x) is Cauchy a.e. in Ω and converges to a limit u(x) so that

|u(x)− un(x)| ≤ g(x), a.e. ∈ Ω, for n ≥ 2, (6.21)

which means u ∈ Lp(Ω). Finally from |un(x)−u(x)| → 0, a.e. in Ω, |un(x)−u(x)|p ≤ |g(x)|pand the Lebesgue dominated convergence theorem implies

‖un − u‖p → 0 as n →∞. ¤ (6.22)


Theorem 6.1.6. Let un ⊂ Lp(Ω) and u ∈ Lp(Ω) such that ‖un − u‖p → 0. Then thereexists a subsequence unk

such that

1. unk(x) → u(x), a.e. in Ω,

2. |unk(x)| ≤ h(x), a.e. in Ω,∀k ∈ N, for some h ∈ Lp(Ω).

Proof: the result is clear for p = ∞. Suppose 1 ≤ p < ∞. From the last theorem we caneasily obtain that |unk

(x) − u(x)| → 0 as k → ∞, a.e. in Ω. To complete the proof, justtake h = u + g, where is defined in the proof of the last theorem. ¤

Theorem 6.1.7. Lp(Ω) is reflexive for all p such that 1 < p < ∞.

Proof: We divide the proof into 3 parts.

1. For 2 ≤ p < ∞ we have that

∥∥∥∥u + v

2

∥∥∥∥Lp(Ω)

+

∥∥∥∥u− v

2

∥∥∥∥Lp(Ω)

≤ 1

2(‖u‖p

Lp(Ω) + ‖v‖pLp(Ω)), ∀u, v ∈ Lp(Ω). (6.23)

Proof: Observe thatαp + βp ≤ (α2 + β2)p/2, ∀α, β ≥ 0. (6.24)

Now taking α =∣∣a+b

2

∣∣ and β =∣∣a−b

2

∣∣ in (6.24), we obtain (using the convexity of tp/2),

|a + b

2|p + |a− b

2|p ≤ (|a + b

2|2 + |a− b

2|2)p/2 = (

a2

2+

b2

2)p/2 ≤ 1

2|a|p +

1

2|b|p. (6.25)

The inequality (6.23) follows immediately.

2. Lp(Ω) is uniformly convex, and therefore reflexive for 2 ≤ p < ∞.

Proof: Suppose ε > 0 and suppose that

‖u‖p ≤ 1, ‖v‖p ≤ 1 and ‖u− v‖p > ε. (6.26)

From part 1, we obtain ∥∥∥∥u + v

2

∥∥∥∥p

p

< 1−(ε

2

)p

, (6.27)

and therefore ∥∥∥∥u + v

2

∥∥∥∥p

< 1− δ, (6.28)

for δ = 1− (1− (ε/2)p)p > 0. Thus Lp(Ω) is uniformly convex and reflexive (TheoremIII.29, Brezis [6]).


3. Lp(Ω) is reflexive for 1 < p ≤ 2. From 2 we can conclude that Lq is reflexive. We willdefine T : Lp(Ω) → (Lq)∗ by

〈Tu, f〉Lq(Ω) =

∫

Ω

ufdx, ∀u ∈ Lp(Ω), f ∈ Lq(Ω). (6.29)

From the Holder inequality, we obtain

|〈Tu, f〉Lq(Ω)| ≤ ‖u‖p‖f‖q, (6.30)

so that‖Tu‖(Lq(Ω))∗ ≤ ‖u‖p. (6.31)

Pick u ∈ Lp(Ω) and define f0(x) = |u(x)|p−2u(x) (f0(x) = 0 if u(x) = 0). Thus, wehave that f0 ∈ Lq(Ω), ‖f0‖q = ‖u‖p−1

p and 〈Tu, f0〉Lq(Ω) = ‖u‖pp. Therefore,

‖Tu‖(Lq(Ω))∗ ≥〈Tu, f0〉Lq(Ω)

‖f0‖q

= ‖u‖p (6.32)

Hence from (6.31) and (6.32) we have

‖Tu‖(Lq(Ω))∗ = ‖u‖p,∀u ∈ Lp(Ω). (6.33)

Thus T is an isometry from Lp(Ω) to a closed subspace of (Lq(Ω))∗. Since from thefirst part Lq(Ω) is reflexive, we have that (Lq(Ω))∗ is reflexive. From proposition III.17in Brezis [6], T (Lp(Ω)) and Lp(Ω) are reflexive. ¤

Theorem 6.1.8 (Riesz Representation Theorem). Let 1 < p < ∞ and let f be a continuouslinear functional on Lp(Ω). Then there exists a unique u0 ∈ Lq such that

f(v) =

∫

Ω

vu0 dx, ∀v ∈ Lp(Ω). (6.34)

Furthermore‖f‖(Lp)∗ = ‖u0‖q. (6.35)

Proof: First we define the operator T : Lq(Ω) → (Lp(Ω))∗ by

〈Tu, v〉Lp(Ω) =

∫

Ω

uv dx, ∀v ∈ Lp(Ω). (6.36)

Similarly to last theorem, we obtain

‖Tu‖(Lp(Ω))∗ = ‖u‖q. (6.37)

We have to show that T is onto. Define E = T (Lq(Ω)). As E is a closed subspace, it sufficesto show that E is dense in (Lp(Ω))∗. Suppose h ∈ (Lp)∗∗ = Lp is such that

〈Tu, h〉Lp(Ω) = 0,∀u ∈ Lq(Ω). (6.38)

Choosing u = |h|p−2h we may conclude that h = 0, which completes the proof. ¤Definition 6.1.9. Let 1 ≤ p ≤ ∞. We say that u ∈ Lp

loc(Ω) if uχK ∈ Lp(Ω) for all compactK ⊂ Ω.


6.1.1 Spaces of Continuous Functions

We introduce some definitions and properties concerning spaces of continuous functions.First, we recall that by a domain we mean an open set in Rn. Thus for a domain Ω ⊂ Rn

and for any nonnegative integer m we define by Cm(Ω) the set of all functions u whichthe partial derivatives Dαu are continuous on Ω for any α such that |α| ≤ m. We defineC∞(Ω) = ∩∞m=0C

m(Ω) and denote C0(Ω) = C(Ω). The sets C0(Ω) and C∞0 (Ω) consist

of functions in C(Ω) and C∞(Ω) respectively, with compact support in Ω. On the otherhand, Cm

B (Ω) denotes the set of functions u ∈ Cm(Ω) for which Dαu is bounded on Ω for0 ≤ |α| ≤ m. Observe that Cm

B (Ω) is a Banach space with the norm denoted by ‖.‖B,m givenby

‖u‖B,m = max0≤|α|≤m

supx∈Ω

|Dαu(x)| .

Also, we define Cm(Ω) as the set of functions u ∈ Cm(Ω) for which Dαu is bounded anduniformly continuous on Ω for 0 ≤ |α| ≤ m. Observe that Cm(Ω) is a closed subspace ofCm

B (Ω) and is also a Banach space with the norm inherited from CmB (Ω). Finally we define

the spaces of Holder continuous functions.

Definition 6.1.10 (Spaces of Holder Continuous Functions). If 0 < λ < 1, for a nonnegativeinteger m we define the space of Holder continuous functions denoted by Cm,λ(Ω), as thesubspace of Cm(Ω) consisting of those functions u for which, for 0 ≤ |α| ≤ m, there exists aconstant K such that

|Dαu(x)−Dαu(y)| ≤ K|x− y|λ,∀x, y ∈ Ω.

Cm,λ(Ω) is a Banach space with the norm denoted by ‖.‖m,λ given by

‖u‖m,λ = ‖u‖B,m + max0≤|α|≤m

supx,y∈Ω

|Dαu(x)−Dαu(y)||x− y|λ , x 6= y

.

Theorem 6.1.11. The space C0(Ω) is dense in Lp(Ω), for 1 ≤ p < ∞.

Proof: For the proof we need the following lemma:

Lemma 6.1.12. Let f ∈ L1loc(Ω) such that

∫

Ω

fu dx = 0,∀u ∈ C0(Ω). (6.39)

Then f = 0 a.e. in Ω.

Suppose f ∈ L1(Ω) and m(Ω) < ∞. Given ε > 0, there exists f1 ∈ C0(Ω) such that‖f − f1‖1 < ε (the proof can be found in [6]) and thus, from (6.39) we obtain

|∫

Ω

f1u dx| ≤ ε‖u‖∞, ∀u ∈ C0(Ω). (6.40)


DefiningK1 = x ∈ Ω | f1(x) ≥ ε, (6.41)

andK2 = x ∈ Ω | f1(x) ≤ −ε. (6.42)

As K1 and K2 are disjoint compact sets, by the Urysohn Theorem there exists u0 ∈ C0(Ω)such that

u0(x) =

+1, if x ∈ K1,−1, if x ∈ K2

(6.43)

and|u0(x)| ≤ 1,∀x ∈ Ω. (6.44)

Also defining K = K1 ∪K2, we may write∫

Ω

f1u0 dx =

∫

Ω−K

f1u0 dx +

∫

K

f1u0 dx. (6.45)

Observe that, from (6.40) ∫

K

|f1| dx ≤∫

Ω

|f1u0| dx ≤ ε (6.46)

so that ∫

Ω

|f1| dx =

∫

K

|f1| dx +

∫

Ω−K

|f1| dx ≤ ε + εm(Ω). (6.47)

Hence‖f‖1 ≤ ‖f − f1‖1 + ‖f1‖1 ≤ 2ε + εm(Ω). (6.48)

Since ε > 0 is arbitrary, we have that f = 0 a.e. in Ω. Finally, if m(Ω) = ∞ , define

Ωn = x ∈ Ω | dist(x, Ωc) > 1/n and |x| < n. (6.49)

It is clear that Ω = ∪∞n=1Ωn and from above f = 0 a.e. on Ωn, ∀n ∈ N, so that f = 0 a.e. inΩ.

Finally, to finish the proof of Theorem 6.1.11, suppose h ∈ Lq(Ω) is such that∫

Ω

hu dx = 0,∀u ∈ C0(Ω). (6.50)

Observe that h ∈ L1loc(Ω) since

∫K|h| dx ≤ ‖h‖qm(K)1/p < ∞. From last lemma h = 0 a.e.

in Ω, which completes the proof. ¤

Theorem 6.1.13. Lp(Ω) is separable for any 1 ≤ p < ∞.

Proof: The result follows from last theorem and from the fact that C0(K) is separable foreach K ⊂ Ω compact (from the Weierstrass theorem, polynomials with rational coefficientsare dense C0(K)). Observe that Ω = ∪∞n=1Ωn, Ωn defined as in (6.49), where Ωn is compact,∀n ∈ N.


Theorem 6.1.14. We denote

u(x) =

u(x), ifx ∈ Ω,0, otherwise.

Let 1 ≤ p < ∞. A bounded set K ≤ Lp(Ω) is pre-compact in Lp(Ω) if and only if for everyε > 0 there exists δ > 0 and a subset A ⊂⊂ Ω such that for every u ∈ K and every h ∈ Rn

with |h| < δ we have ∫

Ω

|u(x + h)− u(x)| dx < εp, (6.51)

and ∫

Ω−A

|u(x)|p dx < εp. (6.52)

Proof: See Adams [1] Theorem 2.21.

6.2 The Sobolev Spaces

Now we define the Sobolev spaces, denoted by Wm,p(Ω).

Definition 6.2.1 (Sobolev Spaces). We say that u ∈ Wm,p(Ω) if u ∈ Lp(Ω) and Dαu ∈Lp(Ω), for all α such that 0 ≤ |α| ≤ m, where the derivatives are understood in the distribu-tional sense.

Definition 6.2.2. We define the norm ‖.‖m,p for Wm,p(Ω), where m ∈ N and 1 ≤ p ≤ ∞,as

‖u‖m,p =

∑

0≤|α|≤m

‖Dαu‖pp

1/p

, if 1 ≤ p < ∞, (6.53)

and‖u‖m,∞ = max

0≤|α|≤m‖Dαu‖∞ . (6.54)

Theorem 6.2.3. Wm,p(Ω) is a Banach space.

Proof: Consider un a Cauchy sequence in Wm,p(Ω). Then Dαun is a Cauchy sequence foreach 0 ≤ |α| ≤ m. Since Lp(Ω) is complete there exist functions u and uα, for 0 ≤ |α| ≤ m, inLp(Ω) such that un → u and Dαun → uα in Lp(Ω) as n →∞. From above Lp(Ω) ⊂ L1

loc(Ω)and so un determines a distribution Tun ∈ D′(Ω). For any φ ∈ D(Ω) we have, by Holder’sinequality

|Tun(φ)− Tu(φ)| ≤∫

Ω

|un(x)− u(x)||φ(x)|dx ≤ ‖φ‖q‖un − u‖p. (6.55)


Hence Tun(φ) → Tu(φ) for every φ ∈ D(Ω) as n → ∞. Similarly TDαun(φ) → Tuα(φ) forevery φ ∈ D(Ω). We have that

Tuα(φ) = limn→∞

TDαun(α) = limn→∞

(−1)|α|Tun(Dαφ) = (−1)Tu(Dαφ) = TDαu(φ), (6.56)

for every φ ∈ D(Ω). Thus uα = Dαu in the sense of distributions, for 0 ≤ |α| ≤ m, andu ∈ Wm,p(Ω). As limn→∞ ‖u− un‖m,p = 0, Wm,p(Ω) is complete. ¤

Remark 6.2.4. Observe that distributional and classical derivatives coincide when the latterexist and are continuous. We define S ⊂ Wm,p(Ω) by

S = φ ∈ Cm(Ω) | ‖φ‖m,p < ∞ (6.57)

Thus, the completion of S concerning the norm ‖.‖m,p is denoted by Hm,p(Ω).

Corollary 6.2.5. Hm,p(Ω) ⊂ Wm,p(Ω)

Proof: Since Wm,p(Ω) is complete we have that Hm,p(Ω) ⊂ Wm,p(Ω). ¤

Theorem 6.2.6. Wm,p(Ω) is separable if 1 ≤ p < ∞, and is reflexive and uniformly convexif 1 < p < ∞. Particularly, Wm,2(Ω) is a separable Hilbert space with the inner product

(u, v)m =∑

0≤|α|≤m

〈Dαu,Dαv〉L2(Ω). (6.58)

Proof: We can see Wm,p(Ω) as a subspace of Lp(Ω,RN), where N =∑

0≤|α|≤m 1. From the

relevant properties for Lp(Ω), we have that Lp(Ω;RN) is a reflexive and uniformly convexfor 1 < p < ∞ and separable for 1 ≤ p < ∞. Given u ∈ Wm,p(Ω), we may associate thevector Pu ∈ Lp(Ω;RN) defined by

Pu = Dαu0≤|α|≤m. (6.59)

Since ‖Pu‖pN = ‖u‖m,p, we have that Wm,p is closed subspace of Lp(Ω;RN). Thus fromtheorem 1.21 in Adams [1], we have that Wm,p(Ω) is separable if 1 ≤ p < ∞ and, reflexiveand uniformly convex, if 1 < p < ∞. ¤

Lemma 6.2.7. Let 1 ≤ p < ∞ and define U = Lp(Ω;RN). For every continuous linearfunctional f on U , there exists a unique v ∈ Lq(Ω;RN) = U∗ such that

f(u) =N∑

i=1

〈ui, vi〉,∀u ∈ U. (6.60)

Moreover,‖f‖U∗ = ‖v‖qN , (6.61)

where ‖.‖qN = ‖.‖Lq(Ω,RN ).


Proof: For u = (u1, ..., un) ∈ Lp(Ω;RN) we may write

f(u) = f((u1, 0, ..., 0)) + ... + f((0, ..., 0, uj, 0, ..., 0)) + ... + f((0, ..., 0, un)), (6.62)

and since f((0, ..., 0, uj, 0, ..., 0)) is continuous linear functional on uj ∈ Lp(Ω), there exists aunique vj ∈ Lq(Ω) such that f(0, ..., 0, uj, 0, ..., 0) = 〈uj, vj〉L2(Ω),∀uj ∈ Lp(Ω), ∀ 1 ≤ j ≤ N ,so that

f(u) =N∑

i=1

〈ui, vi〉,∀u ∈ U. (6.63)

From Holder’s inequality we obtain

|f(u)| ≤N∑

j=1

‖uj‖p‖vj‖q ≤ ‖u‖pN‖v‖qN , (6.64)

and hence ‖f‖U∗ ≤ ‖v‖qN . The equality in (6.64) is achieved for u ∈ Lp(Ω,RN), 1 < p < ∞such that

uj(x) =

|vj|q−2vj, if vj 6= 00, if vj = 0.

(6.65)

If p = 1 choose k such that ‖vk‖∞ = max1≤j≤N ‖vj‖∞. Given ε > 0, there is a measurableset A such that m(A) > 0 and |vk(x)| ≥ ‖vk‖∞ − ε, ∀x ∈ A. Defining u(x) as

ui(x) =

vk/vk, if i = k, x ∈ A and vk(x) 6= 00, otherwise,

(6.66)

we have

f(uk) = 〈u, vk〉L2(Ω) =

∫

A

|vk|dx ≥ (‖(vk‖∞ − ε)‖u‖1 = (‖v‖∞N − ε)‖uk‖1N . (6.67)

Since ε is arbitrary, the proof is complete. ¤

Theorem 6.2.8. Let 1 ≤ p < ∞. Given a continuous linear functional f on Wm,p(Ω), thereexists v ∈ Lq(Ω,RN) such that

f(u) =∑

0≤|α|≤m

〈Dαu, vα〉L2(Ω). (6.68)

Proof: Consider f a continuous linear operator on U = Wm,p(Ω). By the Hahn BanachTheorem, we can extend f to f , on Lp(Ω;RN), so that ‖f‖qN = ‖f‖U∗ and by the lasttheorem, there exists vα ∈ Lq(Ω;RN) such that

f(u) =∑

0≤|α|≤m

〈uα, vα〉L2(Ω),∀v ∈ Lp(Ω;RN). (6.69)


In particular for u ∈ Wm,p(Ω), defining u = Dαu ∈ Lp(Ω;RN) we obtain

f(u) = f(u) =∑

1≤|α|≤m

〈Dαu, vα〉L2(Ω). (6.70)

Finally, observe that, also from the Hahn-Banach theorem ‖f‖U∗ = ‖f‖qN = ‖v‖qN . ¤Now we state some density results. The proofs and more details may be found in Adams [1]Chapters 2 and 3.

Proposition 6.2.9. C∞0 (Ω) is dense in Lp(Ω) if 1 ≤ p < ∞.

Theorem 6.2.10 (Meyers and Serrin). If 1 ≤ p ≤ ∞, then Hm,p(Ω) = Wm,p(Ω).

6.3 The Sobolev Imbedding Theorem

We start with some preliminary definitions.

Definition 6.3.1 (The Cone Condition). The set Ω ⊂ Rn has the cone property if thereexists a finite cone C such that each point x ∈ Ω is the vertex of a finite cone Cx containedin Ω and congruent to C.

Definition 6.3.2 (The Strong Local Lipschitz Property). Ω has the strong local Lipschitzproperty if there exist positive numbers δ and M , a locally finite open cover Uj of bdryΩ,and for each Uj a real valued function fj of n−1 variables, such that the following conditionshold.

1. For some N ∈ N, every collection of N + 1 of the sets Uj has empty intersection.

2. For every x, y ∈ Ωδ = x ∈ Ω | dist(x, bdryΩ) < δ such that |x− y| < δ there existsj such that

x, y ∈ Vj = z ∈ Uj | dist(z, bdryUj) > δ. (6.71)

3. Each function fj satisfies a Lipschitz condition with constant M , that is

|f(ξ1, ..., ξn−1)− f(η1, ..., ηn−1)| ≤ M |(ξ1 − η1, ..., ξn−1 − ηn−1)|. (6.72)

4. For some Cartesian coordinate system (ξj,1, ..., ξj,n) in Uj the set Ω ∩ Uj is representedby the inequality ξj,n < fj(ξj,1, ..., ξj,n−1).


6.3.1 The Statement of Sobolev Imbedding Theorem

Now we present the Sobolev Imbedding Theorem. For a proof see Adams [1], Chapter 3.We recall that for normed spaces X,Y the notation

X → Y

means that X ⊂ Y and there exists a constant K > 0 such that

‖u‖Y ≤ K‖u‖X ,∀u ∈ X.

If in addition the imbedding is compact then for any bounded sequence un ⊂ X thereexists a convergent subsequence unk

, which converges to some u in the norm ‖.‖Y .

Theorem 6.3.3 (The Sobolev Imbedding Theorem). Let Ω be a domain in Rn and, for1 ≤ k ≤ n, let Ωk be the intersection of Ω with a plane of dimension k in Rn (if k=n, thenΩk = Ω). Let j ≥ 0 and m ≥ 1 be integers and let 1 ≤ p < ∞.

1. Part I. Suppose Ω satisfies the cone condition.

(a) Case A If either mp > n or m = n and p = 1 then

W j+m,p(Ω) → CjB(Ω). (6.73)

Moreover, if 1 ≤ k ≤ n, then

W j+m,p(Ω) → W j,q(Ωk), for p ≤ q ≤ ∞, (6.74)

and, in particular

Wm,p(Ω) → Lq(Ω), for p ≤ q ≤ ∞. (6.75)

(b) Case B If 1 ≤ k ≤ n and mp = n, then

W j+m,p(Ω) → W j,q(Ωk), for p ≤ q < ∞, (6.76)

and, in particular

Wm,p(Ω) → Lq(Ω), for p ≤ q < ∞. (6.77)

(c) Case C If mp < n and either n −mp < k ≤ n or p = 1 and n −m ≤ k ≤ n,then

W j+m,p(Ω) → W j,q(Ωk), for p ≤ q ≤ p∗ =kp

n−mp, (6.78)

and, in particular

Wm,p(Ω) → Lq(Ω), for p ≤ q ≤ p∗ =kp

n−mp. (6.79)

The imbedding constants depend only on n,m, p, q, j, k and the dimensions of theC in the cone condition.


2. Part II. Suppose Ω satisfies the strong local Lipschitz condition. Then the target CjB

in the first imbedding above may be replaced by Cj(Ω), and the imbedding can be furtherrefined as follows:If mp > n > (m− 1)p, then

W j+m,p → Cj,λ(Ω), for 0 < λ ≤ m− (n/p), (6.80)

and if n = (m− 1)p, then

W j+m,p → Cj,λ(Ω), for 0 < λ ≤ 1. (6.81)

Also, if n = m− 1 and p = 1, then (6.81) holds for λ = 1 as well.

3. Part III. All imbeddings in Parts A and B are valid for arbitrary domains Ω if theW − space undergoing the imbedding is replaced with the corresponding W0 − space.

6.4 The Rellich-Kondrachov Theorem

In this section we present the Rellich-Kondrachov theorem. We start with the followingresult which is proved in [1].

Theorem 6.4.1. Let m be a non-negative integer and let 0 < ν < λ ≤ 1. Then followingimbeddings exist:

1. Cm+1(Ω) → Cm(Ω),

2. Cm,λ(Ω) → Cm(Ω),

3. Cm,λ(Ω) → Cm,ν(Ω).

If Ω is bounded, then imbeddings 2 and 3 are compact.

Theorem 6.4.2 (Rellich-Kondrachov). Let Ω be a bounded domain. Let j, m be integers,j ≥ 0,m ≥ 1, and let 1 ≤ p < ∞.

1. Part I- If Ω has the cone property and mp ≤ n, then the following imbeddings arecompact:

W j+m,p(Ω) → W j,q(Ω), 0 < n−mp < n and 1 ≤ np/(n−mp), (6.82)

W j+m,p(Ω) → W j,q(Ω), if n = mp, 1 ≤ q < ∞. (6.83)


2. Part II- If Ω has the cone property and mp > n, then the following imbeddings arecompact:

W j+m,p → CjB(Ω), (6.84)

W j+m,p(Ω) → W j,q(Ω), if 1 ≤ q ≤ ∞. (6.85)

3. Part III- If Ω has the strong Lipschitz property, then the following imbeddings arecompact:

W j+m,p(Ω) → Cj(Ω), if mp > n, (6.86)

W j+m,p(Ω) → Cj,λ(Ω), if mp > n ≥ (m− 1)p and 0 < λ < m− n/p. (6.87)

4. Part IV- All the above imbeddings are compact if we replace W j+m,p(Ω) by W j+m,p0 (Ω).

Remark 6.4.3. Given X, Y, Z spaces, for which we have the imbeddings X → Y and Y → Zand if one of these imbeddings is compact then the composite imbedding X → Z is compact.Since the extension operator u → u where u(x) = u(x) if x ∈ Ω and u(x) = 0 otherwise,defines an imbedding W j+m,p

0 (Ω) → W j+m,p(Rn) we have that Part-IV of above theoremfollows from the application of Parts I-III to Rn (despite the fact we are assuming Ω bounded,the general results may be found in Adams [1]).

Remark 6.4.4. To prove the compactness of any of above imbeddings it is sufficient toconsider the case j = 0. Suppose, for example, that the first imbedding has been proved forj = 0. For j ≥ 1 and ui bounded sequence in W j+m,p(Ω) we have that Dαui is boundedin Wm,p(Ω) for each α such that |α| ≤ j. From the case j = 0 it is possible to extract asubsequence (similarly to a diagonal process) uik for which Dαuik converges in Lq(Ω)for each α such that |α| ≤ j, so that uik converges in W j,q(Ω).

Remark 6.4.5. Since Ω is bounded, C0B(Ω) → Lq(Ω) for 1 ≤ q ≤ ∞. In fact

‖u‖0,q,Ω ≤ ‖u‖C0B[vol(Ω)]1/q. (6.88)

Thus the compactness of (6.85) (for j = 0) follows from that of (6.84).

Proof of Part III : If mp > n > (m − 1)p and 0 < λ < (m − n)/p, then there exists µsuch that λ < µ < m − (n/p). Since Ω is bounded, the imbedding C0,µ(Ω) → C0,λ(Ω) iscompact by Theorem 1.31 in Adams [1]. Since by the Sobolev Imbedding Theorem we haveWm,p(Ω) → C0,µ(Ω), we have that imbedding (6.87) is compact.

If mp > n, let j∗ be the non-negative integer satisfying (m− j∗)p > n ≥ (m− j∗−1)p. Thuswe have the chain of imbeddings

Wm,p(Ω) → Wm−j∗,p(Ω) → C0,µ(Ω) → C(Ω), (6.89)

where 0 < µ < m− j∗− (n/p). The last imbedding in (6.89) is compact by Theorem 1.31 inAdams [1], so that (6.86) is compact for j = 0.


Proof of Part II: Supposing that Ω has the cone property, we may write Ω = ∪Mk=1Ωk, where

each Ωk has the strong local Lipschitz property. If mp > n, then Wm,p(Ω) → Wm,p(Ωk) →C(Ωk), the latter imbedding being compact as proved above. If ui is a bounded sequencein Wm,p(Ω), we may select a subsequence uil whose restriction to Ωk converges in C(Ωk),for all k such that 1 ≤ k ≤ M . Thus, uil converges in C0

B(Ω) proving that (6.84) iscompact for j = 0. Therefore from the above remarks, (6.85) is also compact. For the proofof Part I, we need the following lemma:

Lemma 6.4.6. Let Ω be an bounded domain in Rn. Let 1 ≤ q1 ≤ q0 and suppose

Wm,p(Ω) → Lq0(Ω), (6.90)

Wm,p(Ω) → Lq1 . (6.91)

Suppose also that (6.91) is compact. If q1 ≤ q < q0, then the imbedding

Wm,p → Lq(Ω) (6.92)

is compact.

Proof: Define λ = q1(q0−q)/(q(q0−q1)) and µ = q0(q−q1)/(q(q0−q1)). We have that λ > 0and µ ≥ 0. From Holder’s inequality and (6.90) there exists K ∈ R+ such that,

‖u‖0,q,Ω ≤ ‖u‖λ0.q1,Ω‖u‖µ

0,q0,Ω ≤ K‖u‖0,q1,Ω‖u‖µm,p,Ω, ∀u ∈ Wm,p(Ω). (6.93)

Thus considering a sequence ui bounded in Wm,p(Ω), since (6.91) is compact there existsa subsequence unk that converges, and is therefore a Cauchy sequence in Lq1(Ω). From(6.93), unk is also a Cauchy sequence in Lq(Ω), so that (6.92) is compact.

Proof of Part I: Consider j = 0. Define q0 = np/(n−mp). To prove the imbedding

Wm,p(Ω) → Lq(Ω), 1 ≤ q < q0, (6.94)

is compact, by last lemma it suffices to do so only for q = 1. For k ∈ N, define

Ωk = x ∈ Ω | dist(x, ∂Ω) > 2/k. (6.95)

Suppose A is set of functions bounded in Wm,p(Ω). Also, suppose given ε > 0, and define,for u ∈ Wm,p, u(x) = u(x) if x ∈ Ω, u(x) = 0, otherwise. From Holder’s inequality andconsidering that Wm,p(Ω) → Lq0(Ω), we have

∫

Ω−Ωk

|u(x)|dx ≤∫

Ω−Ωk

|u(x)|q0dx

1/q0∫

Ω−Ωk

1dx

1−1/q0

≤ K1‖u‖m,p,Ω[vol(Ω−Ωk)]1−1/q0 ,

(6.96)Thus, since u is bounded in Wm,p, there exists K0 ∈ N such that if k ≥ K0 then

∫

Ω−Ωk

|u(x)|dx < ε, (6.97)


and, for every h ∈ Rn, ∫

Ω−Ωk

|u(x + h)− u(x)|dx < 2ε. (6.98)

Observe that if |h| < 1/k, then x + th ∈ Ω2k provided x ∈ Ωk and 0 ≤ t ≤ 1. If u ∈ C∞(Ω)we have that

∫

Ωk

|u(x + h)− u(x)| ≤∫

Ωk

dx

∫ 1

0

|du(x + th)

dt|dt

≤ |h|∫ 1

0

dt

∫

Ω2k

|∇u(y)|dy ≤ |h|‖u‖1,1,Ω

≤ K2|h|‖u‖m,p,Ω. (6.99)

Since C∞(Ω) is dense in Wm,p(Ω), for |h| sufficiently small

∫

Ω

|u(x + h)− u(x)|dx < 3ε, ∀u ∈ A, (6.100)

which means that A is pre-compact in L1(Ω) and therefore from Theorem 2.21 in Adams [1],the imbedding indicated (6.94) is compact for q = 1, which completes the proof.

Chapter 7

Basic Concepts on Convex Analysis

7.1 Convex Sets and Convex Functions

Let S be a subset of a vector space U . We recall that S is convex if given u, v ∈ S then

λu + (1− λ)v ∈ S, ∀λ ∈ [0, 1]. (7.1)

Definition 7.1.1 (Convex hull). Let S be a subset of a vector space U , we define the convexhull of S, denoted by Co(S) as

Co(S) =

n∑

i=1

λiui | n ∈ N,

n∑i=1

λi = 1, λi ≥ 0, ui ∈ S, ∀i ∈ 1, ..., n

. (7.2)

Definition 7.1.2 (Convex Functional). Let S be convex subset of the vector space U . Afunctional F : S → R = R ∪ +∞,−∞ is said to be convex if

F (λu + (1− λ)v) ≤ λF (u) + (1− λ)F (v), ∀u, v ∈ S, λ ∈ [0, 1]. (7.3)

Definition 7.1.3 (Lower Semi-continuity). Let U be Banach space. We say that F : U → Ris lower semi-continuous (l.s.c.) at u ∈ U , if

lim infn→+∞

F (un) ≥ F (u), (7.4)

wheneverun → u strongly (in norm). (7.5)

Definition 7.1.4 (Weak Lower Semi-Continuity). Let U be Banach space. We say thatF : U → R is weakly lower semi-continuous (w.l.s.c.) at u ∈ U , if

lim infn→+∞

F (un) ≥ F (u), (7.6)

wheneverun u, weakly. (7.7)

58

CHAPTER 7. BASIC CONCEPTS ON CONVEX ANALYSIS 59

Remark 7.1.5. We say that F is a (weak) lower semi-continuous function, if F : U → Ris (weak) lower semi-continuous ∀u ∈ U .

Definition 7.1.6 (Epigraph). Given F : U → R we define its Epigraph, denoted by Epi(F )as

Epi(F ) = (u, a) ∈ U × R | a ≥ F (u).

Now we present a very important result but which we do not prove.

Proposition 7.1.7. A function F : U → R is l.s.c. (lower semi-continuous) if and only ifits epigraph is closed.

Corollary 7.1.8. Every convex l.s.c. function F : U → R is also w.l.s.c. (weakly lowersemi-continuous).

Proof: The result follows from the fact that the epigraph of F is convex and closed convexsets are weakly closed. ¤

Definition 7.1.9 (Affine Continuous Function). Let U be a Banach space. A functionalF : U → R is said to be affine continuous if there exist u∗ ∈ U∗ and α ∈ R such that

F (u) = 〈u, u∗〉U + α, ∀u ∈ U. (7.8)

Definition 7.1.10 (Γ(U)). Let U be a Banach space, we say that F : U → R belongs toΓ(U) and write F ∈ Γ(U) if F can be represented as the point-wise supremum of a family ofaffine continuous functions. If F ∈ Γ(U), F 6= +∞ and F 6= −∞ for some u ∈ U then wewrite F ∈ Γ0(U).

Proposition 7.1.11. Let U be a Banach space, then F ∈ Γ(U) if and only if F is convexand l.s.c., and if F takes the value −∞ then F ≡ −∞.

Definition 7.1.12 (Convex Envelope). Let U be a Banach space. Given F : U → R, wedefine its convex envelope, denoted by CF : U → R as

CF (u) = sup(u∗,α)∈A∗

〈u, u∗〉+ α, (7.9)

whereA∗ = (u∗, α) ∈ U∗ × R | 〈v, u∗〉U + α ≤ F (v),∀v ∈ U (7.10)

Definition 7.1.13 (Polar Functionals). Given F : U → R, we define the related polarfunctional, denoted by F ∗ : U∗ → R, as

F ∗(u∗) = supu∈U

〈u, u∗〉U − F (u), ∀u∗ ∈ U∗. (7.11)


Definition 7.1.14 (Bipolar Functional). Given F : U → R, we define the related bipolarfunctional, denoted by F ∗∗ : U → R, as

F ∗∗(u) = supu∗∈U∗

〈u, u∗〉U − F ∗(u∗), ∀u ∈ U. (7.12)

Proposition 7.1.15. Given F : U → R, then F ∗∗(u) = CF (u) and in particular if F ∈ Γ(U)then F ∗∗(u) = F (u).

Proof: By definition, the convex envelope of F is the supremum of all affine continuousminorants of F . We can consider only the maximal minorants, that functions of the form

u 7→ 〈u, u∗〉U − F ∗(u∗). (7.13)

Thus,CF (u) = sup

u∗∈U∗〈u, u∗〉U − F ∗(u∗) = F ∗∗(u). ¤ (7.14)

Corollary 7.1.16. Given F : U → R, we have F ∗ = F ∗∗∗.

Proof Since F ∗∗ ≤ F we obtainF ∗ ≤ F ∗∗∗. (7.15)

On the other hand, we have

F ∗∗(u) ≥ 〈u, u∗〉U − F ∗(u∗), (7.16)

so thatF ∗∗∗(u∗) = sup

u∈U〈u, u∗〉U − F ∗∗(u) ≤ F ∗(u∗). (7.17)

From (7.15) and (7.17) we obtain F ∗(u∗) = F ∗∗∗(u∗). ¤

Definition 7.1.17 (Gateaux Differentiability). A functional F : U → R is said to be Gateauxdifferentiable at u ∈ U if there exists u∗ ∈ U∗ such that:

limλ→0

F (u + λh)− F (u)

λ= 〈h, u∗〉U , ∀h ∈ U. (7.18)

The vector u∗ is said to be the Gateaux derivative of F : U → R at u and may be denoted asfollows:

u∗ =∂F (u)

∂uor u∗ = δF (u) (7.19)

Definition 7.1.18 (Sub-gradients). Given F : U → R, we define the set of sub-gradients ofF at u, denoted by ∂F (u) as:

∂F (u) = u∗ ∈ U∗, such that 〈v − u, u∗〉U + F (u) ≤ F (v), ∀v ∈ U. (7.20)


Definition 7.1.19 (Adjoint Operator). Let U and Y be Banach spaces and Λ : U → Y acontinuous linear operator. The Adjoint Operator related to Λ, denoted by Λ∗ : Y ∗ → U∗ isdefined through the equation:

〈u, Λ∗v∗〉U = 〈Λu, v∗〉Y , ∀u ∈ U, v∗ ∈ Y ∗. (7.21)

Lemma 7.1.20 (Continuity of Convex Functions). If in a neighborhood of a point u ∈ U , aconvex function F is bounded above by a finite constant, then F is continuous at u.

Proof:By translation, we may reduce the problem to the case where u = θ and F (u) = 0. LetV be a neighborhood of origin such that F (v) ≤ a < +∞,∀v ∈ V. Define W = V ∩ (−V)(which is a symmetric neighborhood of origin). Pick ε ∈ (0, 1). If v ∈ εW, since F is convexand

v

ε∈ V (7.22)

we may infer thatF (v) ≤ (1− ε)F (0) + εF (v/ε) ≤ εa. (7.23)

Also −v

ε∈ V . (7.24)

ThusF (v) ≥ (1 + ε)F (0)− εF (−v/ε) ≥ −εa. (7.25)

Therefore|F (v)| ≤ εa,∀v ∈ εW , (7.26)

that is, F is continuous at u = θ. ¤

Proposition 7.1.21. Let F : U → R be a convex function finite and continuous at u ∈ U .Then ∂F (u) 6= ∅.

Proof: Since F is convex, Epi(F ) is convex, as F is continuous at u, we have that Epi(F )is non-empty. Observe that (u, F (u)) belongs to the boundary of Epi(F ), so that denotingA = Epi(F ), we may separate (u, F (u)) from A by a closed hyper-plane H, which may bewritten as

H = (v, a) ∈ U × R | 〈v, u∗〉U + αa = β, (7.27)

for some fixed α, β ∈ R and u∗ ∈ U∗, so that

〈v, u∗〉U + αa ≥ β, ∀(v, a) ∈ Epi(F ), (7.28)

and〈u, u∗〉U + αF (u) = β, (7.29)

where (α, β, u∗) 6= (0, 0, θ). Suppose α = 0, thus we have

〈v − u, u∗〉U ≥ 0,∀v ∈ U, (7.30)


and thus we obtain u∗ = θ, and β = 0. Therefore we may assume α > 0 (considering (7.28))so that ∀v ∈ U we have

β

α− 〈v, u∗/α〉U ≤ F (v), (7.31)

andβ

α− 〈u, u∗/α〉U = F (u), (7.32)

or〈v − u,−u∗/α〉U + F (u) ≤ F (v),∀v ∈ U, (7.33)

so that−u∗/α ∈ ∂F (u), ¤ (7.34)

Definition 7.1.22 (Caratheodory Mapping ). Let S ⊂ Rn be an open set, we say that thatg : S × Rl → R is a Caratheodory mapping if:

∀ξ ∈ Rl, x 7→ g(x, ξ) is a measurable function,

andfor almost all x ∈ S, ξ 7→ g(x, ξ) is a continuous function.

The proof of next results may be found in Ekeland and Temam [14].

Proposition 7.1.23. Let E and F be two Banach spaces, S a Borel subset of Rn, andg : S×E → F a Caratheodory mapping. For each measurable function u : S → E, let G1(u)be the measurable function x 7→ g(x, u(x)) ∈ F .

If G1 maps Lp(S, E) into Lr(S, F ) for 1 ≤ p, r < ∞, then G1 is continuous in the normtopology.

For the functional G : U → R, defined by G(u) =∫

Sg(x, u(x))dS , where U = U∗ = [L2(S)]l

(this is a especial case of the more general hypothesis presented in [14]) we have the followingresult.

Proposition 7.1.24. Considering the last proposition we can express G∗ : U∗ → R as :

G∗(u∗) =

∫

S

g∗(x, u∗(x))dS, (7.35)

where g∗(x, y) = supη∈Rl

(y · η − g(x, η)), almost everywhere in S.

For non-convex functionals it may be sometimes difficult to express analytically conditionsfor a global extremum. This fact motivates the definition of Legendre Transform, which isestablished through a local extremum.


Definition 7.1.25 (Legendre’s Transform and Associated Functional). Consider a differen-tiable function g : Rn → R. Its Legendre Transform, denoted by g∗L : Rn

L → R is expressedas:

g∗L(y∗) = x0i · y∗i − g(x0), (7.36)

where x0 is the solution of the system:

y∗i =∂g(x0)

∂xi

, (7.37)

and RnL = y∗ ∈ Rn such that equation (7.37) has a unique solution.

Furthermore, considering the functional G : Y → R defined as G(v) =∫

Sg(v)dS, we define

the Associated Legendre Transform Functional, denoted by G∗L : Y ∗

L → R as:

G∗L(v∗) =

∫

S

g∗L(v∗)dS, (7.38)

where Y ∗L = v∗ ∈ Y ∗ | v∗(x) ∈ Rn

L, a.e. in S.

About the Legendre transform we still have the following results:

Proposition 7.1.26. Considering the last definitions, suppose that for each y∗ ∈ RnL at least

in a neighborhood (of y∗) it is possible to define a differentiable function by the expression

x0(y∗) = [

∂g

∂x]−1(y∗). (7.39)

Then, ∀ i ∈ 1, ..., nwe may write:

y∗i =∂g(x0)

∂xi

⇔ x0i =∂g∗L(y∗)

∂y∗i(7.40)

Proof: Suppose firstly that:

y∗i =∂g(x0)

∂xi

, ∀ i ∈ 1, ..., n, (7.41)

thus:g∗L(y∗) = y∗i x0i − g(x0) (7.42)

and taking derivatives for this expression we have:

∂g∗L(y∗)∂y∗i

= y∗j∂x0j

∂y∗i+ x0i − ∂g(x0)

∂xj

∂x0j

∂y∗i, (7.43)

or∂g∗L(y∗)

∂y∗i= (y∗j −

∂g(x0)

∂xj

)∂x0j

∂y∗i+ x0i (7.44)


which from (7.41) implies that:


= x0i , ∀ i ∈ 1, ..., n. (7.45)

This completes the first half of the proof. Conversely, suppose now that:

x0i =∂g∗L(y∗)

∂y∗i, ∀ i ∈ 1, ..., n. (7.46)

As y∗ ∈ RnL there exists x0 ∈ Rn such that:

y∗i =∂g(x0)

∂xi

∀ i ∈ 1, ..., n, (7.47)

and,g∗L(y∗) = y∗i x0i − g(x0) (7.48)

and therefore taking derivatives for this expression we can obtain:


= y∗j∂x0j

∂y∗i+ x0i − ∂g(x0)

∂xj

∂x0j

∂y∗i, (7.49)

∀ i ∈ 1, ..., n, so that:


= (y∗j −∂g(x0)

∂xj

)∂x0j

∂y∗i+ x0i (7.50)

∀ i ∈ 1, ..., n, which from (7.46) and (7.47), implies that:

x0i =∂g∗L(y∗)

∂y∗i= x0i , ∀ i ∈ 1, ..., n, (7.51)

from this and (7.47) we have:

y∗i =∂g(x0)

∂xi

=∂g(x0)

∂xi

∀ i ∈ 1, ..., n. ¤ (7.52)

Theorem 7.1.27. Consider the functional J : U → R defined as J(u) = (GΛ)(u)−〈u, f〉Uwhere Λ(= Λi) : U → Y (i ∈ 1, ..., n) is a continuous linear operator and, G : Y → Ris a functional that can be expressed as G(v) =

∫S

g(v)dS, ∀v ∈ Y (here g : Rn → R is adifferentiable function that admits Legendre Transform denoted by g∗L : Rn

L → R. That is,the hypothesis mentioned at Proposition 7.1.26 are satisfied).

Under these assumptions we have:

δJ(u0) = θ ⇔ δ(−G∗L(v∗0) + 〈u0, Λ

∗v∗0 − f〉U) = θ, (7.53)

where v∗0 = ∂G(Λ(u0))∂v

is supposed to be such that v∗0(x) ∈ RnL, a.e. in S and in this case:

J(u0) = −G∗L(v∗0). (7.54)


Proof: Suppose first that δJ(u0) = θ, that is:

Λ∗∂G(Λu0)

∂v− f = θ (7.55)

which, as v∗0 = ∂G(Λu0)∂v

implies that:

Λ∗v∗0 − f = θ, (7.56)

and

v∗0i =∂g(Λu0)

∂xi

. (7.57)

Thus from the last proposition we can write:

Λi(u0) =∂g∗L(v∗0)

∂y∗i, for i ∈ 1, .., n (7.58)

which means:

Λu0 =∂G∗

L(v∗0)∂v∗

. (7.59)

Therefore from (7.56) and (7.59) we have:

δ(−G∗L(v∗0) + 〈u0, Λ

∗v∗0 − f〉U) = θ. (7.60)

This completes the first part of the proof.

Conversely, suppose now that:

δ(−G∗L(v∗0) + 〈u0, Λ

∗v∗0 − f〉U) = θ, (7.61)

that is:Λ∗v∗0 − f = θ (7.62)

and

Λu0 =∂G∗

L(v∗0)∂v∗

. (7.63)

Clearly, from (7.63), the last proposition and (7.62) we can write:

v∗0 =∂G(Λ(u0))

∂v(7.64)

and

Λ∗∂G(Λu0)

∂v− f = θ, (7.65)

which implies:δJ(u0) = θ. (7.66)

Finally, we have:J(u0) = G(Λu0)− 〈u0, f〉U (7.67)

From this, (7.62) and (7.64) we have

J(u0) = G(Λu0)− 〈u0, Λ∗v∗0〉U = G(Λu0)− 〈Λu0, v

∗0〉Y = −G∗

L(v∗0). ¤ (7.68)


7.2 Duality in Convex Optimization

Let U be a Banach space. Given F : U → R (F ∈ Γ0(U)) we define the problem P as

P : minimize F (u) on U. (7.69)

We say that u0 ∈ U is a solution of problem P if F (u0) = infu∈U F (u). Consider a functionφ(u, p) (φ : U × Y → R) such that

φ(u, 0) = F (u), (7.70)

we define the problem P∗, as

P∗ : maximize − φ∗(0, p∗) on Y ∗. (7.71)

Observe that

φ∗(0, p∗) = sup(u,p)∈U×Y

〈0, u〉U + 〈p, p∗〉Y − φ(u, p) ≥ −φ(u, 0), (7.72)

orinfu∈U

φ(u, 0) ≥ supp∗∈Y ∗

−φ∗(0, p∗). (7.73)

Proposition 7.2.1. Consider φ ∈ Γ0(U × Y ). If we define

h(p) = infu∈U

φ(u, p), (7.74)

then h is convex.

Proof: We have to show that given p, q ∈ Y and λ ∈ (0, 1), we have

h(λp + (1− λ)q) ≤ λh(p) + (1− λ)h(q). (7.75)

If h(p) = +∞ or h(q) = +∞ we are done. Thus let us assume h(p) < +∞ and h(q) < +∞.For each a > h(p) there exists u ∈ U such that

h(p) ≤ φ(u, p) ≤ a, (7.76)

and, if b > h(q), there exists v ∈ U such that

h(q) ≤ φ(v, q) ≤ b. (7.77)

Thush(λp + (1− λ)q) ≤ inf

w∈Uφ(w, λp + (1− λ)q)

≤ φ(λu + (1− λ)v, λp + (1− λ)q) ≤ λφ(u, p) + (1− λ)φ(v, q) ≤ λa + (1− λ)b. (7.78)

Letting a → h(p) and b → h(q) we obtain

h(λp + (1− λ)q) ≤ λh(p) + (1− λ)h(q). ¤ (7.79)


Proposition 7.2.2. For h as above, we have h∗(p∗) = φ∗(0, p∗),∀p∗ ∈ Y ∗, so that

h∗∗(0) = supp∗∈Y ∗

−φ∗(0, p∗). (7.80)

Proof: Observe that

h∗(p∗) = supp∈Y

〈p, p∗〉Y − h(p) = supp∈Y

〈p, p∗〉Y − infu∈U

φ(u, p), (7.81)

so thath∗(p∗) = sup

(u,p)∈U×Y

〈p, p∗〉Y − φ(u, p) = φ∗(0, p∗). ¤ (7.82)

Proposition 7.2.3. The set of solutions of the problem P∗ (the dual problem) is identicalto ∂h∗∗(0).

Proof: Consider p∗0 ∈ Y ∗ a solution of Problem P∗, that is,

−φ∗(0, p∗0) ≥ −φ∗(0, p∗), ∀p∗ ∈ Y ∗, (7.83)

which is equivalent to−h∗(p∗0) ≥ −h∗(p∗),∀p∗ ∈ Y ∗, (7.84)

which is equivalent to

−h(p∗0) = supp∗∈Y ∗

〈0, p∗〉Y − h∗(p∗) ⇔ −h∗(p∗0) = h∗∗(0) ⇔ p∗0 ∈ ∂h∗∗(0). ¤ (7.85)

Theorem 7.2.4. Consider φ : U × Y → R convex. Assume infu∈Uφ(u, 0) ∈ R and thereexists u0 ∈ U such that p 7→ φ(u0, p) is finite and continuous at 0 ∈ Y , then

infu∈U

φ(u, 0) = supp∗∈Y ∗

−φ∗(0, p∗), (7.86)

and the dual problem has at least one solution.

Proof: By hypothesis h(0) ∈ R and as was shown above, h is convex. As the functionp 7→ φ(u0, p) is convex and continuous at 0 ∈ Y , there exists a neighborhood V of zero in Ysuch that

φ(u0, p) ≤ M < +∞, ∀p ∈ V , (7.87)

for some M ∈ R. Thus, we may write

h(p) = infu∈U

φ(u, p) ≤ φ(u0, p) ≤ M, ∀p ∈ V . (7.88)


Hence, from Lemma 7.1.20, h is continuous at 0. Thus by Proposition 7.1.21, h is sub-differentiable at 0, which means h(0) = h∗∗(0). Therefore by Proposition 7.2.3, the dualproblem has solutions and

h(0) = infu∈U

φ(u, 0) = supp∗∈Y ∗

−φ∗(0, p∗) = h∗∗(0). ¤ (7.89)

Now we apply the last results to φ(u, p) = G(Λu+p)+F (u), where Λ : U → Y is a continuouslinear operator whose adjoint operator is denoted by Λ∗ : Y ∗ → U∗. We may enunciate thefollowing theorem.

Theorem 7.2.5. Suppose U is a reflexive Banach space and define J : U → R by

J(u) = G(Λu) + F (u) = φ(u, 0), (7.90)

where lim J(u) = +∞ as ‖u‖U →∞ and F ∈ Γ0(U), G ∈ Γ0(Y ). Also suppose there existsu ∈ U such that J(u) < +∞ with the function p 7→ G(p) continuous at Λu. Under suchhypothesis, there exist u0 ∈ U and p∗0 ∈ Y ∗ such that

J(u0) = minu∈U

J(u) = maxp∗∈Y ∗

−G∗(p∗)− F ∗(−Λ∗p∗) = −G∗(p∗0)− F ∗(−Λ∗p∗0). (7.91)

Proof: The existence of solutions for the primal problem follows from the direct method ofcalculus of variations. That is, considering a minimizing sequence, from above (coercivityhypothesis), such a sequence is bounded and has a weakly convergent subsequence to someu0 ∈ U . Finally, from the lower semi-continuity of primal formulation, we may conclude thatu0 is a minimizer. The other conclusions follow from Theorem 7.2.4 just observing that

φ∗(0, p∗) = supu∈U,p∈Y

〈p, p∗〉Y −G(Λu+p)−F (u) = supu∈U,q∈Y

〈q, p∗〉−G(q)−〈Λu, p∗〉−F (u),(7.92)

so that

φ∗(0, p∗) = G∗(p∗) + supu∈U

−〈u, Λ∗p∗〉U − F (u) = G∗(p∗) + F ∗(−Λ∗p∗). (7.93)

Thus,infu∈U

φ(u, 0) = supp∗∈Y ∗

−φ∗(0, p∗) (7.94)

and solutions u0 and p∗0 for the primal and dual problems, respectively, imply that

J(u0) = minu∈U

J(u) = maxp∗∈Y ∗

−G∗(p∗)− F ∗(−Λ∗p∗) = −G∗(p∗0)− F ∗(−Λ∗p∗0). ¤ (7.95)


7.3 Relaxation for the Scalar Case

In this section, Ω ⊂ RN denotes a bounded open Lipschitz set. The proof of next result isfound in [14].

Theorem 7.3.1. Let r ∈ N and let uk, 1 ≤ k ≤ r be piecewise affine functions from Ω intoR and αk1≥k≤r such that αk > 0, ∀k ∈ 1, ..., r and

∑rk=1 αk = 1. Given ε > 0, there

exists a locally Lipschitz function u : Ω → R and r disjoint open sets Ωk, 1 ≤ k ≤ r, suchthat

|m(Ωk)− αkm(Ω)| < αkε, ∀k ∈ 1, ..., r, (7.96)

∇u(x) = ∇uk(x), a.e. on Ωk, (7.97)

|∇u(x)| ≤ max1≤k≤r

|∇uk(x)|, a.e. on Ω, (7.98)

∣∣∣∣∣u(x)−r∑

k=1

αkuk

∣∣∣∣∣ < ε, ∀x ∈ Ω, (7.99)

u(x) =r∑

k=1

αkuk(x),∀x ∈ ∂Ω. (7.100)

The next result is also found in [14].

Proposition 7.3.2. Let r ∈ N and let uk, 1 ≤ k ≤ r be piecewise affine functions from Ωinto R. We take a finite family F of normal integrands of Ω × RN and a positive functionc ∈ L1(Ω) which satisfy

c(x) ≥ supf(x, ξ) | f ∈ F , |ξ| ≤ max1≤k≤r

‖∇uk‖∞. (7.101)

Given ε > 0, there exists a locally Lipschitz function u : Ω → R such that

∣∣∣∣∣∫

Ω

f(x,∇u)dx−r∑

k=1

αk

∫

Ω

f(x,∇uk)dx

∣∣∣∣∣ < ε, (7.102)

|∇u(x)| ≤ max1≤k≤r

|∇uk(x)|, a.e. in Ω, (7.103)

|u(x)−r∑

k=1

αkuk(x)| < ε, ∀x ∈ Ω (7.104)

u(x) =r∑

k=1

αkuk(x),∀x ∈ ∂Ω. (7.105)


Proof: It is sufficient to establish the result for functions uk affine over Ω, since Ω can bedivided into pieces on which uk are affine, and such pieces can be put together through(7.105). Let ε > 0 be given. We know that simple functions are dense in L1(Ω), concerningthe L1 norm. Thus there exists a partition of Ω into a finite number of open sets Oi,1 ≤ i ≤ N1 and a negligible set, and there exists fk constant functions over each Oi suchthat ∫

Ω

|f(x,∇uk(x))− fk(x)|dx < ε, ∀f ∈ F , 1 ≤ k ≤ r. (7.106)

Now choose δ > 0 such that

δ ≤ ε

N1(1 + max1≤k≤r‖fk‖∞)(7.107)

and if B is a measurable set

m(B) < δ ⇒∫

B

c(x)dx ≤ ε/N1. (7.108)

Now we apply Theorem 7.3.1, to each of the open sets Oi, therefore there exists a locallyLipschitz function u : Oi → R and there exist r open disjoints spaces Ωi

k, 1 ≤ k ≤ r, suchthat

|m(Ωik)− αkm(Oi)| ≤ αkδ, for 1 ≤ k ≤ r, (7.109)

|∇u(x)| ≤ max1≤k≤r

|∇uk(x)|, a.e. Ωi, (7.110)

∣∣∣∣∣u(x)−r∑

k=1

αkuk(x)

∣∣∣∣∣ ≤ δ,∀x ∈ Oi (7.111)

u(x) =r∑

k=1

αkuk(x),∀x ∈ ∂Oi. (7.112)

We can define u =∑r

k=1 αkuk on Ω−∪N1i=1Oi. Therefore u is continuous and locally Lipschitz.

Now observe that∫

Oi

f(x,∇u(x))dx−r∑

k=1

∫

Ωik

f(x,∇uk(x))dx =

∫

Oi−∪rk=1Ω

ik

f(x,∇u(x))dx. (7.113)

From |f(x,∇u(x))| ≤ c(x), m(Oi − ∪rk=1Ω

ik) ≤ δ and (7.108) we obtain

∣∣∣∣∣∫

Oi


k=1

∫

Ωik

f(x,∇uk(x)dx

∣∣∣∣∣ =

∣∣∣∣∣∫

Oi−∪rk=1Ω

ik

f(x,∇u(x))dx

∣∣∣∣∣ ≤ ε/N1.

(7.114)Considering that fk is constant in Oi, from (7.106), (7.108), (7.107) and (7.109) we obtain

r∑

k=1

|∫

Ωik

fk(x)dx− αk

∫

Oi

fk(x)dx| < ε/N1. (7.115)


We recall that Ωk = ∪N1i=1Ω

ik so that

∣∣∣∣∣∫

Ω


k=1

αk

∫

Ω

f(x,∇uk(x))dx

∣∣∣∣∣ ≤

∣∣∣∣∣∫

Ω


k=1

∫

Ωk

f(x,∇uk(x))dx

∣∣∣∣∣ +r∑

k=1

∫

Ωk

|f(x,∇uk(x)− fk(x)|dx+

+

p∑

k=1

∣∣∣∣∫

Ωk

fk(x)dx− αk

∫

Ω

fk(x)dx

∣∣∣∣ +r∑

k=1

αk

∫

Ω

|fk(x)− f(x,∇uk(x))|dx. (7.116)

From (7.114), (7.106),(7.115) and (7.106) again, we obtain

∣∣∣∣∣∫

Ω


k=1

αk

∫

Ω

f(x,∇uk)dx

∣∣∣∣∣ < 4ε. ¤ (7.117)

The next result we do not prove it.

Proposition 7.3.3. If u ∈ W 1,p0 (Ω) there exists a sequence un of piecewise affine functions

over Ω, null on ∂Ω, such thatun → u, in Lp(Ω) (7.118)

and∇un → ∇u, in Lp(Ω;RN). (7.119)

Proposition 7.3.4. For p such that 1 < p < ∞, suppose that f : Ω × RN → R is aCaratheodore function , for which there exist a1, a2 ∈ L1(Ω) and constants c1 ≥ c2 > 0 suchthat

a2(x) + c2|ξ|p ≤ f(x, ξ) ≤ a1(x) + c1|ξ|p,∀x ∈ Ω, ξ ∈ RN . (7.120)

Then, given u ∈ W 1,p(Ω) piecewise affine, ε > 0 and a neighborhood V of zero in the topologyσ(Lp(Ω,RN), Lq(Ω,RN)) there exists a function v ∈ W 1,p(Ω) such that

∇v −∇u ∈ V , (7.121)

u = v on ∂Ω,

‖v − u‖∞ < ε, (7.122)

and ∣∣∣∣∫

Ω

f(x,∇v(x))dx−∫

Ω

f ∗∗(x,∇u(x))dx

∣∣∣∣ < ε. (7.123)


Proof: Suppose given ε > 0, u ∈ W 1,p(Ω) piecewise affine continuous, and a neighborhoodV of zero, which may be expressed as

V = w ∈ Lp(Ω,RN) |∣∣∣∣∫

Ω

hm · wdx

∣∣∣∣ < η, ∀m ∈ 1, ..., M, (7.124)

where M ∈ N, hm ∈ Lq(Ω,RN), η ∈ R+. By hypothesis, there exists a partition of Ωinto a negligible set Ω0 and open subspaces ∆i, 1 ≤ i ≤ r, over which ∇u(x) is constant.From standard results of convex analysis in RN , for each i ∈ 1, ..., r we can obtain αk ≥01≤k≤N+1, and ξk such that

∑N+1k=1 αk = 1 and

N+1∑

k=1

αkξk = ∇u,∀x ∈ ∆i, (7.125)

andN+1∑

k=1

αkf(x, ξk) = f ∗∗(x,∇u(x)). (7.126)

Define βi = maxk∈1,...,N+1|ξk| on ∆i, and ρ1 = maxi∈1,...,rβi, and ρ = maxρ1, ‖∇u‖∞.Now, observe that we can obtain functions hm ∈ C∞

0 (Ω;RN) such that

maxm∈1,...,M

‖hm − hm‖Lq(Ω,RN ) <η

4ρm(Ω). (7.127)

Define C = maxm∈1,...,M ‖div(hm)‖Lq(Ω) and we can also define

ε1 = minε, 1/m(Ω), η/(2Cm(Ω)) (7.128)

We recall that ρ does not depend on ε. Furthermore, for each i ∈ 1, ..., r there exists acompact subset Ki ⊂ ∆i such that

∫

∆i−Ki

[a1(x) + c1(x) max|ξ|≤ρ

|ξ|p]dx <ε1

r. (7.129)

Also, observe that the restrictions of f and f ∗∗ to Ki× ρB are continuous, so that from thisand from the compactness of ρB, for all x ∈ Ki, we can find an open ball ωx with center inx and contained in Ω, such that

|f ∗∗(y,∇u(x))− f ∗∗(x,∇u(x))| < ε1

m(Ω),∀y ∈ ωx ∩Ki, (7.130)

and|f(y, ξ)− f(x, ξ)| < ε1

m(Ω),∀y ∈ ωx ∩Ki,∀ξ ∈ ρB. (7.131)

Therefore, we may write∣∣∣∣∣f∗∗(y,∇u(x))−

N+1∑

k=1

αkf(y, ξk)

∣∣∣∣∣ <2ε1

m(Ω),∀y ∈ ωx ∩Ki. (7.132)


We can cover the compact set Ki with a finite number of those open balls ωx, denoted byωj, 1 ≤ j ≤ l. Consider the open sets ω′j = ωj − ∪j−1

i=1 ωi, we have that ∪lj=1ω

′j = ∪l

j=1ωj.

Defining functions uk, for 1 ≤ k ≤ n + 1 such that ∇uk = ξk and u =∑N+1

k=1 αkuk we mayapply Proposition 7.3.2 to each of the open sets ω′j, so that we obtain functions vi ∈ W 1,p(Ω)such that ∣∣∣∣∣

∫

ω′j

f(x,∇vi(x)dx−n+1∑

k=1

αk

∫

ω′j

f(x, ξk)dx

∣∣∣∣∣ <ε1

rl, (7.133)

|∇vi| < ρ, ∀x ∈ ω′j, (7.134)

|vi(x)− u(x)| < ε1,∀x ∈ ω′j, (7.135)

andvi(x) = u(x), ∀x ∈ ∂ω′j. (7.136)

Finally we setvi = u on ∆i − ∪l

j=1ωj. (7.137)

We may define a continuous mapping v : Ω → R by

v(x) = vi(x), if x ∈ ∆i, (7.138)

v(x) = u(x), if x ∈ Ω0. (7.139)

We have that v(x) = u(x),∀x ∈ ∂Ω and ‖∇v‖∞ < ρ. Also, from (7.129)

∫

∆i−Ki

|f ∗∗(x,∇u(x)|dx <ε1

r(7.140)

and ∫

∆i−Ki

|f(x,∇v(x)|dx <ε1

r. (7.141)

On the other hand, from (7.132) and (7.133)

∣∣∣∣∣∫

Ki∩ω′j


Ki∩ω′j


∣∣∣∣∣ ≤ε1

rl+

ε1m(ω′j ∩Ki)

m(Ω)(7.142)

so that

|∫

Ki


Ki

f ∗∗(x,∇u(x))dx| ≤ ε1

r+

ε1m(Ki)

m(Ω). ¤ (7.143)

Now summing up in i and considering (7.140) and (7.141) we obtain (7.123). Also, observethat from above, we have

‖v − u‖∞ < ε1, (7.144)


and thus∣∣∣∣∫

Ω

hm · (∇v(x)−∇u(x))dx

∣∣∣∣ =

∣∣∣∣−∫

Ω

div(hm)(v(x)− u(x))dx

∣∣∣∣≤ ‖div(hm)‖Lq(Ω)‖v − u‖Lp(S)

≤ Cε1m(Ω)

<η

2. (7.145)

Also we have that∣∣∣∣∫

Ω

(hm − hm) · (∇v −∇u)dx

∣∣∣∣ ≤ ‖hm − hm‖Lq(Ω,RN )‖∇v −∇u‖Lp(Ω,RN ) ≤η

2. (7.146)

Thus ∣∣∣∣∫

Ω

hm · (∇v −∇u)dx

∣∣∣∣ < η, ∀m ∈ 1, ...,M. ¤ (7.147)

Theorem 7.3.5. Assuming the hypothesis of last theorem, given a function u ∈ W 1,p0 (Ω),

given ε > 0 and a neighborhood of zero V in σ(Lp(Ω,RN), Lq(Ω,RN)), we have that thereexists a function v ∈ W 1,p

0 (Ω) such that

∇v −∇u ∈ V , (7.148)

and ∣∣∣∣∫

Ω


Ω


∣∣∣∣ < ε. (7.149)

Proof: We can approximate u by a function w which is piecewise affine and null on theboundary. Thus, there exists δ > 0 such that we can obtain w ∈ W 1,p

0 (Ω) piecewise affinesuch that

‖u− w‖1,p < δ (7.150)

so that

∇w −∇u ∈ 1

2V , (7.151)

and ∣∣∣∣∫

Ω

f ∗∗(x,∇w(x))dx−∫

Ω


∣∣∣∣ <ε

2. (7.152)

From Proposition we may obtain v ∈ W 1,p(Ω)0 such that

∇v −∇w ∈ 1

2V , (7.153)

and ∣∣∣∣∫

Ω

f ∗∗(x,∇w(x))dx−∫

Ω

f(x,∇v(x))dx

∣∣∣∣ <ε

2. (7.154)


From (7.152) and (7.154)∣∣∣∣∫

Ω

f ∗∗(x,∇u(x))dx−∫

Ω

f(x,∇v(x))dx

∣∣∣∣ < ε. (7.155)

and from (7.151) and (7.153) we have

∇v −∇u ∈ V . ¤ (7.156)

To finish this chapter, we present two theorems which summarize the last results.

Theorem 7.3.6. Let f be a Caratheodory function from Ω× RN into R which satisfies

a2(x) + c2|ξ|p ≤ f(x, ξ) ≤ a1(x) + c1|ξ|p (7.157)

where a1, a2 ∈ L1(Ω), 1 < p < +∞, b ≥ 0 and c1 ≥ c2 > 0. Under such assumptions,defining U = W 1,p

0 (Ω), we have

infu∈U

∫

Ω

f(x,∇u)dx

= min

u∈U

∫

Ω

f ∗∗(x,∇u)dx

(7.158)

The solutions of relaxed problem are weak cluster points in W 1,p0 (Ω) of the minimizing se-

quences of primal problem.

Proof: The existence of solutions for the convex relaxed formulation is a consequence ofthe reflexivity of U and coercivity hypothesis, which allows an application of the directmethod of calculus of variations. That is, considering a minimizing sequence, from above(coercivity hypothesis), such a sequence is bounded and has a weakly convergent subsequenceto some u ∈ W 1,p(Ω). Finally, from the lower semi-continuity of relaxed formulation, we mayconclude that u is a minimizer. The relation (7.158) follows from last theorem. ¤

Theorem 7.3.7. Let f be a Caratheodory function from Ω× RN into R which satisfies

a2(x) + c2|ξ|p ≤ f(x, ξ) ≤ a1(x) + c1|ξ|p (7.159)

where a1, a2 ∈ L1(Ω), 1 < p < +∞, b ≥ 0 and c1 ≥ c2 > 0. Let u0 ∈ W 1,p(Ω). Under suchassumptions, defining U = u | u− u0 ∈ W 1,p

0 (Ω), we have

infu∈U

∫

Ω

f(x,∇u)dx

= min

u∈U

∫

Ω

f ∗∗(x,∇u)dx

(7.160)

The solutions of relaxed problem are weak cluster points in W 1,p(Ω) of the minimizing se-quences of primal problem.

Proof: Just apply the last theorem to the integrand g(x, ξ) = f(x, ξ +∇u0). For details see[14]. ¤


7.4 Duality Suitable for the Vectorial Case

Definition 7.4.1 (A Cone and its Partial Order Relation). Let U be a Banach space andm > 0. We define C(m) as

C(m) = (u, a) ∈ U × R | a + m‖u‖U ≤ 0. (7.161)

Also, we define an order relation for the cone C(m), namely

(u, a) ≤ (v, b) ⇔ (v − u, b− a) ∈ C(m). (7.162)

Proposition 7.4.2. Let S ⊂ U × R be a closed set such that

infa | (u, a) ∈ S > −∞. (7.163)

Then S has a maximal element under the order relation of last definition.

Proof: See Ekeland and Temam [14], page 28.

The next result is particularly relevant for non-convex functionals.

Theorem 7.4.3. Let F : U → R be lower semi-continuous functional such that −∞ <infu∈UF (u) < +∞. Given ε > 0, suppose u ∈ U is such that

F (u) ≤ infu∈U

F (u)+ ε, (7.164)

then, for each λ > 0, there exists uλ ∈ U such that

‖u− uλ‖U ≤ λ and F (uλ) ≤ F (u), (7.165)

andEpi(F ) ∩ (uλ, F (uλ)) + C(ε/λ) = (uλ, F (uλ)). (7.166)

Proof: We will apply the last proposition to S = Epi(F ), which is a closed set. For theorder relation associated with C(ε/λ), there exists a maximal element, which we denote by(uλ, aλ). Thus (uλ, aλ) ≥ (u, F (u)). Since (uλ, aλ, ) is maximal, we have aλ = F (uλ) andhence (7.166) is satisfied. Also observe that

(u, F (u)) ≤ (uλ, F (uλ)), (7.167)

so thatε

λ‖u− uλ‖U ≤ F (u)− F (uλ). (7.168)

From this and (7.164) we obtain

0 ≤ F (u)− F (uλ) ≤ ε, (7.169)

and therefore‖u− uλ‖U ≤ λ. ¤ (7.170)


Remark 7.4.4. Observe that

F (uλ)− ε

λt‖v‖U ≤ F (uλ + tv),∀t ∈ [0, 1], v ∈ U, (7.171)

so that, if F is Gateaux differentiable, we obtain

− ε

λ‖v‖U ≤ 〈δF (uλ), v〉U . (7.172)

Thus‖δF (uλ)‖U∗ ≤ ε/λ. (7.173)

Now, for λ =√

ε we obtain the following result.

Theorem 7.4.5. Let F : U → R be a Gateaux differentiable functional. Given ε > 0 supposethat u ∈ U is such that

F (u) ≤ infu∈U

F (u)+ ε. (7.174)

Then there exists v ∈ U such thatF (v) ≤ F (u), (7.175)

‖u− v‖U ≤√

ε, (7.176)

and‖δF (v)‖U∗ ≤

√ε. ¤ (7.177)

The next theorem easily follows from above results.

Theorem 7.4.6. Let J : U → R, be defined by

J(u) = G(∇u)− 〈f, u〉L2(S;RN ), (7.178)

whereU = W 1,2

0 (S;RN), (7.179)

We suppose G is Gateaux-differentiable and J bounded from below. Then, given ε > 0, thereexists uε ∈ U such that

J(uε)− infu∈U

J(u) < ε, (7.180)

and‖δJ(uε)‖U∗ <

√ε. ¤ (7.181)

Now we establish a general duality principle that is applicable to more complex situationsconcerning vectorial problems in the calculus of variations. In fact, it is very simple result,given by the following theorem:


Theorem 7.4.7. Consider (G Λ) : U → R (not necessarily convex) such that J : U → Rdefined by

J(u) = G(Λu)− 〈u, f〉U ,∀u ∈ U,

is bounded from below (here as usual Λ : U → Y is a continuous linear operator). Undersuch assumptions, we have

infu∈U

J(u) = supv∗∈A∗

−(G Λ)∗(Λ∗v∗)

whereA∗ = v∗ ∈ Y ∗ | Λ∗v∗ − f = 0.

Proof: The proof is simple, just observe that

−(G Λ)∗(Λ∗v∗) = −(G Λ)∗(f) = − supu∈U

〈u, f〉U −G(Λu),∀v∗ ∈ A∗. ¤

Remark 7.4.8. What seems to be relevant is that, when computing (GΛ)∗(Λ∗v∗), we obtaina duality which is perfect concerning the convex envelope of the primal formulation, that is,no duality gap, as may be seen in the next chapters.

We finish this Chapter with the most important result we have obtained for vectorial prob-lems in the Calculus of Variations, namely:

Theorem 7.4.9. Consider (G Λ) : U → R and (F Λ1) : U → R convex l.s.c. functionalssuch that J : U → R defined as

J(u) = (G Λ)(u)− (F Λ1)(u)− 〈u, f〉Uis below bounded. (Here Λ : U → Y and Λ1 : U → Y1 are continuous linear operators whoseadjoint operators are denoted by Λ∗ : Y ∗ → U∗ and Λ∗1 : Y ∗ → U∗, respectively). Also wesuppose the existence of L : Y1 → Y continuous and linear operator such that L∗ is onto and

Λ(u) = L(Λ1(u)),∀u ∈ U.

Under such assumptions, we have

infu∈U

J(u) ≥ supv∗∈A∗

infz∗∈Y ∗1

F ∗(L∗z∗)−G∗(v∗ + z∗),

whereA∗ = v∗ ∈ Y ∗ | Λ∗v∗ = f.

Proof: Observe that

G∗(v∗ + z∗) ≥ 〈Λu, v∗〉Y + 〈Λu, z∗〉Y −G(Λu),∀u ∈ U,


that is,

−F ∗(L∗z∗) + G∗(v∗ + z∗) ≥ 〈u, f〉U − F ∗(L∗z∗) + 〈Λ1u, L∗z∗〉Y1 −G(Λu),∀u ∈ U, v∗ ∈ A∗

so that

supz∗∈Y ∗1

−F ∗(L∗z∗) + G∗(v∗ + z∗) ≥ supz∗∈Y ∗1

〈u, f〉U − F ∗(L∗z∗) + 〈Λ1u, L∗z∗〉Y1 −G(Λu),

for v∗ ∈ A∗, and therefore

G(Λu)− F (Λ1u)− 〈u, f〉U ≥ infz∗∈Y ∗1

F ∗(L∗z∗)−G∗(v∗ + z∗), if v∗ ∈ A∗,

which meansinfu∈U

J(u) ≥ supv∗∈A∗

infz∗∈Y ∗1

F ∗(L∗z∗)−G∗(v∗ + z∗),

whereA∗ = v∗ ∈ Y ∗ | Λ∗v∗ = f. ¤

Chapter 8

Constrained Variational Optimization

8.1 Basic Concepts

We start with the definition of cone:

Definition 8.1.1 (Cone). Given U a Banach space, we say that C ⊂ U is a cone with vertexat origin, if given u ∈ C, we have that λu ∈ C, ∀λ ≥ 0. By analogy we define a cone withvertex at p ∈ U as P = p + C, where C is any cone with vertex at origin.

Definition 8.1.2. Let P be a convex cone in U . For u, v ∈ U we write u ≥ v (with respectto P ) if u− v ∈ P . In particular u ≥ θ if and only if u ∈ C. Also

P+ = u∗ ∈ U∗ | 〈u, u∗〉U ≥ 0, ∀u ∈ P. (8.1)

If u∗ ∈ P+ we write u∗ ≥ θ∗.

Proposition 8.1.3. Let U be a Banach space and P be a closed cone in U . If u ∈ U satisfies〈u, u∗〉U ≥ 0, ∀u∗ ≥ θ∗, then u ≥ θ.

Proof: We repeat here the proof found in Luenberger [23], page 215. Assume u is not inP . Then by the separating hyperplane theorem there is an u∗ ∈ U∗ such that 〈u, u∗〉U <〈p, u∗〉U ,∀p ∈ P . Since P is cone we must have 〈p, u∗〉U ≥ 0, otherwise we would have〈u, u∗〉 > 〈αp, u∗〉U for some α > 0. Thus u∗ ∈ P+. Finally, since infp∈P〈p, u∗〉U = 0, weobtain 〈u, u∗〉U < 0 which completes the proof. ¤

Definition 8.1.4 (Convex Mapping). Let U,Z be vector spaces. Let P ⊂ Z be a cone. Amapping G : U → Z is said to be convex if the domain of G is convex and

G(αu1 + (1− α)u2) ≤ αG(u1) + (1− α)G(u2),∀u1, u2 ∈ U, α ∈ [0, 1]. (8.2)

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CHAPTER 8. CONSTRAINED VARIATIONAL OPTIMIZATION 81

Consider the problem P , defined as

Problem P : Minimize F : U → R subject to u ∈ Ω, and G(u) ≤ θ

Defineω(z) = infF (u) | u ∈ Ω and G(u) ≤ z. (8.3)

For such a functional we have the following result.

Proposition 8.1.5. If F is a real convex functional and G is convex, then ω is convex.

Proof: Observe that

ω(αz1 + (1− α)z2) = infF (u) | u ∈ Ω and G(u) ≤ αz1 + (1− α)z2(8.4)

≤ infF (u) | u = αu1 + (1− α)u2 u1, u2 ∈ Ω

and G(u1) ≤ z1, G(u2) ≤ z2(8.5)

≤ α infF (u1) | u1 ∈ Ω, G(u1) ≤ z1+(1− α) infF (u2) | u2 ∈ Ω, G(u2) ≤ z2

(8.6)

≤ αω(z1) + (1− α)ω(z2). ¤ (8.7)

Now we establish the Lagrange multiplier theorem for convex global optimization.

Theorem 8.1.6. Let U be a vector space, Z a Banach space, Ω a convex subset of U , Pa positive cone of Z. Assume that P contains a interior point. Let F be a real convexfunctional on Ω and G a convex mapping from Ω into Z. Assume the existence of u1 ∈ Ωsuch that G(u1) < θ. Defining

µ0 = infu∈ΩF (u) | G(u) ≤ θ, (8.8)

then there exists z∗0 ≥ θ, z∗0 ∈ Z∗ such that

µ0 = infu∈ΩF (u) + 〈G(u), z∗0〉Z. (8.9)

Furthermore, if the infimum in (8.8) is attained by u0 ∈ U such that G(u0) ≤ θ, it is alsoattained in (8.9) by the same u0 and also 〈G(u0), z

∗0〉V = 0. We refer to z∗0 as the Lagrangian

Multiplier.

Proof: Consider the space W = R× Z and the sets A, B where

A = (r, z) ∈ (R, Z) | r ≥ F (u), z ≥ G(u) for some u ∈ Ω, (8.10)


andB = (r, z) ∈ (R, Z) | r ≤ µ0, z ≤ θ, (8.11)

where µ0 = infu∈ΩF (u) | G(u) ≤ θ. Since F and G are convex, A and B are convex sets.It is clear that A contains no interior point of B, and since N = −P contains an interiorpoint , the set B contains an interior point. Thus, from the separating hyperplane theorem,there is a non-zero element w∗

0 = (r0, z∗0) ∈ W ∗ such that

r0r1 + 〈z1, z∗0〉Z ≥ r0r2 + 〈z2, z

∗0〉Z , ∀(r1, z1) ∈ A, (r2, z2) ∈ B. (8.12)

From the nature of B it is clear that w∗0 ≥ θ. That is, r0 ≥ 0 and z∗0 ≥ θ. We will show that

r0 > 0. The point (µ0, θ) ∈ B, hence

r0r + 〈z, z∗0〉Z ≥ r0µ0, ∀(r, z) ∈ A. (8.13)

If r0 = 0 then 〈G(u1), z∗0〉Z ≥ 0 and z∗0 6= θ. Since G(u1) < θ and z∗ ≥ θ we have a

contradiction. Therefore r0 > 0 and, without loss of generality we may assume r0 = 1. Sincethe point (µ0, θ) is arbitrarily close to A and B, we have

µ0 = inf(r,z)∈A

r + 〈z, z∗0〉Z ≤ infu∈ΩF (u)+ 〈G(u), z∗0〉Z ≤ infF (u) | u ∈ Ω, G(u) ≤ θ = µ0.

(8.14)Also, if there exists u0 such that G(u0) ≤ θ, µ0 = F (u0), then

µ0 ≤ F (u0) + 〈G(u0), z∗0〉Z ≤ F (u0) = µ0. (8.15)

Hence〈G(u0), z

∗0〉Z = 0. ¤ (8.16)

Corollary 8.1.7. Let the hypothesis of the last theorem hold. Suppose

F (u0) = infu∈ΩF (u) | G(u) ≤ θ. (8.17)

Then there exists z∗0 ≥ θ such that the Lagrangian L : U × Z∗ → R defined by

L(u, z∗) = F (u) + 〈G(u), z∗〉Z (8.18)

has a saddle point at (u0, z∗0). That is

L(u0, z∗) ≤ L(u0, z

∗0) ≤ L(u, z∗0),∀u ∈ Ω, z∗ ≥ θ. (8.19)

Proof: For z∗0 obtained in the last theorem, we have

L(u0, z∗0) ≤ L(u, z∗0),∀u ∈ Ω. (8.20)

As G(u0, z∗0) = 0, we have

L(u0, z∗)− L(u0, z

∗0) = 〈G(u0), z

∗〉Z − 〈G(u0), z∗0〉Z = 〈G(u0), z

∗〉Z ≤ 0. ¤ (8.21)

We now prove two theorems relevant to develop the subsequent section.


Theorem 8.1.8. Let F : Ω ⊂ U → R and G : Ω → Z. Let P ⊂ Z be a cone. Suppose thereexist (u0, z

∗0) ∈ U × Z∗ where z∗0 ≥ θ and u0 ∈ Ω such that

F (u0) + 〈G(u0), z∗0〉Z ≤ F (u) + 〈G(u), z∗0〉Z ,∀u ∈ Ω. (8.22)

ThenF (u0) + 〈G(u0), z

∗0〉Z = infF (u) | u ∈ Ω and G(u) ≤ G(u0). (8.23)

Proof: Suppose there is a u1 ∈ Ω such that F (u1) < F (u0) and G(u1) ≤ G(u0). Thus

〈G(u1), z∗0〉Z ≤ 〈G(u0), z

∗0〉Z (8.24)

so thatF (u1) + 〈G(u1), z

∗0〉Z < F (u0) + 〈G(u0), z

∗0〉Z , (8.25)

which contradicts the hypothesis of the theorem. ¤Theorem 8.1.9. Let F be a convex real functional and G : Ω → Z convex and let u0 andu1 be solutions to the problems P0 and P1 respectively, where

P0 : minimize F (u) subject to u ∈ Ω and G(u) ≤ z0, (8.26)

andP1 : minimize F (u) subject to u ∈ Ω and G(u) ≤ z1. (8.27)

Suppose z∗0 and z∗1 are the Lagrange multipliers related to these problems. Then

〈z1 − z0, z∗1〉Z ≤ F (u0)− F (u1) ≤ 〈z1 − z0, z

∗0〉Z . (8.28)

Proof: For u0, z∗0 we have

F (u0) + 〈G(u0)− z0, z∗0〉Z ≤ F (u) + 〈G(u)− z0, z

∗0〉Z ,∀u ∈ Ω, (8.29)

and, particularly for u = u1 and considering that 〈G(u0)− z0, z∗0〉Z = 0, we obtain

F (u0)− F (u1) ≤ 〈G(u1)− z0, z∗0〉Z ≤ 〈z1 − z0, z

∗0〉Z . (8.30)

A similar argument applied to u1, z∗1 provides us the other inequality. ¤

8.2 Duality

Consider the basic convex programming problem:

Mininize F (u) subject to G(u) ≤ θ, u ∈ Ω, (8.31)

where F : U → R is a convex functional, G : U → Z is convex mapping, and Ω is a convexset. We define ϕ : Z∗ → R by

ϕ(z∗) = infu∈ΩF (u) + 〈G(u), z∗〉Z. (8.32)


Proposition 8.2.1. ϕ is concave and

ϕ(z∗) = infz∈Γω(z) + 〈z, z∗〉Z, (8.33)

whereω(z) = inf

u∈ΩF (u) | G(u) ≤ z, (8.34)

andΓ = Range(G).

Proof: Observe that

ϕ(z∗) = infu∈ΩF (u) + 〈G(u), z∗〉Z

≤ infu∈ΩF (u) + 〈z, z∗〉Z | G(u) ≤ z

= ω(z) + 〈z, z∗〉Z ,∀z∗ ≥ θ, z ∈ Γ. (8.35)

On the other hand, for any u1 ∈ Ω, defining z1 = G(u1), we obtain

F (u1) + 〈G(u1), z∗〉Z ≥ inf

u∈ΩF (u) + 〈z1, z

∗〉Z | G(u) ≤ z1 = ω(z1) + 〈z1, z∗〉Z , (8.36)

so thatϕ(z∗) ≥ inf

z∈Γω(z) + 〈z, z∗〉Z. ¤ (8.37)

Theorem 8.2.2 (Lagrange Duality). Consider F : Ω ⊂ U → R a convex functional, Ω aconvex set, and G : U → Z a convex mapping. Suppose there exists a u1 such that G(u1) < θand that infu∈ΩF (u) | G(u) ≤ θ < ∞. Under such assumptions, we have

infu∈ΩF (u) | G(u) ≤ θ = max

z∗≥θϕ(z∗). (8.38)

If the infimum on the left side in (12.50) is achieved at some u0 ∈ U and the max on theright side at z∗0 ∈ Z∗, then

〈G(u0), z∗0〉Z = 0 (8.39)

and u0 minimizes F (u) + 〈G(u), z∗0〉Z on Ω.

Proof: For z∗0 ≥ θ we have

infu∈ΩF (u) + 〈G(u), z∗〉Z ≤ inf

u∈Ω,G(u)≤θF (u) + 〈G(u), z∗〉Z ≤ inf

u∈Ω,G(u)≤θF (u) ≤ µ0. (8.40)

orϕ(z∗) ≤ µ0. (8.41)

The result follows from Theorem 8.1.6. ¤


8.3 Lagrange Multiplier Theorems

Definition 8.3.1 (Regular Point). Let U, V be Banach spaces and consider T : D ⊂ U → V(D open) a continuously Frechet differentiable mapping. We say that u0 ∈ U is a regularpoint of T if T ′(u0) : U → V is onto.

Now we present the version for Banach spaces of Inverse Function Theorem. For a proof seeLuenberger [23].

Theorem 8.3.2 (Generalized Inverse Function Theorem). Let U, V be Banach spaces andconsider T : D ⊂ U → V (D open) a continuously Frechet differentiable mapping. Letu0 ∈ U be a regular point of T . Then there is a neighborhood Vv0 of v0 = T (u0) and K ∈ R+,such that for each v ∈ Vv0, there exists u ∈ U such that

T (u) = v and ‖u− u0‖U ≤ K‖v − v0‖V . (8.42)

Before the final result, we need the lemma:

Lemma 8.3.3. Suppose the functional F : U → R achieves a local extremum under the con-straint H(u) = θ at the point u0. Also assume that F and H are continuously Frechetdifferentiable in an open set containing u0 and that u0 is a regular point of H. Then〈h, F ′(u0)〉U = 0 for all h satisfying H ′(u0)h = θ.

Proof: Without loss of generality, suppose the local extremum is a minimum. Consider thetransformation T : U → R × Z defined by T (u) = (F (u), H(u)). Suppose there exists a hsuch that H ′(u0)h = θ, F ′(u0)h 6= 0, then T ′(u0) = (F ′(u0), H

′(u0)) : U → R × Z is ontosince H ′(u0) is onto Z. By the inverse function theorem, given ε > 0 there exists u ∈ U andδ > 0 with ‖u−u0‖U < ε such that T (u) = (F (u0)− δ, θ), which contradicts the assumptionthat u0 is a local minimum. ¤Theorem 8.3.4 (Lagrange Multiplier). Suppose F is a continuously Frechet differentiablefunctional which has a local extremum under the constraint H(u) = θ at the regular pointu0, then there exists a Lagrangian multiplier z∗0 ∈ Z∗ such that the Lagrangian functional

L(u) = F (u) + 〈H(u), z∗0〉Z (8.43)

is stationary at u0, that is, F ′(u0) + H ′(u0)∗z∗0 = θ.

Proof: From last lemma we have that F ′(u0) is orthogonal to the null space of H ′(u0). Sincethe the range of H ′(u0) is closed, it follows that

F ′(u0) ∈ R[H ′(u0)]∗, (8.44)

therefore there exists z∗0 ∈ Z∗ such that

F ′(u0) = −H ′(u0)∗z∗0 . ¤ (8.45)

Chapter 9

Duality Applied to a Plate Model

9.1 Introduction

The main objective of the present chapter is to develop systematic approaches for obtain-ing dual variational formulations for systems originally modeled by non-linear differentialequations.

Duality for linear systems is well established and is the main subject of classical convexanalysis, since in case of linearity, both primal and dual formulations are generally convex. Incase of non-linear differential equations, some complications occur and the standard modelsof duality for convex analysis must be modified and extended.

In particular in the case of Kirchhoff-Love plate model, there is a non-linearity concerningthe strain tensor (that is, a geometric non-linearity). To apply the classical results of convexanalysis and obtain the complementary formulation is possible only for a special class ofexternal loads. This leads to non-compressed plates, please see Telega [33], Gao [18] andother references therein.

We now describe the primal formulation and related duality principles. Consider a platewhose middle surface is represented by an open bounded set S ⊂ R2, whose boundary isdenoted by Γ, subjected to a load to be specified. We denote by uα : S → R (α = 1, 2) thehorizontal displacements and by w : S → R, the vertical displacement field. The boundaryvalue form of the Kirchhoff-Love model can be expressed by the equations:

Nαβ,β = 0,Qα,α + Mαβ,αβ + P = 0, a.e. in S

(9.1)

86

CHAPTER 9. DUALITY APPLIED TO A PLATE MODEL 87

and

Nαβ.nβ − Pα = 0,

(Qα + Mαβ,β)nα +∂(Mαβtαnβ)

∂s− P = 0,

Mαβnαnβ −Mn = 0, on Γt,

(9.2)

where,Nαβ = Hαβλµγλµ,

Mαβ = hαβλµκλµ

and,

γαβ(u) =1

2(uα,β + uβ,α + w,αw,β),

καβ(u) = −w,αβ,

with the boundary conditions

uα = w =∂w

∂n= 0, on Γu.

Here, Nαβ denote the membrane forces, Mαβ denote the moments and Qα = Nαβw,βstand for functions related to the rotation work of membrane forces, P ∈ L2(S) is a fieldof vertical distributed forces applied on S, (Pα, P ) ∈ (L2(Γt))

3 denote forces applied to Γt

concerning the horizontal directions defined by α = 1, 2 and vertical direction respectively.Mn are distributed moments applied also to Γt, where Γ is such that Γu ⊂ Γ, Γ = Γu ∪ Γt

and Γu ∩ Γt = ∅. Finally, the matrices Hαβλµ and hαβλµ are related to the coefficientsof Hooke’s Law.

The corresponding primal variational formulation to this boundary value model is repre-sented by the functional J : U → R, where

J(u) =1

2

∫

S

HαβλµγαβγλµdS +1

2

∫

S

hαβλµκαβκλµdS−∫

S

PwdS−∫

Γt

(Pw+ Pαuα−Mn∂w

∂n)dΓ

and

U = (uα, w) ∈ W 1,2(S)×W 1,2(S)×W 2,2(S), uα = w =∂w

∂n= 0 on Γu.

The first duality principle presented is the classical one (again we mention the earlier similarresults in Telega [33], Gao [18]) , and is obtained by applying a little change of Rockafellar’sapproach for convex analysis. We have developed a different proof from the one found in[33], by using the definition of Legendre Transform and related properties. Such a resultmay be summarized as

infu∈U

J(u) = supv∗∈A∗∩C∗

−G∗L(v∗) (9.3)


The dual functional, denoted by −G∗L : A∗ ∩ C∗ → R is expressed as

G∗L(v∗) =

1

2

∫

S

HαβλµNαβNλµdS +1

2

∫

S

hαβλµMαβMλµdS +1

2

∫

S

NαβQαQβdS

,

where C∗ is defined by equations (9.1) and (9.2) and,

A∗ = v∗ ∈ Y ∗| N11 > 0, N22 > 0, and N11N22 −N212 > 0, a.e. in S, (9.4)

here v∗ = Nαβ,Mαβ, Qα ∈ Y ∗ = L2(S;R10) ≡ L2(S)

Therefore, since the functional G∗L(v∗) is convex in A∗, the duality is perfect if the optimum

solution for the primal formulation satisfies the constraints indicated in (9.4), however it isimportant to emphasize that such constraints imply no compression along the plate.

For the second and third principles, we emphasize that our dual formulations remove or relaxthe constraints on the external load, and are valid even for compressed plates.

Still for these two principles, we use a theorem (Toland [35]) which does not require convexityof primal functionals. Such a result can be summarized as:

infu∈U

G(u)− F (u) = infu∗∈U∗

F ∗(u∗)−G∗(u∗)

Here G : U → R and F : U → R and, F ∗ : U∗ → R and G∗ : U∗ → R denote the primal anddual functionals respectively.

In particular for the second principle, we modify the above result by applying it to a notone to one relation between primal and dual variables, obtaining the final duality principleexpressed as follows

inf(u,p)∈U×Y

JK(u, p) ≤ inf(u,v∗)∈U×Y ∗

J∗K(u, v∗)

where

JK(u, p) = G(Λu + p)− F (u) +K

2〈p, p〉L2(S)

and

J∗K(u, v∗) = F ∗(Λ∗v∗)−GL(v∗) + K

∥∥∥∥Λu− ∂g∗L(v∗)∂y∗

∥∥∥∥2

L2(S)

+1

2K〈v∗, v∗〉L2(S).

Here K ∈ R is a positive constant and we are particularly concerned with the fact that

JK(uK , pK) → J(u0), as K → +∞

andJ∗K(uK , v∗K) → J(u0), as K → +∞


whereJK(uK , pK) = inf

(u,p)∈U×YJK(u, p),

J∗K(uK , v∗K) = inf(u,v∗)∈U×Y ∗

J∗K(u, v∗)

andJ(u0) = inf

u∈UJ(u) = G(Λu)− F (u).

Even though we do not prove it in the present article, postponing a more rigorous analysisconcerning the behavior of uK indicated above as K → +∞, for a future work.

For the third duality principle, the dual variables must satisfy the following constraints :

N11 + K > 0, N22 + K > 0 and (N11 + K)(N22 + K)−N212 > 0, a.e. in S. (9.5)

Such a principle may be summarized by the following result,

infu∈U

G(Λu)− F (Λ1u)− 〈u, p〉U ≤ infz∗∈Y ∗

supv∗∈B∗(z∗)

F ∗(z∗)−G∗L(v∗),

whereB∗(z∗) = v∗ ∈ Y ∗ | Λ∗v∗ − Λ∗1z

∗ − p = 0

Therefore the constant K > 0 must be chosen so that the optimum point of the primalformulation satisfies the constraints indicated in (9.5). This is because these relations alsodefine an enlarged region in which the analytical expression of the functional G∗

L : Y ∗ → Ris convex, so that, in this case, negative membrane forces are allowed.

In Section 9.7, we present a convex dual variational formulation which may be expressedthrough the following duality principle:

infu∈U

J(u) = sup(v∗,z∗,w)∈E∗∩B∗

−G∗(v∗) + 〈z∗α, z∗α〉L2(S)/(2K)

where,

G∗(v∗) = G∗L(v∗) =

1

2

∫

S


2

∫

S


2

∫

S

NKαβQ,αQ,βdS

if v∗ ∈ E∗, where

v∗ = Nαβ, Mαβ, Qα ∈ E∗ ⇔ v∗ ∈ L2(S,R10) and

N11 + K > 0 N22 + K > 0 and (N11 + K)(N22 + K)−N212 > 0, a.e. in S,

where

NKαβ =

N11 + K N12

N12 N22 + K

−1

(9.6)


and

(v∗, z∗) ∈ B∗ ⇔

Nαβ,β + Pα = 0,

Qα,α + Mαβ,αβ − z∗α,α + P = 0,

h1212M12 + z∗1,2/K = 0,

z∗1,2 = z∗2,1, a.e. in S, and, z∗ = θ on Γ.

Here we are assuming the existence of u0 ∈ U such that δJ(u0) = θ, and so that there existsK > 0 for which N11(u0)+K > 0, N22(u0)+K > 0 , (N11(u0)+K)(N22(u0)+K)−N12(u0)

2 >0 (a.e in S) and h1212/(2K0) > K where K0 is the constant related to Poincare Inequalityand,

Nαβ(u0) = Hαβλµγλµ(u0).

Finally, in the last section, we prove a result similar to those obtained through the trialitycriterion introduced in Gao [20] and establish sufficient conditions for the existence of aminimizer for the primal formulation. Such conditions may be summarized by δJ(u0) = θand

1

2

∫

S

Nαβ(u0)w,αw,βdS +1

2

∫

S

hαβλµw,αβw,λµdS ≥ 0,∀w ∈ W 2,20 (S). ¤

For this last result, our proof is new. The statement of results themselves follows those ofGao [20].

We are now ready to state the result of Toland [35], through which will be constructed threeduality principles.

Theorem 9.1.1. Let J : U −→ R be a functional defined as J(u) = G(u)− F (u), ∀u ∈ U ,where there exists u0 ∈ U such that J(u0) = inf

u∈UJ(u) and ∂F (u0) 6= ∅, then

infu∈U

G(u)− F (u) = infu∗∈U∗

F ∗(u∗)−G∗(u∗)

and for u∗0 ∈ ∂F (u0) we have,

F ∗(u∗0)−G∗(u∗0) = infu∗∈U∗

F ∗(u∗)−G∗(u∗).

Furthermore u∗0 ∈ ∂G(u0).

9.2 The Primal Variational Formulation

Let S ⊂ R2 be an open bounded set (with a boundary denoted by Γ) which represents themiddle surface of a plate of thickness h. The vectorial basis related to the Cartesian system


x1, x2, x3 is denoted by (aα, a3), where α = 1, 2 (in general Greek indices stand for 1 or 2),a3 denotes the vector normal to S, t is the vector tangent to Γ and n is the outer normal toS. The displacements will be denoted by:

u = uα, u3 = uαaα + u3a3,

The Kirchhoff-Love relations are

uα(x1, x2, x3) = uα(x1, x2)− x3w(x1, x2),α and u3(x1, x2, x3) = w(x1, x2),

where − h/2 ≤ x3 ≤ h/2 so that we have u = (uα, w) ∈ U where

U =

(uα, w) ∈ W 1,2(S)×W 1,2(S)×W 2,2(S), uα = w =

∂w

∂n= 0 on Γu

.

We divide the boundary into two parts, so that Γu ⊂ Γ, Γ = Γu ∪ Γt and Γu ∩ Γt = ∅. Thestrain tensors are denoted by

γαβ(u) =1

2[Λ1αβ(u) + Λ2α(u)Λ2β(u)] (9.7)

andκαβ(u) = Λ3αβ(u) (9.8)

where: Λ = Λ1αβ, Λ2α, Λ3αβ : U → Y = Y ∗ = L2(S;R10) ≡ L2(S) is defined by:

Λ1αβ(u) = uα,β + uβ,α, (9.9)

Λ2α(u) = w,α (9.10)

andΛ3αβ(u) = −w,αβ. (9.11)

The constitutive relations are expressed as

Nαβ = Hαβλµγλµ, (9.12)

Mαβ = hαβλµκλµ (9.13)

where: Hαβλµ and hαβλµ = h2

12Hαβλµ, are positive definite matrices and such that

Hαβλµ = Hαβµλ = Hβαλµ = Hβαµλ. Furthermore Nαβ denote the membrane forces andMαβ the moments. The plate stored energy, denoted by (G Λ) : U → R is expressed as

(G Λ)(u) =1

2

∫

S

NαβγαβdS +1

2

∫

S

MαβκαβdS (9.14)


and the external work, denoted as F : U → R, is given by

F (u) =

∫

S

PwdS +

∫

Γt

(Pw + Pαuα −Mn∂w

∂n)dΓ, (9.15)

where P denotes a vertical distributed load applied in S and P , Pα are forces applied onΓt ⊂ Γ related to directions defined by a3 and aα respectively, and, Mn denote moments alsoapplied on Γt. The potential energy, denoted by J : U → R is expressed as:

J(u) = (G Λ)(u)− F (u)

It is important to emphasize that conditions for the existence of a minimizer (here denotedby u0) related to G(Λu) − F (u) were presented in Ciarlet [11]. Such u0 ∈ U satisfies theequation:

δ(G(Λu0)− F (u0)) = θ

and we should expect at least one minimizer if ‖Pα‖L2(Γt) is small enough and m(Γu) > 0(here m stands for the Lebesgue measure) and with no restrictions concerning the magnitudeof ‖Pα‖L2(S) if m(Γ) = m(Γu), so that in the latter case, we consider a field of distributedforces Pα applied on S.

Some inequalities of Sobolev type are necessary to prove the above result, and in this workwe assume some regularity hypothesis concerning S and its boundary, namely: in additionto S being open and bounded, also we assume it is connected with a Lipschitz continuousboundary Γ, so that S is locally on one side of Γ.

The formal proof of existence of a minimizer for J(u) = G(Λu)− F (u) is obtained throughthe Direct Method of Calculus of variations. We do not repeat this procedure here, we justrefer to Ciarlet [11] for details.

9.3 The Legendre Transform

In this section we determine the Legendre Transform related to the function g : R10 → Rwhere:

g(y) =1

2Hαβλµ[(y1αβ + y1βα + y2αy2β)/2][(y1λµ + y1µλ + y2λy2µ)/2] +

1

2hαβλµy3αβy3λµ (9.16)

and we recall that

G(Λu) =

∫

S

g(Λu)dS.

From Definition 7.1.25 we may write

g∗L(y∗) = 〈y0, y∗〉R10 − g(y0)


where y0 is the unique solution of the system,

y∗i =∂g(y0)

∂yi

which for the above function g, implies:

y∗1αβ = Hαβλµ(y1λµ + y2λy2µ/2)

y∗2α = Hαβλµ(y1λµ , +y2λy2µ/2)y2β = y∗1αβy2β ,

andy∗3αβ = hαβλµy3λµ .

Inverting this system we obtain

y021 = (y∗122 .y∗21 − y∗112 .y∗22)/∆,

y022 = (−y∗112 .y∗21 + y∗111 .y∗22)/∆,

andy01αβ = Hαβλµy

∗1λµ − y02α .y02β/2

whereHαβλµ = Hαβλµ−1,

∆ = y∗111y∗122 − (y∗112)2 (we recall that y∗112 = y∗121 , as a result of the symmetries of Hαβλµ).By analogy,

y03αβ = hαβλµv∗3λµ

where:hαβλµ = hαβλµ−1.

Thus we can define the set RnL, concerning Definition 7.1.25 as

RnL = y∗ ∈ R10 | ∆ 6= 0. (9.17)

After some simple algebraic manipulations we obtain the expression for g∗L : RnL → R, that

is,

g∗L(y∗) =1

2Hαβλµy

∗1αβy∗1λµdS +

1

2hαβλµy

∗3αβy∗3λµdS +

1

2y∗1αβy02αy02βdS. (9.18)

Also from Definition 7.1.25, we have

Y ∗L = v∗ ∈ Y ∗ = L2(S;R10) ≡ L2(S) | v∗(x) ∈ Rn

L a.e. in S

so that G∗L : Y ∗

L → R may be expressed as

G∗L(v∗) =

∫

S

g∗L(v∗)dS.


Or, from (9.18),

G∗L(v∗) =

1

2

∫

S

Hαβλµv∗1αβv

∗1λµdS +

1

2

∫

S

hαβλµv∗3αβv

∗3λµdS +

1

2

∫

S

v∗1αβv02αv02βdS ¤

Changing the notation, as indicated below,

v∗1αβ = Nαβ, v∗2α = Qα = v∗1αβv02β = Nαβv02β , v∗3αβ = Mαβ

we could express G∗L : Y ∗

L → R as

G∗L(v∗) =

1

2

∫

S


2

∫

S


2

∫

S

NαβQαQβdS,

whereNαβ = Nαβ−1.

Remark 9.3.1. Also we can use the transformation

Qα = Nαβw,β

and obtain

G∗L(v∗) =

1

2

∫

S


2

∫

S


2

∫

S

Nαβw,αw,βdS.

The term denoted by Gp : Y ∗ × U → R and expressed as

Gp(v∗, w) =

1

2

∫

S

Nαβw,αw,βdS

is known as the gap function.

9.4 The Classical Dual Formulation

In this section we establish the dual variational formulation in the classical sense.

We recall that J : U → R is expressed by

J(u) = (G Λ)(u)− F (u),

where (G Λ) : U → R and F : U → R were defined by equations (9.14) and (9.15)respectively. It is known and easy to see that

infu∈U

G(Λu) + F (u) ≥ supv∗∈Y ∗

−G∗(v∗)− F ∗(−Λ∗v∗). (9.19)

Now we prove a result concerning the representation of the polar functional, namely:


Proposition 9.4.1. Considering the earlier definitions and assumptions on G : Y → R (seesection 9.2), expressed by G(v) =

∫S

g(v)dS, where g : R10 → R is indicated in (9.16), wehave

v∗ ∈ A∗ ⇒ G∗(v∗) = G∗L(v∗)

where

G∗L(v∗) =

1

2

∫

S


2

∫

S


2

∫

S

NαβQαQβdS (9.20)

and

A∗ = v∗ = Nαβ, Mαβ, Qα ∈ Y ∗| N11 > 0, N22 > 0, and N11N22−N212 > 0, a.e. in S

(9.21)

Proof: First, consider the quadratic inequality in x as indicated below,

ax2 + bx + c ≤ 0, ∀x ∈ R,

which is equivalent to

(a < 0 and b2 − 4ac ≤ 0) or (a = 0, b = 0 and c ≤ 0). (9.22)

Consider now the inequality

a1x2 + b1xy + c1y

2 + d1x + e1y + f1 ≤ 0,∀x, y ∈ R2 (9.23)

and the quadratic equation related to the variable x, for

a = a1, b = b1y + d1 and c = c1y2 + e1y + f1,

and for a1 < 0, from (9.22) the inequality (9.23) is equivalent to

(b21 − 4a1c1)y

2 + (2b1d1 − 4a1e1)y + d21 − 4a1f1 ≤ 0, ∀y ∈ R.

Finally, fora = b2

1 − 4a1c1 < 0, b = 2b1d1 − 4a1e1 and c = d21 − 4a1f1,

also from (9.22), the last inequality is equivalent to

−c1d21 − a1e

21 + b1d1e1 − (b2

1 − 4a1c1)f1 ≤ 0. (9.24)

In order to represent the polar functional related to the plate stored energy, we first considerthe polar functional related to g1(y), where

g1(y) =1

2Hαβλµ(y1αβ +

1

2y2αy2β)(y1λµ +

1

2y2λy2µ),


g(y) = g1(y) + g2(y)

and

g2(y) =1

2hαβλµy3αβy3λµ .

In fact we determine a set in which the polar functional is represented by the LegendreTransform g∗1L(y∗), where, from (9.18),

g∗1L(y∗) =1

2Hαβλµy

∗1αβy∗1λµ +

y∗111(y∗22)2 − 2.y∗112y∗21y∗22 + y∗122(y∗21)2

2[y∗111y∗122 − (y∗112)2]. (9.25)

Thus, sinceg∗1(y

∗) = supy∈R6

y∗1αβy1αβ + y∗2αy2α − g1(y)

we can write

g∗1L(y∗) = g∗1(y∗) ⇔ g∗1L(y∗) ≥ y∗1αβy1αβ + y∗2αy2α − g1(y),∀y ∈ R6.

Or

y∗1αβy1αβ + y∗2αy2α − 1

2Hαβλµ(y1αβ +

1

2y2αy2β)(y1λµ +

1

2y2λy2µ)− g∗1L(y∗) ≤ 0,∀y ∈ R6. (9.26)

However, considering the transformation

y1αβ = y1αβ +1

2y2αy2β ,

y1αβ = y1αβ − 1

2y2αy2β , (9.27)

and substituting such relations into (9.26), we obtain

g∗1L(y∗) = g∗1(y∗) ⇔

y∗1αβ(y1αβ − 1

2y2αy2β) + y∗2αy2α − 1

2Hαβλµy1αβ y1λµ − g∗1L(y∗) ≤ 0,∀y1αβ , y2α ∈ R6. (9.28)

On the other hand, since Hαβλµ is a positive definite matrix we have

supy

1αβ ∈R4

y∗1αβ y1αβ − 1

2Hαβλµy1αβ y1λµ =

1

2Hαβλµy

∗1αβy

∗1λµ . (9.29)

Thus considering (9.29) and the expression of g∗L(y∗) indicated in (9.25) , inequality (9.28)is satisfied if

−1

2y∗1αβy2αy2β + y∗2αy2α − y∗111 .(y∗22)2 − 2.y∗112 .y∗21y∗22 + y∗122 .(y∗21)2

2[y∗111y∗122 − (y∗112)2]≤ 0,∀y2α ∈ R2. (9.30)


So, for

a1 = −1

2y∗111 , b1 = −y∗112 , c1 = −1

2y∗122 , , d1 = y∗21 , e1 = y∗22

and

f1 = −y∗111 .(y∗22)2 − 2.y∗112 .y∗21y∗22 + y∗122 .(y∗21)2

2[y∗111y∗122 − (y∗112)2]

we obtain−c1d

21 − a1e

21 + b1d1e1 − (b2

1 − 4a1c1)f1 = 0

Therefore from (9.24), the inequality (9.26) is satisfied if a1 < 0 (y∗111 > 0) and b21−4a1c1 < 0

(y∗111y∗122 − (y∗112)2 > 0 which implies y∗122 > 0).

Thus we have shown thaty∗ ∈ A∗ ⇒ g∗1(y

∗) = g∗1L(y∗), (9.31)

whereA∗ = y∗ ∈ R6| y∗111 > 0, y∗122 > 0, y∗111y∗122 − (y∗112)2 > 0.

On the other hand, by analogy to above results, it can easily be proved that

g∗2(y∗) = g∗2L(y∗),∀y∗3αβ ∈ R3 (9.32)

where

g∗2L(y∗) =1

2hαβλµy

∗3αβy

∗3λµ (9.33)

and

g∗2(y∗) = sup

y∈R3

y∗3αβy3αβ − 1

2hαβλµy3αβy3λµ.

From (9.31) and (9.32), we can write

if y∗ ∈ A∗ then g∗1(y∗) + g∗2(y

∗) = g∗1L(y∗) + g∗2L(y∗) ≤ (g1 + g2)∗(y∗).

As (g1 + g2)∗(y∗) ≤ g∗1(y

∗) + g∗2(y∗) we have

if y∗ ∈ A∗ then g∗L(y∗) = g∗1L(y∗) + g∗2L(y∗) = (g1 + g2)∗(y∗) = g∗(y∗). (9.34)

However, from Proposition 7.1.24

G∗(v∗) =

∫

S

g∗(v∗)dS (9.35)

so that from (9.34) and (9.35) we obtain

v∗ ∈ A∗ ⇒ G∗(v∗) =

∫

S

g∗L(v∗)dS = G∗L(v∗)

where,A∗ = v∗ ∈ Y ∗ | v∗(x) ∈ A∗, a.e. in S.


Alternatively,

A∗ = v∗ ∈ Y ∗ | v∗111 > 0, v∗122 > 0, and v∗111v∗122 − (v∗112)2 > 0, a.e. in S. (9.36)

Thus, through the notation

v∗1αβ = Nαβ, v∗2α = Qα = v∗1αβv02β = Nαβv02β , v∗3αβ = Mαβ

we have

A∗ = v∗ = Nαβ, Mαβ, Qα ∈ Y ∗| N11 > 0, N22 > 0, and N11N22−N212 > 0, a.e. in S ¤

(9.37)

9.4.1 The Polar Functional Related to F : U → R

We are concerned with the evaluation of the extremum,

F ∗(−Λ∗v∗) = supu∈U

〈u,−Λ∗v∗〉U − F (u),or

F ∗(−Λ∗v∗) = supu∈U

〈Λu,−v∗〉Y − F (u).Considering

F (u) = −(∫

S

PwdS +

∫

Γt

(Pw + Pαuα −Mn∂w

∂n)dΓ

)= 〈u, f〉U

we have

F ∗(−Λ∗v∗) =

0, if v∗ ∈ C∗,+∞, otherwise,

(9.38)

where v∗ ∈ C∗ ⇔ v∗ ∈ Y ∗ andv∗

1αβ,β = 0,v∗2α,α + v∗

3αβ,αβ + P = 0, a.e. in S,(9.39)

and

v∗1αβ .nβ − Pα = 0,

(v∗2α + v∗3αβ,β).nα +

∂(v∗3αβ tαnβ)

∂s− P = 0,

v∗3αβnαnβ −Mn = 0, on Γt. ¤

(9.40)

Remark 9.4.2. We can also denote

C∗ = v∗ ∈ Y ∗ | Λ∗v∗ = f, (9.41)

where the relation Λ∗v∗ = f is defined by (9.39) and (9.40).


9.4.2 The First Duality Principle

Considering inequality (9.19), the expression of G∗(v∗), and the set C∗ above defined, wecan write

infu∈U

(G Λ)(u)− F (u) ≥ supv∗∈A∗∩C∗

−G∗L(v∗) (9.42)

so that the final form of the concerned duality principle results from the following theorem.

Theorem 9.4.3. Let (G Λ) : U → R and F : U → R be defined by (9.14) and (9.15)respectively (and here we express F as F (u) = 〈u, f〉U). If −G∗

L : Y ∗L → R attains a local

extremum at v∗0 ∈ A∗ under the constraint Λ∗v∗ − f = 0, then

infu∈U

(G Λ)(u) + F (u) = supv∗∈A∗∩C∗

−G∗L(v∗)

and u0 ∈ U and v∗0 ∈ Y ∗ such that:

δ−G∗L(v∗0) + 〈u0, Λ

∗v∗0 − f〉U = θ

are also such thatJ(u0) = −G∗

L(v∗0) and δJ(u0) = θ.

The proof of above theorem is consequence of the standard necessary conditions for a localextremum for −G∗

L : Y ∗L → R under the constraint Λ∗v∗ − f = θ, the inequality (9.42) plus

an application of Theorem 7.1.27.

Therefore, in a more explicit format we would have

infu∈U

1

2

∫

S

HαβλµγαβγλµdS +1

2

∫

S

hαβλµκαβκλµdS

−(∫

S

PwdS +

∫

Γt

PwdS +

∫

Γt

PαuαdΓ−∫

Γt

Mn∂w

∂ndΓ

)

= supv∗∈A∗∩C∗

−1

2

∫

S

HαβλµNαβNλµdS − 1

2

∫

S

hαβλµMαβMλµdS

− 1

2

∫

S

NαβQαQβdS

(9.43)

where v∗ ∈ C∗ ⇔ v∗ ∈ Y ∗ and,



and



∂s− P = 0,


with the set A∗ defined by (9.21) and

Nαβ = Nαβ−1. ¤

9.5 The Second Duality Principle

The next result is a extension of Theorem 9.1.1 and, instead of calculating the polar func-tional related to the main part of primal formulation, it is determined its Legendre Transformand associated functional.

Theorem 9.5.1. Consider Gateaux differentiable functionals G Λ : U → R and F Λ1 : U → R where only the second one is necessarily convex, through which is defined thefunctional JK : U × Y → R expressed as

JK(u, p) = G(Λu + p) + K〈p, p〉L2(S) − F (Λ1u)− K〈p, p〉L2(S)

2− 〈u, u∗0〉U .

Suppose there exists (u0, p0) ∈ U × Y such that

JK(u0, p0) = inf(u,p)∈U×Y

JK(u, p)

and δJK(u0, p0) = θ. Here Λ = Λi : U → Y and Λ1 : U → Y are continuous linear opera-tors whose adjoint operators are denoted by Λ∗ : Y ∗ → U∗ and Λ∗1 : Y ∗ → U∗ respectively.

Furthermore assume there exists a differentiable function denoted by g : Rn → R so thatG : Y → R may be expressed as G(v) =

∫Ω

g(v)dS, ∀v ∈ Y where g admits differentiableLegendre transform denoted by g∗L : Rn

L → R.

Under these assumptions we have

inf(u,p)∈U×Y

JK(u, p) ≤ inf(z∗,v∗,u)∈E∗

J∗K(z∗, v∗, u),

where

J∗K(z∗, v∗, u) = F ∗(z∗) + (1/2K)〈v∗, v∗〉L2(S) −G∗L(v∗) + K

n∑i=1

∥∥∥∥Λiu− ∂g∗L(v∗)∂y∗i

∥∥∥∥2

L2(S)

,


andE∗ = (z∗, v∗, u) ∈ Y ∗ × Y ∗

L × U | − Λ∗1z∗ + Λ∗v∗ − u∗0 = θ.

Also, the functions z∗0 , v∗0, and u0, defined by

z∗0 =∂F (Λ1u0)

∂v,

v∗0 =∂G(Λu0 + p0)

∂v,

andu0 = u0

are such that−Λ∗1z

∗0 + Λ∗v∗0 − u∗0 = θ,

and thus

JK(u0, p0) ≤ inf(z∗,v∗,u)∈E∗

J∗K(z∗, v∗, u) ≤ JK(u0, p0) + 2K〈p0, p0〉L2(S) (9.44)

where we are assuming that v∗0 ∈ Y ∗L .

Proof: Defining α = inf(u,p)∈U×Y JK(u, p), G1(u, p) = G(Λu + p) + K〈p, p〉L2(S) andG2(u, p) = F (Λ1u) + (K/2)〈p, p〉L2(S) + 〈u, u∗0〉U we have:

G1(u, p) ≥ G2(u, p) + α, ∀(u, p) ∈ U × Y.

Thus, ∀ v∗ ∈ Y ∗L , we have

sup(u,p)∈U×Y

〈v∗, Λu + p〉L2(S) −G2(u, p) ≥ 〈v∗, Λu + p〉L2(S) −G1(u, p) + α, ∀(u, p) ∈ U × Y.

(9.45)From Theorem 7.2.5:

sup(u,p)∈U×Y

〈v∗, Λu + p〉L2(S) −G2(u, p) = infz∗∈C∗(v∗)

F (z∗) + (1/2K)〈v∗, v∗〉L2(Ω) (9.46)

whereC∗(v∗) = z∗ ∈ Y ∗ | − Λ∗1z

∗ + Λ∗v∗ − u∗0 = θ.

Furthermore

〈v∗, Λu + p〉L2(S) −G1(u, p) = 〈v∗, Λu + p〉L2(S) −G(Λu + p)−K〈p, p〉L2(S).

Choosing u = u and p satisfying the equations


v∗i =∂G(Λu + p)

∂vi

,

from a well known Legendre Transform property, we obtain

pi =∂GL(v∗)

∂v∗i− Λiu

so that

〈v∗, Λu + p〉L2(S) −G1(u, p) = G∗L(v∗)−K

n∑i=1

∥∥∥∥Λiu− ∂g∗L(v∗)∂y∗i

∥∥∥∥2

L2(S)

.

From last results and inequality (9.45) we have

infz∗∈C∗(v∗)

F ∗(z∗) + (1/2K)〈v∗, v∗〉L2(S) −G∗L(v∗) + K

n∑i=1

∥∥∥∥Λiu− ∂g∗L(v∗)∂y∗i

∥∥∥∥2

L2(S)

≥ α

= inf(u,p)∈U×Y

JK(u, p) (9.47)

that is,

F ∗(z∗) +1

2K〈v∗, v∗〉L2(S) −G∗

L(v∗) + K

n∑i=1

∥∥∥∥Λiu− ∂g∗L(v∗)∂y∗i

∥∥∥∥2

L2(S)

≥ α

= inf(u,p)∈U×Y

JK(u, p), if z∗ ∈ C∗(v∗). (9.48)

Hence

inf(z∗,v∗,u)∈E∗

F ∗(z∗) + (1/2K)〈v∗, v∗〉L2(S) −G∗L(v∗) + K

n∑i=1

∥∥∥∥Λiu− ∂g∗L(v∗)∂y∗i

∥∥∥∥2

L2(S)

≥ α

= inf(u,p)∈U×Y

JK(u, p) (9.49)

so that:inf

(z∗,v∗,u)∈E∗J∗K(z∗, v∗, u) ≥ inf

(u,p)∈U×YJK(u, p)

where E∗ = C∗(v∗)× Y ∗L × U, and the remaining conclusions follow from the expressions of

JK(u0, p0) and J∗K(z∗0 , v∗0, u0). ¤


Remark 9.5.2. We conjecture that the duality gap between the primal and dual formulations,namely 2K〈p0, p0〉L2(S), goes to zero as K → +∞, since p0 ∈ Y satisfies the extremalcondition:

1

K

∂G(Λu0 + p0)

∂v+ p0 = 0,

and JK(u, p) is bounded from below. We do not prove it in the present work.

In the application of last theorem to the Kirchhoff-Love plate model, we would have F (Λ1u) =θ, and therefore the variable z∗ is not present in the dual formulation. Also,

〈u, u∗0〉U =

∫

S

PwdS +

∫

Γt

(Pαuα + Pw −Mn

∂w

∂n

)dΓ (9.50)

and thus the relevant duality principle could be expressed as

infu∈U

G(Λu + p) + K〈p, p〉L2(S) − 〈u, u∗0〉U −

K

2〈p, p〉L2(S)

≤

inf(v∗,u)∈E∗

−1

2

∫

S


2

∫

S

hαβλµMαβMλµdS − 1

2

∫

S

NαβQαQβdS+

1

2K

∫

S

NαβNαβdS +1

2K

∫

S

MαβMαβdS

+K

2∑

α,β=1

∥∥∥∥1

2(uα,β + uβ,α)− HαβλµNλµ +

1

2v02αv02β

∥∥∥∥2

L2(S)

+K

2∑α=1

‖w,α − v02α‖2L2(S) + K

2∑

α,β=1

‖ − w,αβ − hαβλµMλµ‖2L2(S)

(9.51)

where (v∗, u) ∈ E∗ = C∗ × U ⇔ (v∗, u) ∈ Y ∗L × U and,


and



∂s− P = 0,


where v02α is defined through the equations

Qα = Nαβv02β


and,Nαβ = Nαβ−1.

Finally, we recall that

Y ∗L = v∗ ∈ Y ∗ | ∆ = N11N22 − (N12)

2 6= 0, a.e in S. ¤

9.6 The Third Duality Principle

Now we establish the third result, which may be summarized by the following theorem:

Theorem 9.6.1. Let U be a reflexive Banach space, (G Λ) : U → R a convex Gateauxdifferentiable functional and (F Λ1) : U → R convex, coercive and lower semi-continuous(l.s.c.) such that the functional

J(u) = (G Λ)(u)− F (Λ1u)− 〈u, p〉Uis bounded from below , where Λ : U → Y and Λ1 : U → Y are continuous linear operators.

Then we may write:

infz∗∈Y ∗


F ∗(z∗)−G∗(v∗) ≥ infu∈U

J(u)

where B∗(z∗) = v∗ ∈ Y ∗ such that Λ∗v∗ − Λ∗1z∗ − p = 0

Proof: By hypothesis there exists α ∈ R (α = infu∈UJ(u)) so that J(u) ≥ α, ∀u ∈ U .

That is,(G Λ)(u) ≥ F (Λ1u) + 〈u, p〉U + α, ∀u ∈ U.

The above inequality clearly implies that

supu∈U

〈u, u∗〉U − F (Λ1u)− 〈u, p〉U ≥ supu∈U

〈u, u∗〉U − (G Λ)(u)+ α

∀u∗ ∈ U∗. Since F is convex, coercive and l.s.c., by Theorem 7.2.5 we may write

supu∈U

〈u, u∗〉U − F (Λ1u)− 〈u, p〉U = infz∗∈A∗(u∗)

F ∗(z∗),

where,A∗(u∗) = z∗ ∈ Y ∗ | Λ∗1z

∗ + p = u∗.Since G also satisfies the hypothesis of Theorem 7.2.5, we have

supu∈U

〈u, u∗〉U − (G Λ)(u) = infv∗∈D∗(u∗)

G∗(v∗),


whereD∗(u∗) = v∗ ∈ Y ∗ | Λ∗v∗ = u∗.

Therefore we may summarize the last results as

F (z∗) + supv∗∈D∗(u∗)

−G∗(v∗) ≥ α, ∀z∗ ∈ A∗(u∗).

This inequality impliesF (z∗) + sup

v∗∈B∗(z∗)−G∗(v∗) ≥ α,

so that we can write

infz∗∈Y ∗


F ∗(z∗)−G∗(v∗) ≥ infu∈U

J(u)

where B∗(z∗) = v∗ ∈ Y ∗ | Λ∗v∗ − Λ∗1z∗ − p = 0. ¤

We will apply the last theorem to a changed functional concerning the primal formulationrelated to the Kirchhoff-Love plate model. We redefine (GΛ) : U → R and (F Λ1) : U → Ras

(G Λ)(u) =1

2

∫

S

Hαβλµγαβ(u)γλµ(u)dS +1

2

∫

S

hαβλµκαβ(u)κλµ(u)dS +1

2K

∫

S

w,α w,α dS

if N11(u) + K > 0, N22(u) + K > 0 and (N11(u) + K)(N22(u) + K)−N12(u)2 > 0 and, +∞otherwise.

Remark 9.6.2. Notice that (G Λ) : U → R is convex and Gateaux differentiable on itseffective domain, which is sufficient for our purposes, since the concerned Fenchel conjugatemay be easily expressed through the region of interest.

Also, we define

F (Λ1u) =1

2K

∫

S

w,α w,α dS

〈u, p〉U =

∫

S

PwdS +

∫

S

PαuαdS

whereu = (uα, w) ∈ U = W 1,2

0 (S)×W 1,20 (S)×W 2,2

0 (S).

These boundary conditions refer to a clamped plate. Furthermore,

Λ1(u) = w,1 , w,2


andΛ = Λ1αβ , Λ2α , Λ3αβ

as indicated in (9.9), (9.10) and (9.11).

Calculating G∗ : Y ∗ → R and F ∗ : Y ∗ → R we would obtain

G∗(v∗) =1

2

∫

S


2

∫

S

hαβλµMαβMλµdS+

+1

2

∫

S

Nαβw,α w,β dS +1

2K

∫

S

w,α w,α dS (9.52)

if v∗ ∈ E∗. Herev∗ = Nαβ,Mαβ, w,α,

E∗ = v∗ ∈ Y ∗ | N11+K > 0, N22+K > 0 and (N11+K)(N22+K)−N212 > 0, a.e. in S

and,

F ∗(z∗) =1

2K

∫

S

(z∗1)2dS +

1

2K

∫

S

(z∗2)2dS.

Furthermore, v∗ ∈ B∗(z∗) ⇔ v∗ ∈ Y ∗ and,

Nαβ,β + Pα = 0,−(z∗α),α + (Nαβw,β),α + Mαβ,αβ + Kw,αα + P = 0, a.e. in S.

Finally, we can express the application of last theorem as:

infz∗∈Y ∗


⋂E∗ 1

2K

∫

S

(z∗1)2dS +

1

2K

∫

S

(z∗2)2dS−

−1

2

∫

S


2

∫

S

hαβλµMαβMλµdS+

−1

2

∫

S

Nαβw,α w,β dS − 1

2K

∫

S

w,α w,α dS≥ inf

u∈UJ(u) ¤ (9.53)

The above inequality can in fact represents an equality if the positive real constant K ischosen so that the point of local extremum v∗0 = ∂G(Λu0)

∂v∈ E∗ (which means N11(u0)+K > 0,

N22(u0)+K > 0, and (N11(u0)+K)(N22(u0)+K)−N12(u0)2 > 0). The mentioned equality

is a result of a little change concerning Theorem 7.1.27.

Remark 9.6.3. For the determination of G∗(v∗) in (9.52) we have used the transformation

Qα = Nαβw,β + Kw,α,

similarly as indicated in Remark 9.3.1.


9.7 A Convex Dual Formulation

Remark 9.7.1. In this section we assume

Hαβλµ = h 4λ0µ0

λ0 + 2µ0

δαβδλµ + 2µ0(δαλδβµ + δαµδβλ),

and

hαβλµ =h2Hαβλµ

12,

where δαβ denotes the Kronecker delta and λ0, µ0 are appropriate constants.

The next result may be summarized by the following Theorem:

Theorem 9.7.2. Consider the functionals (G Λ) : U → R, (F Λ1) : U → R and 〈u, p〉Udefined as

(G Λ)(u) =1

2

∫

S


2

∫

S

hαβλµκαβ(u)κλµ(u)dS +1

2K

∫

S

w,α w,α dS,

F (Λ1u) =1

2K

∫

S

w,α w,α dS,

and

〈u, p〉U =

∫

S

PwdS +

∫

S

PαuαdS,

whereu = (uα, w) ∈ U = W 1,2

0 (S)×W 1,20 (S)×W 2,2

0 (S).

The operators γαβ and καβ are defined in (9.7) and (9.8), respectively. Furthermore, wedefine J(u) = (G Λ)(u)− F (Λ1u)− 〈u, p〉U , and

Λ1(u) = w,1 , w,2 .

Suppose there exists u0 ∈ U such that δJ(u0) = 0, and that there exists K > 0 for whichN11(u0) + K > 0, N22(u0) + K > 0 , (N11(u0) + K)(N22(u0) + K)−N12(u0)

2 > 0 (a.e in S)and h1212/(2K0) > K where K0 is the constant related to Poincare Inequality and,

Nαβ(u0) = Hαβλµγλµ(u0).

Then,

J(u0) = minu∈U

J(u) = max(v∗,z∗,w)∈E∗∩B∗

−G∗(v∗) + 〈z∗α, z∗α〉L2(S)/(2K)= −G∗(v∗0) + 〈z∗0α , z∗0α〉L2(S)/(2K) (9.54)


where,

v∗0 =∂G(Λu0)

∂vand z∗0α = Kw0,α,

G∗(v∗) = G∗L(v∗) =

1

2

∫

S


2

∫

S


2

∫

S

NKαβQ,αQ,βdS

if v∗ ∈ E∗, where v∗ = Nαβ, Mαβ, Qα ∈ E∗ ⇔ v∗ ∈ L2(S,R10) and

N11 + K > 0 N22 + K > 0 and (N11 + K)(N22 + K)−N212 > 0, a.e. in S

and,

(v∗, z∗) ∈ B∗ ⇔

Nαβ,β + Pα = 0,

Qα,α + Mαβ,αβ − z∗α,α + P = 0,

h1212M12 + z∗1,2/K = 0,

z∗1,2 = z∗2,1, a.e. in S, and, z∗ = θ on Γ,

being NKαβ as indicated in (9.6).

Proof: Similarly to Proposition 9.4.1, we may obtain the following result. If v∗ ∈ E∗ then

G∗L(v∗) = G∗(v∗) ≥ 〈v∗, Λu〉Y −G(Λu), ∀u ∈ U,

so that

G∗L(v∗)− 1

2K〈z∗α, z∗α〉L2(S) ≥ 〈v∗, Λu〉Y − 1

2K〈z∗α, z∗α〉L2(S) −G(Λu), ∀u ∈ U,

and thus, as Λ∗v∗ − Λ∗1z∗ − p = 0 (see the definition of B∗) we obtain

Qα,α + Mαβ,αβ − z∗α,α + P = 0 a.e. in S.

Through this equation we may symbolically write

M12 = Λ−1312(−Qα,α + z∗α,α − Mαβ,αβ − P )/2, (9.55)

where Mαβ,αβ denotes M11,11 + M22,22, in S, so that substituting such a relation in the lastinequality we have

1

2

∫

S


2

∫

S

h1111M211dS +

∫

S

h1122M11M22dS +1

2

∫

S

h2222M222dS

+ 2

∫

S

h1212(Λ−1312(v

∗, z∗))2dS +1

2

∫

S

NKαβQ,αQ,βdS

− 1

2K〈z∗α, z∗α〉L2(S)

≥ 〈Λ1u, z∗〉L2(S;R2) − 1

2K〈z∗α, z∗α〉L2(S) −G(Λu)

+ 〈u, p〉U , ∀u ∈ U, (9.56)


where M12 is made explicit through equation (9.55). This equation makes z∗ an independentvariable, so that evaluating the supremum concerning z∗, particularly for the left side ofabove inequality, the global extremum is achieved through the equation :

−([Λ−1312 ]

∗[h1212Λ−1312(v

∗, z∗)]),α − z∗α/K = 0, a.e. in S.

This means

−h1212Λ−1312(v

∗, z∗)− z∗α,β/K = 0, a.e. in S and z∗1 = z∗2 = 0 on Γ

orh1212M12 + z∗α,β/K = 0, a.e. in S and z∗1 = z∗2 = 0 on Γ

for (α, β) = (1, 2) and (2, 1). Therefore, after evaluating the suprema in both sides of (9.56),we may write

G∗L(v∗)− 1

2K〈z∗α, z∗α〉L2(S) ≥ F (Λ1u)−G(Λu) + 〈u, p〉U , ∀u ∈ U, and (v∗, z∗) ∈ B∗ ∩ E∗.

and it seems to be clear that the condition h1212/(2K0) > K guarantees coercivity for theexpression of left side in the last inequality (see the next remark), so that the unique localextremum concerning z∗ is also a global extremum. The equality and remaining conclusionsresults from the Gateaux differentiability of primal and dual formulations and an application(with little changes) of Theorem 7.1.27. ¤

Remark 9.7.3. Observe that the dual functional could be expressed as

G∗L(v∗)− 1

2K〈z∗α, z∗α〉L2(S) =

1

2

∫

S


2

∫

S

h1111M211dS +

∫

S

h1122M11M22dS

+1

2

∫

S

h2222M222dS +

1

2

∫

S

NKαβQ,αQ,βdS +

∫

S

h1212(z∗1,2)

2/K2dS

+

∫

S

h1212(z∗2,1)

2/K2dS − 1

2K〈z∗α, z∗α〉L2(S). (9.57)

Thus, through the relation h1212/(2K0) > K (where K0 is the constant related to Poincareinequality), it is now clear that the dual formulation is convex on E∗ ∩B∗.

9.8 A Final Result, Other Sufficient Conditions of Op-

timality

This final result is developed similarly to the triality criterion introduced in Gao [20], whichdescribes, in some situations, sufficient conditions for optimality.

We prove the following result


Theorem 9.8.1. Consider J : U → R where J(u) = G(Λu) + F (u),

G(Λu) =1

2

∫

S


2

∫

S

hαβλµw,αβw,λµdS.

Here the operators γαβ are defined as in (9.7),

F (u) = −∫

S

PwdS ≡ −〈u, f〉U ,

and,U = W 1,2

0 (S)×W 1,20 (S)×W 2,2

0 (S).

Then, if u0 ∈ U is such that δJ(u0) = θ and

1

2

∫

S


2

∫

S

hαβλµw,αβw,λµdS ≥ 0,∀w ∈ W 2,20 (S), (9.58)

we haveJ(u0) = min

u∈UJ(u).

Proof: It is clear that

G(Λu) + F (u) ≥ −(G Λ)∗(u∗)− F ∗(−u∗), ∀u ∈ U, u∗ ∈ U∗,

so thatG(Λu) + F (u) ≥ −(G Λ)∗(Λ∗v∗)− F ∗(−Λ∗v∗), ∀u ∈ U, v∗ ∈ Y ∗. (9.59)

Consider u0 for which δJ(u0) = θ and such that (9.58) is satisfied.

Defining

v∗0 =∂G(Λu0)

∂v,

from Theorem 7.1.27 we have that

δ(−G∗L(v∗0) + 〈u0, Λv∗0 − f〉U) = θ,

J(u0) = −GL(v∗0),

andΛ∗v∗0 = f.

This meansF ∗(−Λ∗v∗0) = 0.

On the other hand(G Λ)∗(Λ∗v∗0) = sup

u∈U〈Λu, v∗0〉Y −G(Λu),


or

(G Λ)∗(Λv∗0) = supu∈U

〈uα,β + uβ,α

2, Nαβ(u0)〉L2(S) + 〈−w,αβ,Mαβ(u0)〉L2(S)

+〈w,αQα(u0)〉L2(S) − 1

2

∫

S

Hαβλµγαβ(u)γλµ(u)dS

−1

2

∫

S

hαβλµw,αβw,λµdS . (9.60)

Since

γαβ(u) =uα,β + uβ,α

2+

1

2w,αw,β,

from the last equality, we may write

(G Λ)∗(Λ∗v∗0) = supu∈U

〈γαβ(u), Nαβ(u0)〉L2(S) − 〈w,αw,β

2, Nαβ(u0)〉L2(S)

+〈−w,αβ,Mαβ(u0)〉L2(S) + 〈w,α, Qα(u0)〉L2(S)

−1

2

∫

S

Hαβλµγαβ(u)γλµ(u)dS − 1

2

∫

S

hαβλµw,αβw,λµdS

. (9.61)

As (Qα(u0)),α + (Mαβ(u0)),αβ + P = 0, we obtain

(G Λ)∗(Λ∗v∗0) ≤ supu∈U

−

⟨w,αw,β

2, Nαβ(u0)

⟩L2(S)

− 1

2

∫

S

hαβλµw,αβw,λµdS

+

∫

S

PwdS

+

1

2

∫

S

HαβλµNαβ(u0)Nλµ(u0)dS. (9.62)

Therefore, from hypothesis (9.58) the extremum indicated in (9.62) is attained for functionssatisfying

(Nαβ(u0)wβ),α − (hαβλµwλµ),αβ + P = 0. (9.63)

From δJ(u0) = θ and boundary conditions we obtain

w = w0, a.e. in S,

so that

(G Λ)∗(Λ∗v∗0) ≤ 〈w0,αw0,β

2, Nαβ(u0)〉L2(S) +

1

2

∫

S

hαβλµw0,αβw0,λµdS

+1

2

∫

S


However, sinceQα(u0) = Nαβ(u0)w0,β ,

andMαβ(u0) = −hαβλµw0,λµ


from (9.64) we obtain

(G Λ)∗(Λ∗v∗0) ≤ 1

2

∫Nαβ(u0)Qα(u0)Qβ(u0)dS +

1

2

∫

S

hαβλµMαβ(u0)Mλµ(u0)dS

+1

2

∫

S


Hence(G Λ)∗(Λ∗v∗0) ≤ GL(v∗0) = −J(u0),

and thus as F ∗(−Λ∗v∗0) = 0, we have that

J(u0) ≤ −(G Λ)∗(Λ∗v∗0)− F ∗(−Λ∗v∗0),

which, from (9.59) completes the proof. ¤

9.9 Final Remarks

In this chapter we presented four different dual variational formulations for the Kirchhoff-Love plate model. Earlier results (see references [33],[18]) present a constraint concerning thegap functional to establish the complementary energy (dual formulation). In the present workthe dual formulations are established on the hypothesis of existence of a global extremum forthe primal functional and the results are applicable even for compressed plates. In particularthe second duality principle is obtained through an extension of a theorem met in [35], andin this case we are concerned with the solution behavior as K → +∞, even though a rigorousand complete analysis of such behavior has been postponed for a future work. However, whatseems to be interesting is that the dual formulation as indicated in (9.51) is represented bya natural extension of the results found in [35] (particularly Theorem 9.1.1), plus a kind ofpenalization concerning the inversion of constitutive equations.

It is worth noting that the third dual formulation was based on the same theorem, despitethe fact such a result had not been directly used, we followed a similar idea to prove theduality principle. For this last result, the membrane forces are allowed to be negative sinceit is observed the restriction N11 + K > 0, N22 + K > 0 and (N11 + K)(N22 + K) −N2

12 >0, a.e. in S, where K ∈ R is a positive suitable constant.

In section 9.7, we obtained a convex dual variational formulation for the plate model, whichallows non positive definite membrane force matrices. In this formulation, the Poincareinequality plays a fundamental role.

Finally, in the last section, we developed a result similar to Gao’s triality criterion presentedin [20]. In the plate application this gives sufficient conditions for optimality. We present anew proof of sufficient conditions of existence of a global extremum for the primal problem.


As earlier mentioned, such conditions may be summarized by δJ(u0) = θ and

1

2

∫

S


2

∫

S

hαβλµw,αβw,λµdS ≥ 0,∀w ∈ W 2,20 (S). ¤

Chapter 10

Duality Applied to Elasticity

10.1 Introduction and Primal Formulation

Our first objective in the present chapter is to establish a dual variational formulation fora finite elasticity model. Even though existence of solutions for this model has been provenin Ciarlet [10], the concept of complementary energy, as a global optimization approach, ispossible to be defined only if the stress tensor is positive definite at a critical point. Thus wehave the goal of relaxing such constraints and start by describing the primal formulation.

Consider S ⊂ R3 an open, bounded, connected set, which represents the reference volumeoccupied by an elastic solid under the load f ∈ L2(S;R3). We denote by Γ the boundaryof S. The field of displacements under the action of f is denoted by u ≡ (u1, u2, u3) ∈ U ,where u1, u2, and u3 denotes the displacements related to directions x, y, and z respectively,on the cartesian basis (x, y, z). Here U is defined as

U = u = (u1, u2, u3) ∈ W 1,4(S;R3) | u = (0, 0, 0) ≡ θ on Γ (10.1)

Denoting the stress tensor by σij, where

σij = Hijkl

(1

2(uk,l + uk,l + um,kum,l)

)(10.2)

and Hijkl is a positive definite matrix related to the coefficients of Hooke’s Law, theboundary value form of the finite elasticity model is given by

σij,j + (σimum,j),j + fi = 0, a.e. in S,

u = θ, on Γ.(10.3)

114

CHAPTER 10. DUALITY APPLIED TO ELASTICITY 115

The corresponding primal variational formulation is represented by J : U → R, where

J(u) =1

2

∫

S

Hijkl

(1

2(ui,j + ui,j + um,ium,j)

) (1


)dx− 〈u, f〉L2(S;R3)

(10.4)

10.2 The Duality Principles

In this section we establish the duality principles. We start with the following theorem.

Theorem 10.2.1. Define J : U → R as

J(u) = G∗∗(Λu)− F1(u) (10.5)

where

G(Λu) =1

2

∫

S

Hijkl

(1

2(ui,j + ui,j + um,ium,j)

)(1


)dx

+K

2〈um,i, um,i〉L2(S), (10.6)

F1(u) = F (u)− 〈u, f〉L2(S;R3), (10.7)

and

F (u) =K

2〈um,i, um,i〉L2(S). (10.8)

Here Λ : U → Y = Y1 × Y2 ≡ L2(S;R9)× L2(S,R9) is given by

Λu = Λ1u, Λ2u ≡ 1

2(ui,j + ui,j), um,i. (10.9)

Thus we can write

infu∈U

J(u) ≤ infz∗∈Y ∗2

supv∗∈C∗(z∗)

F ∗(z∗)−G∗(v∗), (10.10)

wherev∗ ≡ σ,Q, (10.11)

C∗(z∗) = (σ,Q) ∈ Y ∗ | σij,j + Qij,j − z∗ij,j + fi = 0, a.e. in S, (10.12)

F ∗(z∗) =1

2K〈z∗im, z∗im〉L2(S) (10.13)

and

G∗(v∗) = G∗L(v∗) =

1

2

∫

S

Hijklσijσkldx +1

2

∫

S

σijQmiQmjdx (10.14)


if σK is positive definite in S, where

σK =

σ11 + K σ12 σ13

σ21 σ22 + K σ23

σ31 σ32 σ33 + K

(10.15)

and also σij = σK−1. Finally, Hijkl = Hijkl−1.

Proof: Defining α ≡ infu∈UJ(u) we have

G∗∗(Λu)− F1(u) ≥ α, ∀u ∈ U, (10.16)

or−F1(u) ≥ −G∗∗(Λu) + α, ∀u ∈ U, (10.17)

so thatsupu∈U

〈u, u∗〉U − F1(u) ≥ supu∈U

〈u, u∗〉U −G(Λu)+ α. (10.18)

However, from Theorem 7.2.5

F ∗1 (u∗) = sup

u∈U〈u, u∗〉U − F1(u) = inf

z∗∈C∗(u∗)F ∗(z∗) (10.19)

whereC∗(u∗) = z∗ ∈ Y ∗

2 | z∗ij,j − u∗i − fi = 0, a.e. in S. (10.20)

On the other hand, also from Theorem 7.2.5

(G∗∗ Λ)∗(u∗) = supu∈U

〈u, u∗〉U −G∗∗(Λu) = infv∗∈D∗(u∗)

G∗(v∗) (10.21)

whereD∗(u∗) = v∗ ∈ Y ∗ | σij,j + Qij,j − u∗i = 0, a.e. in S. (10.22)

We can summarize the last results by

infu∈U

J(u) = α ≤ F ∗1 (u∗)− (G∗∗ Λ)∗(u∗) ≤ F ∗(z∗) + sup

v∗∈D∗(u∗)−G∗(v∗), (10.23)

∀z∗ ∈ C∗(u∗).

Hence we can write

infu∈U


supv∗∈D∗(z∗)

F ∗(z∗)−G∗(v∗) (10.24)

whereD∗(z∗) = v∗ ∈ Y ∗ | σij,j + Qij,j − z∗ij,j + fi = 0, a.e. in S. (10.25)


Finally, we have to prove that G∗L(v∗) = G∗(v∗) if σK is positive definite in S. We start by

formally calculating g∗L(y∗), the Legendre transform of g(y), where

g(y) = Hijkl

(y1ij +

1

2y2miy2mj

)(y1kl +

1

2y2mky2ml

)+

K

2y2miy2mi . (10.26)

We recall thatg∗L(y∗) = 〈y, y∗〉R18 − g(y) (10.27)

where y ∈ R18 is solution of equation

y∗ =∂g(y)

∂y. (10.28)

Thus

y∗1ij = σij = Hijkl

(y1kl +

1

2y2mky2ml

)(10.29)

and

y∗2mi = Qmi = Hijkl

(y1kl +

1

2y2oky2ol

)y2mj + Ky2mi (10.30)

so thatQmi = σijy2mj + Ky2mi . (10.31)

Inverting these last equations, we have

y2mi = σijQmj (10.32)

where

σij = σ−1K =

σ11 + K σ12 σ13

σ21 σ22 + K σ23

σ31 σ32 σ33 + K

−1

(10.33)

and also

y1ij = Hijklσkl − 1

2y2miy2mj . (10.34)

Finally

g∗L(σ,Q) =1

2Hijklσijσkl +

1

2σijQmiQmj. (10.35)

Now we will prove that g∗L(v∗) = g∗(v∗) if σK is positive definite. First observe that

g∗(v∗) = supy∈R18

〈y1, σ〉R9 + 〈y2, Q〉R9 − 1

2Hijkl

(y1ij +

1

2y2miy2mj

)(y1kl +

1

2y2mky2ml

)

−K

2y2miy2mi

= sup(y1,y2)∈R9×R9

〈y1ij − 1

2y2miy2mj , σij〉R + 〈y2, Q〉R9 − 1

2Hijkl[y1ij ][y1kl ]− K

2y2miy2mi

.


The result follows just observing that

supy1∈R9

〈y1ij , σij〉R − 1

2Hijkl[y1ij ][y1kl ]

=

1

2Hijklσijσkl (10.36)

and

supy2∈R9

〈−1

2y2miy2mj , σij〉R + 〈y2, Q〉R9 − K

2y2miy2mi

=

1

2σijQmiQmj (10.37)

if σK is positive definite. ¤Through the next theorem we obtain a convex primal dual formulation for the finite elasticitymodel.

Theorem 10.2.2. The solution of the boundary value problem indicated in (10.3) minimizesthe functional J : U × Y → R under the constraint

σij,j + Qij,j −Kui,jj + fi = 0, a.e. in S, (10.38)

whereJ(u, v∗) = G∗(v∗)− 〈Λu, v∗〉Y + G∗∗(Λu), (10.39)

G(Λu) =1

2

∫

S

Hijkl

(1

2(ui,j + ui,j) +

1

2um,ium,j

)(1

2(uk,l + uk,l) +

1

2um,kum,l

)dx

+K

2〈um,i, um,i〉L2(S) (10.40)

and

G∗(v∗) = G∗L(v∗) =

1

2

∫

S

Hijklσijσkldx +1

2

∫

S

σijQmiQmjdx (10.41)

if σK is positive definite in S. Here

σK =

σ11 + K σ12 σ13

σ21 σ22 + K σ23

σ31 σ32 σ33 + K

(10.42)

and also σij = σK−1.

Proof: Consider u0 ∈ U a solution for the boundary value problem. If K > 0 is big enough,defining

v∗0 =∂G(Λu0)

∂v,

we have that (u0, v∗0) minimizes J and satisfies (10.38). ¤


10.3 Final Results, Sufficient Conditions of Optimality

Our final result also establishes a criterium of optimality.

Theorem 10.3.1. Consider the functionals (G Λ) : U → R and (F Λ1) : U → R given by

G(Λu) =1

2

∫

S

Hijkl

(1

2(ui,j + uj,i + um,ium,j)

)(1

2(uk,l + ul,k + um,kum,l)

)dx,

F (Λu) =K

2

∫

S

Λij(u)Λij(u)dx.

HereU = W 1,2

0 (S;R3),

and Λ : U → Y = Y ∗ = L4(S;R9) is expressed as

Λu = Λij(u) =

1

2(ui,j + uj,i + um,ium,j)

.

DefineJ(u) = G(Λu)− 〈u, f〉U

where f ∈ L2(S;R3) is such that that J is bounded below.

Then,

infu∈U

J(u) ≥ supσ∈Y ∗

inf

σ∈Y ∗

J(σ, σ)

, (10.43)

whereJ(σ, σ) = F (σ)−G∗(σ + σ)− Ff (−σ),

F (σ) =1

2K

∫

S

σijσijdx,

G∗(σ + σ) =1

2

∫

S

Hijkl(σij + σij)(σkl + σkl)dx,

Ff (σ) = supu∈U

〈Λij(u), σij〉L2(S) − F (Λu) + 〈u, f〉U.

Furthermore, if there exists (σ0, σ0) ∈ Y ∗ × Y ∗ such that δJ(σ0, σ0) = θ, and K > 0 is suchthat

J(σ) = F (σ)−G∗(σ0 + σ)

is coercive, so that the infimum indicated in right side of (10.43) is attained at σ0, we have

J(σ0, σ0) = maxσ∈Y ∗

minσ∈Y ∗

J(σ, σ)

.


Finally, if there exists a corresponding u0 ∈ U such that δJ(u0) = θ,

σ0 + σ0 =∂G(Λu0)

∂v

andFf (−σ0) = 〈Λij(u0), σ0ij〉L2(S) − F (Λu0) + 〈u0, f〉U,

it is also such thatJ(u0) = min

u∈UJ(u).

Proof: First observe that

G∗(σ + σ) + Ff (−σ) ≥ 〈Λij(u), σij〉L2(S) + 〈Λij(u), σij〉L2(S)

−〈Λij(u), σij〉L2(S) − F (Λu) + 〈u, f〉U −G(Λu), (10.44)

or

−F (σ) + G∗(σ + σ) + Ff (−σ) ≥ −F (σ) + 〈Λij(u), σij〉L2(S)

−F (Λu) + 〈u, f〉U −G(Λu). (10.45)

Thus, taking the supremum in σ in both sides of last inequality, we obtain

supσ∈Y ∗

−F (σ) + G∗(σ + σ) + Ff (−σ) ≥ supσ∈Y ∗

−F (σ) + 〈Λij(u), σij〉L2(S)

−F (Λu) + 〈u, f〉U −G(Λu). (10.46)

That is,supσ∈Y ∗

−F (σ) + G∗(σ + σ) + Ff (−σ) ≥ 〈u, f〉U −G(Λu),∀u ∈ U.

so that

infu∈U

G(Λu)− 〈u, f〉U ≥ supσ∈Y ∗

inf

σ∈Y ∗

F (σ)−G∗(σ + σ)− Ff (−σ)

.

Finally, an application of a little change of Theorem 7.1.27, now for the non-linear operatorcase, completes the proof (we recall that such a theorem establishes a equality between theprimal and dual formulations at a critical point). ¤

Chapter 11

Duality Applied to a Membrane ShellModel


In this chapter we establish dual variational formulations for the elastic membrane shellmodel presented in [12] (P.Ciarlet). Ciarlet proves existence of solutions, however, the com-plementary energy as developed in [33]( J.J.Telega), is possible only for a special class ofexternal loads, that generate a critical point with positive definite membrane tensor. In thischapter we relax such constraints, and in fact our final result is a primal dual formulationwhich is convex. Now we describe the primal formulation.

Consider a domain S ⊂ R2 and a injective mapping ~θ : S× [−ε, ε] → R3 such that S0 = ~θ(S)denotes the middle surface of a shell of thickness 2ε.

The mapping ~θ may be expressed as:

~θ(x) = ~θ1(y) + x3a3(y)

where a3 = (a1 × a2)/‖a1 × a2‖ and aα = ∂α~θ, y = (y1, y2) denote the curvilinear coordi-

nates, x = (y, x3) and −ε ≤ x3 ≤ ε.

The contravariant basis of the tangent plane to S in y, denoted by aα is defined throughthe relations

aα · aβ = δβα, and a3 = a3

The covariant components of the metric tensor, denoted by aαβ, are defined as

aαβ = aα · aβ

and the concerned contravariant components are denoted by aαβ and expressed as

aαβ = aα · aβ

121

CHAPTER 11. DUALITY APPLIED TO A MEMBRANE SHELL MODEL 122

It is not difficult to show that aα = aαβaβ, aαβ = aαβ−1 and aα = aαβaβ. We also

define√

a(y) = ‖a1(y)× a2(y)‖.The curvature tensor denoted by bαβ is expressed as

bαβ = a3 · ∂αaβ.

Concerning the displacements due to external loads action, we denote them by η = ηiai,

and the admissible displacements field is denoted by U , where

U = η ∈ W 1,4(S;R3) | η = (0, 0, 0) on Γ0

and here Γ = Γ0 ∪ Γ1 (Γ0 ∩ Γ1 = ∅) denotes the boundary of S.

We now state the Theorem 9-1-1 of reference [12] (Mathematical Elasticity , Vol. III -Theoryof shells), by P.Ciarlet.

Theorem 11.1.1. Let S ⊂ R2 a domain and ~θ ∈ C2(S;R3) be an injective mapping such

that the two vectors aα = ∂α~θ are linearly independent in all points of S, let a3 = (a1 ×

a2)/‖a1 × a2‖, and let the vectors ai be defined by ai.aj = δij. Given a displacement field

ηiai of the surface S0 = ~θ(S), let the covariant components of the change of metric tensor

associated with this displacement field be defined by

Gαβ(η) = (aαβ(η)− aαβ),

where aαβ and aαβ(η) denote the covariant components of the metric tensors of the surfaces~θ(S) and (~θ + ηia

i)(S) respectively. Then

Gαβ(η) = (ηα‖β + ηβ‖α + amnηm‖αηn‖β)/2

where ηα‖β = ∂βηα − Γσαβησ − bαβη3 and η3‖β = ∂βη3 + bσ

βησ. ¤

We define the constitutive relations as

Nαβ = aαβστε Gστ (η) (11.1)

where Nαβ denotes the membrane forces, and

aαβστε =

ε

4(

4λµ

λ + 2µaαβaστ + 2µ(aασaβτ + aατaβσ)) (11.2)

The potential energy (stored energy plus external work) is expressed by the functional J :U → R where,

J(η) =1

2

∫

S

aαβστε Gστ (η)Gαβ(η)

√a dy −

∫

S

piηi

√a dy


where pi =∫ ε

−εf i dx3, here f i ∈ L2(S × [−ε, ε];R3) denotes the external load density.

We will define (G Λ) : U → R and F : U → R as

(G Λ)(η) =1

2

∫

S


√a dy

and,

F (η) =

∫

S

piηi

√a dy, (11.3)

where Λ = Λ1αβ , Λ2mα, Λ1αβ(η) = (ηα‖β +ηβ‖α)/2 and Λ2mα(η) = ηm‖α. Thus, the primalvariational formulation is given by J : U → R, where

J(η) = G(Λη)− F (η). (11.4)


We will be concerned with the Legendre Transform related to the function g : R10 → Rexpressed as

g(y) =1

2aαβστ

ε (y1αβ +1

2amny2mαy2nβ)(y1στ +

1

2akly2kσy2lτ ) (11.5)

Its Legendre Transform, denoted by g∗L : R10L → R is given by

g∗L(y∗) = 〈y, y∗〉R10 − g(y) (11.6)

where y ∈ R10 is solution of the system

y∗ =∂g(y)

∂y. (11.7)

That is,

y∗1αβ = aαβστε (y1στ +

1


and

y∗2mα = aαβστε (y1στ +

1

2akly2kσy2lτ )amny2nβ (11.9)

ory∗2mα = y∗1αβa

mny∗2nβ . (11.10)

Therefore, after simple algebraic manipulations we would obtain

g∗L(y∗) =1

2aαβστy

∗1στ y∗1αβ +

1

2Rαβmny

∗2αmy∗2nβ , (11.11)


whereRαβmn = y∗1αβa

mn−1. (11.12)

Thus, denoting v∗ = Nαβ, Qmα we have

G∗L(v∗) =

∫

S

g∗L(v∗)√

a dy =

∫

S

aαβστNστNαβ

√a dy +

1

2

∫

w

RαβmnQmαQnβ

√a dy. (11.13)

We now obtain the polar functional related to the external load.

11.3 The Polar Functional Related to F : U → R

The polar functional related to F : U → R, for v∗ = Nαβ, Qmα is expressed by

F ∗(Λ∗v∗) = supη∈U

〈η, Λ∗v∗〉U − F (u). (11.14)

That is,

F ∗(Λ∗v∗) = supη∈U

〈Λ1αβη,Nαβ√

a〉L2(S) + 〈Λ2mαη, Qmα√

a〉L2(S) −∫

S

piηi

√a dy. (11.15)

Thus

F ∗(Λ∗v∗) =

0, if v∗ ∈ C∗,+∞, otherwise,

(11.16)

where

v∗ ∈ C∗ ⇔

−(Nαβ + Qαβ)|β + bαβQ3β = pα in S,

−bαβ(Nαβ + Qαβ)−Q3β|β = p3 in S,

(Nαβ + Qαβ)νβ = 0 on Γ1,

Q3β|βνβ = 0 on Γ1.

(11.17)

11.4 The Final Format of First Duality Principle

The duality principle presented in Theorem 9.5.1 is applied to the present case. Thus weobtain

infη∈UJ(η) ≤ inf

v∗∈C∗ 1

2K〈v∗, v∗〉L2(S,R10) −G∗

L(v∗) + K

∥∥∥∥Λu− ∂g∗L(v∗)∂y∗

∥∥∥∥2

L2(S)

.


More explicitly

infη∈U

1

2

∫

S


√a dy −

∫

S

piηi

√a dy

≤

infv∗∈C∗

1

2K

(∫

S

NαβNαβ√

a dy +

∫

S

QmαQmα√

a dy

)

−1

2

∫

S

aαβστNστNαβ

√a dy − 1

2

∫

S

RαβmnQmαQnβ√

a dy

+2∑

α,β=1

3∑m=1

(K

∥∥ηα‖β − aαβστNστ − aklv02kαv02lβ

∥∥L2(S)

+ K‖ηm‖α − v02mα‖L2(S)

)

11.5 The Second Duality Principle

Our objective now is to establish a convex primal-dual formulation. First we will writethe primal formulation as the difference of two convex functionals and then will obtain thesecond duality principle, as indicated in the next theorem.

Theorem 11.5.1. Define J : U → R as

J(η) = G(Λη)− F (Λ2η) (11.18)

whereG(Λη) = G∗∗(Λη)− 〈η, p〉, (11.19)

G(Λη) =1

2

∫

S


√a dy +

K

2

∫

S

ηm‖αηm‖α√

a dy, (11.20)

F (Λ2η) =K

2

∫

S

ηm‖αηm‖α√

a dy, (11.21)

〈η, p〉 =

∫

S

piηi

√a dy, (11.22)

and Λ : U → Y ∗1 × Y ∗

2 ≡ L2(S;R4)× L2(S;R6) is defined as

Λη = Λ1η, Λ2η ≡

1

2(ηα‖β + ηβ‖α)

, ηm‖α

. (11.23)

Thus, we may write

infη∈UJ(η) ≤ inf

u∗∈U∗sup

v∗∈C∗(u∗)F ∗(u∗)−G∗(v∗) (11.24)


wherev∗ ≡ N, Q, (11.25)

F ∗(u∗) =1

2K

∫

S

u∗miu∗mi

√a dy, (11.26)

G∗(v∗) = G∗L(v∗) =

∫

S

g∗L(v∗)√

ady =1

2

∫

S

aαβστNστNαβ

√ady +

1

2

∫

S

RαβmnQmαQnβ

√a dy

(11.27)if NK = Nαβamn + Kδαβδmn is positive definite in S. Furthermore

Rαβmn = N−1K (11.28)

andaαβστ = aαβστ

ε −1. (11.29)

Finally,

v∗ ≡ N,Q ∈ C∗(u∗) ⇔

−(Nαβ + Qαβ − u∗αβ)|β + bαβ(Q3β − u∗3β) = pα in S,

−bαβ(Nαβ + Qαβ − u∗αβ)− (Q3β − u∗3β)|β = p3 in S,

(Nαβ + (Qαβ − u∗αβ))νβ = 0 on Γ1,

(Q3β − u∗3β)|βνβ = 0 on Γ1.(11.30)

Proof: Defining α ≡ infη∈UJ(η) we have

G(Λη)− F (Λ2η) ≥ α, ∀η ∈ U, (11.31)

or−F (Λ2η) ≥ −G(Λη) + α, ∀η ∈ U, (11.32)

so thatsupη∈U

〈Λ2η, u∗〉 − F (η) ≥ supη∈U

〈Λ2η, u∗〉 − G(Λη)+ α. (11.33)

However,F ∗(u∗) ≥ sup

η∈U〈Λ2η, u∗〉 − F (Λ2η). (11.34)

On the other hand, from Theorem 7.2.5

(G Λ)∗(Λ∗2u∗) = sup

η∈U〈Λ2η, u∗〉 − G(Λη) = inf

v∗∈C∗(u∗)G∗(v∗) (11.35)


where

v∗ ≡ N, Q ∈ C∗(u∗) ⇔

−(Nαβ + Qαβ − u∗αβ)|β + bαβ(Q3β − u∗3β) = pα in S,

−bαβ(Nαβ + Qαβ − u∗αβ)− (Q3β − u∗3β)|β = p3 in S,

(Nαβ + (Qαβ − u∗αβ))νβ = 0 on Γ1,

(Q3β − u∗3β)|βνβ = 0 on Γ1.(11.36)

We can summarize the last results by

infη∈UJ(η) = α ≤ F ∗(u∗)− (G Λ)∗(Λ∗2u

∗) = F ∗(u∗) + supv∗∈C∗(u∗)

−G∗(v∗), (11.37)

∀u∗ ∈ U∗.

Finally, we have to prove that G∗L(v∗) = G∗(v∗) if NK is positive definite in S. We start by

formally calculating g∗L(y∗), the Legendre transform of g(y), where g : R10 → R is expressedas:

g(y) =1

2aαβστ

ε (y1αβ +1

2amny2mαy2nβ)(y1στ +

1

2akly2kσy2lτ ) +

K

2y2mαy2mα . (11.38)

Observe that g∗L : R10L → R is given by

g∗L(y∗) = 〈y, y∗〉R10 − g(y) (11.39)

where y ∈ R10 is solution of the system

y∗ =∂g(y)

∂y. (11.40)

That is,

y∗1αβ = aαβστε (y1στ +

1


and,

y∗2mα = aαβστε (y1στ +

1

2akly2kσy2lτ )amny2nβ + Ky2mα (11.42)

ory∗2mα = y∗1αβa

mny∗2nβ + Ky2mα . (11.43)

Therefore, after simple algebraic manipulations we would obtain

g∗L(y∗) =1

2aαβστy

∗1στ y1αβ +

1

2Rαβmny

∗2αmy∗2nβ (11.44)

whereRαβmn = y∗1αβa

mn + Kδαβδmn−1. (11.45)


Thus, through the notation v∗ = Nαβ, Qmα we would obtain

G∗L(v∗) =

∫

w

g∗L(v∗)√

a dy =1

2

∫

S

aαβστNστNαβ

√a dy +

1

2

∫

S

RαβmnQmαQnβ√

a dy (11.46)

whereRαβmn = Nαβamn + Kδαβδmn−1. (11.47)

Now we will prove that g∗L(v∗) = g∗(v∗) if NK is positive definite. First observe that

g∗(v∗) = supy∈R10

〈y1, N〉R4 + 〈y2, Q〉R6

−1

2aαβστ

ε [y1αβ +1

2amny2mαy2nβ ][y1στ +

1

2akly2kσy2lτ ]− K

2y2mαy2mα =

sup(y1,y2)∈R4×R6

〈y1αβ − 1

2amny2mαy2mβ , Nαβ〉R + 〈y2, Q〉R6 − 1

2aαβστ

ε [y1αβ ][y1στ ]− K

2y2mαy2mα.

(11.48)The result follows just observing that

supy1∈R4

〈y1αβ , Nαβ〉R − 1

2aαβστ

ε [y1αβ ][y1στ ] =1

2aαβστN

στNαβ (11.49)

and

supy2∈R6

〈−1

2amny2mαy2mβ , Nαβ〉R + 〈y2, Q〉R6 − K

2y2mαy2mα =

1

2RαβmnQmαQnβ (11.50)

if NK is positive definite. ¤

11.6 The Convex Primal Dual Formulation

The next theorem is concerned with the convex primal-dual formulation.

Theorem 11.6.1. The solution of the boundary value problem of shell model described aboveminimizes the functional J∗ : U × Y ∗ → R under the constraint (η, v∗) ∈ A∗, where

J∗(η, v∗) = G∗(v∗)− 〈Λη, v∗〉+ G∗∗(Λη) (11.51)

where

G(Λη) =1

2

∫

S

aαβστε Gστ (~η)Gαβ(~η)

√a dy +

K

2

∫

S

ηm‖αηm‖α√

a dy, (11.52)

G∗(v∗) = G∗L(v∗) =

∫

S

g∗L(v∗)√

a dy =1

2

∫

S

aαβστNστNαβ

√a dy +

1

2

∫

S

RαβmnQmαQnβ

√a dy

(11.53)


if NK = Nαβamn + Kδαβδmn is positive definite in S. Furthermore

Rαβmn = N−1K , (11.54)

aαβστ = aαβστε −1 (11.55)

and

〈Λη, v∗〉 =

∫

S

1

2(ηα‖β + ηβ‖α)Nαβ

√a dy +

∫

S

ηm‖αQmα√

a dy. (11.56)

Finally,

v∗ ≡ N, Q ∈ A∗ ⇔

−(Nαβ + Qαβ −Kηα‖β)|β + bαβ(Q3β −Kη3‖β) = pα in S,

−bαβ(Nαβ + Qαβ −Kηα‖β)− (Q3β −Kη3‖β)|β = p3 in S,

(Nαβ + (Qαβ −Kηα‖β))νβ = 0 on Γ1,

(Q3β −Kη3‖β)|βνβ = 0 on Γ1.(11.57)

Proof: For the solution η ∈ U of the boundary value problem, if K is big enough, we candefine

v∗ =∂G(Λη)

∂v, (11.58)

so that (η, v∗) minimizes J∗ and satisfies A∗. ¤

11.7 Conclusion

We obtained three different dual variational formulations for the non-linear elastic shell modelstudied in P.Ciarlet [12] (a membrane shell model).The first duality principle presented isan extension of a theorem found in Toland [35]. The solution behavior as K → +∞ is ofparticular interest for a future work.

The second duality principle relaxes the condition of positive definite membrane tensor,and thus the constant K must be chosen so that the matrix NK is positive definite at theequilibrium point, where NK = Nαβamn + Kδαβδmn.Finally, we also obtain a convex primal dual variational formulation. In fact it was our longterm objective to obtain a convex variational formulation for such a class of non-convexproblems.

Chapter 12

Duality Applied to Phase TransitionProblems

12.1 Introduction

In this chapter, our first objectives are to show existence and develop dual formulationsconcerning the semi-linear Ginzburg-Landau equation. We start by describing the primalformulation.

By S ⊂ R3 we denote an open connected bounded set with a sufficiently regular boundaryΓ = ∂S (regular enough so that the Sobolev Imbedding Theorem holds). The Ginzburg-Landau equation is given by:

−∇2u + α(u2

2− β)u− f(x) = 0 a.e in S,

u = 0 on Γ.(12.1)

Here u : S → R denotes the primal field, f(x) ∈ L2(S), and α, β are real positive constants.

The corresponding variational formulation is given by the functional J : U → R where,

J(u) =1

2

∫

S

|∇u|2dS +α

2

∫

S

(u2

2− β)2dx−

∫

S

fudx (12.2)

and U = u ∈ W 1,2(S) | u = 0 on Γ = W 1,20 (S).

Equations indicated in (12.1) are necessary conditions for the solution of Problem P , where

Problem P : to determine u0 ∈ U such that J(u0) = minu∈U

J(u).

For this problem, our first result is a convex primal-dual formulation. It is not difficultto show that the solution of Ginzburg-Landau equations indicated in (12.1), minimizes thefunctional J : U × Y ∗ → R where

130

CHAPTER 12. DUALITY APPLIED TO PHASE TRANSITION PROBLEMS 131

J(u, v∗) =1

2

∫

S

|v∗1|2dx +3

4(α/2)1/3

∫

S

(v∗2)43 dx− 〈∇u, v∗1〉L2(S;R3)

−〈u, v∗2〉L2(S) +1

2

∫

S

|∇u|2dx +α

8

∫

S

u4 dx

under the constraint−div(v∗1) + v∗2 − αβu = f, a.e. in S.

The second duality principle presented gives us another convex dual variational formula-tion, through which optimality conditions for the primal problem may be obtained. Such aprinciple is expressed as

infu∈U

J(u) ≥ sup(z∗,v∗1 ,v∗0)∈B∗

− 1

2K2

∫

S

|∇z∗|2dx +1

2K

∫

S

(z∗)2dx

− 1

2

∫

S

(v∗1)2

v∗0 + Kdx− 1

2α

∫

S

(v∗0)2dx− β

∫

S

v∗0dx

, (12.3)

where

B∗ =(z∗, v∗1, v

∗0) ∈ L2(S;R3) | − 1

K∇2z∗ + v∗1 − z∗ = f,

v∗0 + K > 0, a.e. in S, z∗ = 0 on Γ . (12.4)

Sufficient conditions for optimality concerning the primal formulation are given by δJ(u0) = θand v∗0(u0) + K > 0 a.e. in S where K = 1/K0, here K0 is the constant related to Poincareinequality, and v∗0(u0) = α(u2

0/2− β).

As earlier mentioned, our second objective in this work is to provide, through the tools ofconvex analysis, duality principles which are valid even for the vectorial case in the calculus ofvariations. We obtain a simple result, namely Theorem 12.6.1, through which we establish aduality principle for a phase transition problem. Thus, considering the phases e1, e2, ..., eN ,here ek ∈ R3×3,∀k ∈ 1, ..., N, that a elastic solid may present, we denote the primalfunctional as J : U → R, where

J(u) =

∫

S

mink∈1,...,N

gk(∇u)dx +α

2〈ui, ui〉L2(S) − 〈u, f〉L2(S;R3),

∇u =

∂ui

∂xj

,

gk(∇u) =1

2

(∂ui

∂xj

− ekij

)Ck

ijlm

(∂ul

∂xm

− eklm

),

and,U = u ∈ W 1,2(S;R3) | u = (0, 0, 0) on Γ.


Here Ckijlm are positive definite matrices for each k ∈ 1, ..., N and f ∈ L2(S;R3) is a

external load. We have obtained a duality principle expressed as

infu∈U


infλ∈B

supσ∈C∗

−

∫

S

(v∗1ij + σ1ij)(Dijlm)(v∗1lm + σ1lm)dx

−∫

S

(σ1ij + v∗1ij)DijlmλkCkmlope

kopdx− 1

2α〈v∗2i + σ2i , v∗2i + σ2i〉L2(S)

+1

2

∫

S

(Dijop(σop + v∗op + ηop)− ekij)(λkC

kijlm)(Dlmrs(σrs + v∗rs + ηrs)− ek

lm)dx

,

whereηij = λkC

kijlmek

lm,Dijlm = λkC

kijlm−1,

C∗ = σ ∈ Y ∗ | σ1ij,j − σ2i = 0, a.e. in S, ∀i ∈ 1, 2, 3,A∗ = v∗ ∈ Y ∗ | v∗1ij,j − v∗2i + fi = 0, a.e. in S, ∀i ∈ 1, 2, 3,

and

B =

(λ1(x), ..., λN(x)), | λk(x) ≥ 0,∀k ∈ 1, ..., N and

N∑

k=1

λk(x) = 1, a.e. in S

.

Remark 12.1.1. The dual formulation is concave in v∗ and, as mentioned above the solutionof dual problem reflects the average behavior of minimizing sequences for the primal problem,when this latter problem has not solutions in the classical sense.

12.2 Existence of Solution for the Ginzburg-Landau

Equation

Remark 12.2.1. From the Sobolev Imbedding Theorem (Adams [1]) for

mp < n, n−mp < n, p ≤ q ≤ p∗ = np/(n−mp),

we haveW j+m,p(Ω) → W j,q(Ω).

Therefore, considering n = 3, m = 1, j = 0, p = 2, and q = 4, we obtain

W 1,2(Ω) ⊂ L4(Ω) ⊂ L2(Ω)

and thus‖u‖L4(Ω) → +∞⇒ ‖u‖W 1,2(Ω) → +∞.


Furthermore, from above and the Poincare Inequality it is clear that for J given by (12.2),we have

J(u) → +∞ as ‖u‖W 1,2(S) → +∞,

that is, J is coercive.

Now we establish the existence of a minimizer for J : U → R. It is a well known procedure(the direct method of calculus of variations). We present it here for the sake of completeness.

Theorem 12.2.2. For α, β ∈ R+, f ∈ L2(S) there exists at least one u0 ∈ U such that

J(u0) = minu∈U

J(u)

where

J(u) =1

2

∫

S

|∇u|2dx +α

2

∫

S

(u2

2− β)2dx−

∫

S

fudx

and U = u ∈ W 1,2(S)| u = 0 on Γ = W 1,20 (S).

Proof: From Remark 12.2.1 we have

J(u) → +∞ as ‖u‖U → +∞.

Thus as J is strongly continuous, there exists α1 ∈ R such that α1 = infu∈UJ(u), so that,for un minimizing sequence, in the sense that

J(un) → α1 as n → +∞ (12.5)

we have that ‖un‖U is bounded, and thus, as W 1,20 (S) is reflexive, there exists u0 ∈ W 1,2

0 (S)and a subsequence unj ⊂ un such that

unj u0, weakly in W 1,20 (S). (12.6)

From (12.6), by the Rellich-Kondrachov theorem, up to a subsequence, which is also denotedby unj, we have

unj → u0, strongly in L2(S). (12.7)

Furthermore, defining J1 : U → R as

J1(u) =1

2

∫

S

|∇u|2dx +α

8

∫

S

u4dx−∫

S

fudx

we have that J1 : U → R is convex and strongly continuous, therefore weakly lower semi-continuous, so that

lim infj→+∞

J1(unj) ≥ J1(u0). (12.8)


On the other hand, from (12.7)

∫

S

(unj)2dx →

∫

S

u20dx, as j → +∞ (12.9)

and thus, from (12.8) and (12.9) we may write

α1 = infu∈U

J(u) = lim infj→+∞

J(unj) ≥ J(u0). ¤

12.3 Convex Dual Formulations for the Ginzburg-Landau

Equation

We start this section by stating the following theorem which was proved in F.Botelho [3].

Theorem 12.3.1. Let U be a reflexive Banach space, (G Λ) : U → R a convex Gateauxdifferentiable functional and (F Λ1) : U → R convex, coercive and lower semi-continuous(l.s.c.) such that the functional

J(u) = (G Λ)(u)− F (Λ1u)− 〈u, u∗0〉Uis bounded from below , where Λ : U → Y and Λ1 : U → Y1 are continuous linear operators.

Then we may write

infz∗∈Y ∗1


F ∗(z∗)−G∗(v∗) ≥ infu∈U

J(u)

where B∗(z∗) = v∗ ∈ Y ∗ such that Λ∗v∗ − Λ∗1z∗ − u∗0 = 0.

We may apply the last theorem to the variational formulation of Ginzburg-Landau equation.

Just defining

(G Λ)(u) =1

2

∫

S

|∇u|2dx +α

8

∫

S

u4dx, (12.10)

(F Λ1)(u) =αβ

2

∫

S

u2dx, (12.11)

〈u, u∗0〉U = 〈u, f〉L2(S) (12.12)

and,Λu ≡ ∇u, u and Λ1u ≡ u.

Thus,

G∗(v∗) =1

2

∫

S

|v∗1|2dx +

∫

S

3

4(α/2)1/3(v∗2)

43 dx,


F ∗(z∗) =1

2αβ

∫

S

(z∗)2dx

andB∗(z∗) = v∗ ∈ Y ∗ | − div(v∗1) + v∗2 − z∗ = f, a.e. in S.

Finally,

infu∈U



1

2αβ

∫

S

(z∗)2dx− 1

2

∫

S

|v∗1|2dx− 3

4(α/2)1/3

∫

S

(v∗2)43 dx

.

We are now ready to establish the convex primal dual formulation. First, observe that

G∗(v∗) ≥ 〈Λu, v∗〉Y −G(Λu), ∀u ∈ U, v∗ ∈ Y ∗,

or

− 1

2αβ

∫

S

(z∗)2 + G∗(v∗) ≥ − 1

2αβ

∫

S

(z∗)2dx + 〈Λu, v∗〉Y −G(Λu)

subject to−div(v∗1) + v∗2 − z∗ = f, a.e. in S.

Therefore we can write

αβ

2

∫

S

u2dx− 〈u, z∗〉L2(S) +1

2αβ

∫

S

(z∗)2dx + G∗(v∗) ≥ 〈Λu, v∗〉Y −G(Λu). (12.13)

In particular for z∗ = αβu, from (12.13) we have

G∗(v∗)− 〈Λu, v∗〉Y + G(Λu) ≥ 0,

subject to

(v∗, u) ∈ C∗ ≡ (v∗, u) ∈ Y ∗ × U | − div(v∗1) + v∗2 − αβu = f, a.e. in S.

Thus, we can state the next theorem.

Theorem 12.3.2. The solution of Ginzburg-Landau equations indicated in (12.1), minimizesthe functional J : U × Y ∗ → R where

J(u, v∗) =1

2

∫

S

|v∗1|2dx +3

4(α/2)1/3

∫

S

(v∗2)43 dx− 〈∇u, v∗1〉L2(S;R3)

−〈u, v∗2〉L2(S) +1

2

∫

S

|∇u|2dx +α

8

∫

S

u4dx (12.14)

under the constraint−div(v∗1) + v∗2 − αβu = f, a.e. in S. (12.15)


Proof: Just consider a solution u0 for the boundary value problem related to Ginzburg-Landau equation. Defining v∗0 = ∂G(Λu0)

∂v, we have that (u0, v

∗0) minimizes the functional

above indicated in (12.14) and satisfies (12.15). ¤

Remark 12.3.3. Observe that the primal-dual formulation presented in (12.14) is convex,but one shortcoming of such a variational approach is that any critical point of the originalproblem works as (global) minimizer. However, now we will indicate a procedure that leadsto optimization of our primal variational formulation through the dual one.

Our next result refers to a convex dual variational formulation, through which we obtainsufficient conditions for optimality.

Theorem 12.3.4. Consider J : U → R, where

J(u) =

∫

S

1

2|∇u|2dx +

∫

S

α

2(u2

2− β)2dx−

∫

S

fudx,

and U = W 1,20 (S). For K = 1/K0, where K0 stands for the constant related to Poincare

inequality, we have the following duality principle

infu∈U

J(u) ≥ sup(z∗,v∗1 ,v∗0)∈B∗

−G∗L(v∗, z∗)

where

G∗L(v∗, z∗) =

1

2K2

∫

S

|∇z∗|2dx− 1

2K

∫

S

(z∗)2dx+1

2

∫

S

(v∗1)2

v∗0 + Kdx+

1

2α

∫

S

(v∗0)2dx+β

∫

S

v∗0dx,

and

B∗ =(z∗, v∗1, v

∗0) ∈ L2(S;R3) | − 1

K∇2z∗ + v∗1 − z∗ = f,

v∗0 + K > 0, a.e. in S, z∗ = 0 on Γ .(12.16)

If in addition there exists u0 ∈ U such that δJ(u0) = θ and v∗0 + K = (α/2)u20 − β + K >

0, a.e. in S, then

J(u0) = minu∈U

J(u) = max(z∗,v∗1 ,v∗0)∈B∗

−G∗L(v∗, z∗) = −G∗

L(v∗, z∗),

wherev∗0 =

α

2u2

0 − β,

v∗1 = (v∗0 + K)u0

andz∗ = Ku0.


Proof : Observe that we may write

J(u) = G(Λu)− F (Λ1u)−∫

S

fudx,

where

G(Λu) =

∫

S

1

2|∇u|2dx +

∫

S

α

2(u2

2− β + 0)2dx +

K

2

∫

S

u2dx,

F (Λ1u) =K

2

∫

S

u2dx,

whereΛu = Λ0u, Λ1u, Λ2u,

andΛ0u = 0, Λ1u = u, Λ2u = ∇u.

From Theorem 12.3.1 (here this is an auxiliary theorem through which we obtain A∗, indi-cated below), we have

infu∈U

J(u) = infz∗∈Y ∗1

supv∗∈A∗

F ∗(z∗)−G∗(v∗).

Here

F ∗(z∗) =1

2K

∫

S

(z∗)2dx,

and

G∗(v∗) =1

2

∫

S

|v∗2|2dx +1

2

∫

S

(v∗1)2

v∗0 + Kdx +

1

2α

∫

S

(v∗0)2dx + β

∫

S

v∗0dx,

if v∗0 + K > 0, a.e. in S, and

A∗ = v∗ ∈ Y ∗ | Λ∗v∗ − Λ∗1z∗ − f = 0,

or

A∗ = (z∗, v∗) ∈ L2(S)× L2(S;R5) | − div(v∗2) + v∗1 − z∗ − f = 0, a.e. in S.Observe that

G∗(v∗) ≥ 〈Λu, v∗〉Y −G(Λu), ∀u ∈ U, v∗ ∈ Y ∗,

and thus

−F ∗(z∗) + G∗(v∗) ≥ −F ∗(z∗) + 〈Λ1u, z∗〉L2(S) + 〈u, f〉U −G(Λu). (12.17)

Hence, making z∗ an independent variable through A∗, from (12.17) we may write

supz∗∈L2(S)

−F ∗(z∗) + G∗(v∗2(v∗1, z

∗), v∗1, v∗0)

≥ supz∗∈L2(S)

−F ∗(z∗) + 〈Λ1u, z∗〉L2(S) +

∫

S

fu dx−G(Λu).(12.18)


Thus,

supz∗∈L2(S)

− 1

2K

∫

S

(z∗)2 dx +1

2

∫

S

(v∗2(z∗, v∗1))

2 dx +1

2

∫

S

(v∗1)2

v∗0 + Kdx

+1

2α

∫

S

(v∗0)2 dx + β

∫

S

v∗0 dx

≥ F (Λ1u) +

∫

S

fu dx−G(Λu),∀u ∈ U. (12.19)

Therefore if K ≤ 1/K0 (here K0 denotes the constant concerning Poincare Inequality), thesupremum in the left side of (12.19) is attained through the relations

v∗2 =∇z∗

Kand z∗ = 0 on Γ.

Hence the final format of our duality principle is given by

infu∈U

J(u) ≥ sup(z∗,v∗1 ,v∗0)∈B∗

− 1

2K2

∫

S

|∇z∗|2dx +1

2K

∫

S

(z∗)2dx

− 1

2

∫

S

(v∗1)2

v∗0 + Kdx− 1

2α

∫

S

(v∗0)2dx− β

∫

S

v∗0dx

, (12.20)

where

B∗ =(z∗, v∗1, v

∗0) ∈ L2(S;R3) | − 1

K∇2z∗ + v∗1 − z∗ = f,

v∗0 + K > 0, a.e. in S, z∗ = 0 on Γ . (12.21)

The remaining conclusions follow from an application of Theorem 7.1.27. ¤Remark 12.3.5. The relations

v∗2 =∇z∗

Kand z∗ = 0 on Γ,

are sufficient for attainability of supremum indicated in (12.19) but just partially necessary,however we assume them because the expression of dual problem is simplified without violat-ing inequality (12.20) (in fact the difference between the primal and dual functionals evenincreases under such relations).

In a similar fashion, we have also the following result.

Theorem 12.3.6. Considering the functionals (G Λ) : U → R and (F Λ1) : U → Rdefined in (12.10) and (12.11) respectively, we can write

infu∈U

G(Λu)− F (Λ1u)− 〈u, f〉L2(S) ≥ sup(v∗1 ,z∗)∈D∗

1

2αβ

∫

S

(z∗)2dx

− 1

2

∫

S

|v∗1|2dx− 3α/2

4(αβ)4

∫

S

(z∗)4dx

(12.22)


where

D∗ =

(v∗, z∗) ∈ L2(S;R3)× L2(S) | α

2(z∗

αβ)3 − (div(v∗1) + z∗ + f) = 0, a.e. in S

.

The equation in the definition of D∗ represents the attainment of supremum in z∗ for theleft side of (12.25), indicated below.

Proof: Again, we have

G∗(v∗) ≥ 〈Λu, v∗〉Y −G(Λu), ∀u ∈ U, v∗ ∈ Y ∗

or

− 1

2αβ

∫

S

(z∗)2dx +1

2

∫

S

|v∗1|2dx +3

4(α/2)1/3

∫

S

(v∗2)4/3dx

≥ − 1

2αβ

∫

S

(z∗)2dx + 〈Λu, v∗〉Y −G(Λu) (12.23)

subject to−div(v∗1) + v∗2 − z∗ = f, a.e. in S

orv∗2 = div(v∗1) + z∗ + f. (12.24)

Thus, replacing equation (12.24) into (12.23) we obtain

− 1

2αβ

∫

S

(z∗)2dx +1

2

∫

S

|v∗1|2dx +3

4(α/2)1/3

∫

S

(div(v∗1) + z∗ + f)4/3dx

≥ − 1

2αβ

∫

S

(z∗)2dx + 〈u, z∗〉L2(S) + 〈u, f〉L2(S) −G(Λu) (12.25)

and taking the supremum in z∗ in both sides of (12.25) we have

− 1

2αβ

∫

S

(z∗)2dx +1

2

∫

S

|v∗1|2dx +3α/2

4(αβ)4

∫

S

(z∗)4dx ≥ αβ

2

∫

S

(u)2dx + 〈u, f〉L2(S) −G(Λu)

(12.26)subject to

α

2(z∗

αβ)3 − (div(v∗1) + z∗ + f) = 0 (12.27)

where the relation (12.27), as above mentioned, represents the solution of supremum in z∗

concerning the left side of inequality (12.25). ¤The cubic equation (12.27) has three roots. Each one leads to a different local extremum ofprimal variational problem.

Numerical results seems to indicated that the global minimum is obtained through the onlyroot that is always real.


12.4 Applications to Phase Transition in Polymers

We consider now a variational problem closely related to the Ginzburg-Landau formulation.See [9] and other references therein for more information how this applies to phase transitionin polymers). For an open bounded S ⊂ R3 with a sufficient regular boundary denoted byΓ, let us define J : U × V → R, as

J(u, v) =ε

2

∫

S

|∇u|2dx +1

4ε

∫

S

(u2 − 1)2dx +γ

2

∫

S

|∇v|2dx,

under the constraints,1

|S|∫

S

udx = m, (12.28)

−∇2v = u−m, (12.29)

and ∫

S

vdx = 0. (12.30)

Here U = V = W 1,2(S), ε is a small constant and −1 < m < 1. We may rewrite the primalfunctional, now denoting it by J : U × V → R = R ∪ +∞ as

J(u, v) = G1(Λ(u, v))− F (u).

Here Λ : U × V → Y ≡ L2(S)× L4(S)× L2(S;R3)× L2(S;R3) is defined as

Λ(u, v) = Λ0u = 0, Λ1u = u, Λ2u = ∇u, Λ3v = ∇v,also

G1(Λ(u, v)) = G(Λ(u, v)) + Ind1(u, v) + Ind2(u, v) + Ind3(u, v).

Hence

G(Λ(u, v)) =ε

2

∫

S

|∇u|2dx +1

4ε

∫

S

(u2 − 1 + 0)2dx +K

2

∫

S

u2dx +γ

2

∫

S

|∇v|2dx,

Ind1(u, v) =

0, if 1

|S|∫

Sudx = m,

+∞, otherwise,

Ind2(u, v) =

0, if ∇2v + u−m = 0, a.e. in S,+∞, otherwise,

Ind3(u, v) =

0, if

∫S

vdx = 0,+∞, otherwise,

and,

F (u) =K

2

∫

S

u2dx.


Similarly to Theorem 12.3.1, through appropriate Lagrange multipliers for the constraints,we may write

inf(u,v)∈U×V

J(u, v) ≤ infz∗∈Y ∗1

sup(u∗,v∗,λ)∈A∗

F ∗(z∗)−G∗(u∗, v∗) + λ1m +

∫

S

λ2mdx

,

where

G∗(u∗, v∗) =1

2ε

∫

S

|u∗1|2dx +1

2

∫

S

(u∗2)2

2u∗0 + Kdx + ε

∫

S

(u∗0)2dx +

∫

S

u∗0dx +1

2γ

∫

S

|v∗|2dx,

if 2u∗0 + K > 0, a.e. in S, and

F (z∗) =1

2K

∫

S

(z∗)2dx.

Also, defining Y ∗ = L2(S)× Y ∗ × R× L2(S)× R, we have

A∗ = A∗1 ∩ A∗

2 ∩ A∗3,

A∗1 = (z∗, u∗, λ) ∈ Y ∗ | div(u∗1)−u∗2 + z∗− λ1

|S| −λ2 = 0 a.e. in S and u∗1.n = 0 on Γ,

A∗2 = (z∗, u∗, λ) ∈ Y ∗ | ∇2λ2 +λ3−div(v∗) = 0 a.e. in S, v∗.n+

∂λ2

∂n= λ2 = 0, on Γ,

andA∗

3 = (z∗, u∗, λ) ∈ Y ∗ | 2u∗0 + K > 0, a.e. in S.Similarly to the case of Ginzburg-Landau formulation we can obtain

inf(u,v)∈U×V

J(u, v) ≥

sup(z∗,u∗,v∗,λ)∈C∗

− ε

2K2

∫

S

|∇z∗|2dx +1

2K

∫

S

(z∗)2dx− 1

2

∫

S

(u∗2)2

2u∗0 + Kdx−

−ε

∫

S

(u∗0)2dx +

∫

S

u∗0dx− 1

2γ

∫

S

|v∗|2dx− λ1m−∫

S

λ2mdx,where

C∗ = C∗1 ∩ A∗

2 ∩ A∗3,

C∗1 = (z∗, u∗, λ) ∈ Y ∗ | ε∇2z∗

K− u∗2 + z∗− λ1

|S| − λ2 = 0 a.e. in S and∂z∗

∂n= 0 on Γ.

Remark 12.4.1. It is important to emphasize that by analogy to Section 12.3, we mayobtain sufficient conditions for optimality. That is, if there exists a critical point for thedual formulation for which 2u∗0 + K > 0 a.e. in S and K = ε/K0 (where K0 stands forthe constant related to Poincare inequality), then the corresponding primal point throughLegendre transform relations is also a global minimizer and the last inequality is in fact anequality, as has been pointed out in Theorem 12.3.4.


12.4.1 Another Two Phase Model in Polymers

The following problem has applications in two phase models in Polymers. To minimize thefunctional J : U × V → R (here consider S ⊂ R3 as above), where

J(u, v) = |Du|(S) +γ

2

∫

S

|∇v|2dx,

under the constraints ∫

S

udx = m,

−∇2v = u−m, (12.31)

whereU = BV (S, −1, 1), (12.32)

and V = W 1,2(S) (here BV denotes the space of functions with bounded variation in S and|Du|(S) denotes the total variation of u in S).

Redefining U as U = W 1,1(S) ∩ L2(S), we rewrite the primal formulation, through suitableLagrange multipliers, now denoting it by J(u, v, λ1, λ2, λ3) (J : U×V ×L2(S)×R×L2(S) →R), as

J(u, v, λ) =

∫

S

|∇u|dx +γ

2

∫

S

|∇v|2dx +

∫

S

λ1

2(u2 − 1)dx

+ λ2

(∫

S

udx−m

)+

∫

S

λ3(∇2v + u−m)dx, (12.33)

where λ = (λ1, λ2, λ3). We may write

J(u, v, λ) = G(Λ(u, v)) + F (u, v, λ),

where

G(Λ(u, v)) =

∫

S

|∇u|dx +γ

2

∫

S

|∇v|2dx,

here Λ : U × V → Y = L2(S;R3)× L2(S;R3) is defined as

Λ(u, v) = Λ1u = ∇u, Λ2v = ∇v

and

F (u, v, λ) = λ2

(∫

S

udx−m

)+

∫

S

λ3(∇2v + u−m)dx +

∫

S

λ1

2(u2 − 1)dx.


Defining

U =

(u, v) ∈ U × V |

∫

S

udx = m, ∇2v + u−m = 0, u2 = 1 a.e. in S

,

from the Lagrange Multiplier version of Theorem 7.2.5 we have that

inf(u,v)∈U

G(Λ(u, v)) + F (u, v, θ) = sup(v∗,λ)∈Y ∗×B∗

−G∗(v∗)− F ∗(−Λ∗v∗, λ),

where

G∗(v∗) = supv∈Y

〈v, v∗〉Y −G(v) =1

2γ

∫

S

|v∗2|2dx + Ind0(v∗1),

and

Ind0(v∗1) =

0, if |v∗1|2 ≤ 1, a.e. in S,+∞, otherwise.

We may define

Ind1(v∗1) =

0, if v∗1.n = 0, on Γ,+∞, otherwise,

and

Ind2(v∗2) =

0, if div(v∗2)−∇2λ3 = 0, a.e. in S, v∗2.n + ∂λ3

∂n= λ3 = 0 on Γ,

+∞, otherwise,

so that

F ∗(−Λ∗v∗, λ) =

∫

S

|div(v∗1)− λ2 − λ3|dx + Ind2(v∗2) + Ind1(v

∗1) + λ2m +

∫

S

λ3mdx.

Therefore, we can summarize the last results by the following duality principle,

inf(u,v)∈U

G(Λ(u, v)) + F (u, v, θ) = sup(v∗,λ)∈A∗∩B∗

− 1

2γ

∫

S

|v∗2|2dx

−∫

S

|div(v∗1)− λ2 − λ3|dx− λ2m−∫

S

λ3mdx

,

where

A∗ = v∗ ∈ Y ∗ = L2(S;R3)× L2(S;R3) | |v∗1(x)|2 ≤ 1, a.e. in S and v∗1.n = 0, on Γ.and

B∗ = (v∗2, λ3) ∈ L2(S;R3)× L2(S) | div(v∗2)−∇2λ3 = 0, a.e. in S,

v∗2.n +∂λ3

∂n= λ3 = 0 on Γ. (12.34)

Remark 12.4.2. It is worth noting that the last dual formulation represents a standardconvex non-smooth optimization problem. The non-smoothness is the responsible by a pos-sible micro-structure formation. Furthermore such a formulation seems to be amenable tonumerical computation (in a simpler way as compared to the primal approach). ¤


12.5 The Multi-Well Problem

This section is dedicated to analysis of the Multi-well problem via duality. We start withthe scalar case, for which the results may be obtained through the theory developed inEkeland and Temam [14] (These authors surely deserves most of the credit on the analysisfor the scalar case. In fact, what we do here is to connect parts I and III of mentionedbook). In Section 12.5.3 we present an example which is completely solved through the dualformulation.

12.5.1 The Primal Variational Formulation

Consider an open bounded set S ⊂ Rn with a regular boundary denoted by Γ. For i ∈1, ..., N, also consider the convex differentiable functions gi : Rn → R and (g ∇) non-convex defined by

g(∇u) = mini∈1,...,N

gi(∇u), (12.35)

such thatG(∇u)

‖u‖U

→ +∞ as ‖u‖U →∞, (12.36)

where

G(∇u) =

∫

S

mini∈1,...,N

gi(∇u)dx =

∫

S

g(∇u)dx

andU = u ∈ W 1,2(S)| u = u0 on Γ. (12.37)

As a preliminary result, we recall Theorem 7.3.6.

Theorem 12.5.1. Let f be a Caratheodory function from Ω×(R×Rn) into R which satisfies

a2(x) + c2|ξ|α ≤ f(x, s, ξ) ≤ a1(x) + b|s|α + c1|ξ|α

where a1, a2 ∈ L1(Ω), 1 < α < +∞, b ≥ 0 and c1 ≥ c2 > 0. Let u0 ∈ W 1,α(Ω). Under suchassumptions, defining U = u | u− u0 ∈ W 1,2

0 (Ω), we have

infu∈U

∫

Ω

f(x, u,∇u)dx

= min

u∈U

∫

Ω

f ∗∗(x, u;∇u)dx

The solutions of relaxed problem are weak cluster points in W 1,α(Ω) of the minimizing se-quences of primal problem.

Now we can state the following result.


Theorem 12.5.2. Let (G ∇) : U → R satisfies (12.35) and (12.36). Also assume thehypothesis of Theorem 12.5.1. Then,

infu∈U

G(∇u)− 〈u, f〉L2(S) = infu∈U

G∗∗(∇u)− 〈u, f〉L2(S),

and there exists u ∈ U such that

minu∈U

G∗∗(∇u)− 〈u, f〉L2(S) = G∗∗(∇u)− 〈u, f〉L2(S).

The proof follows directly from Theorem 12.5.1.

Our next proposition is very important to establish the duality principle. It is a well knowresult in convex analysis so we do not prove it.

Proposition 12.5.3. Consider g : Rn → R defined as

g(v) = mini∈1,...,N

gi(v)

where gi : Rn → R are not necessarily convex functions. Under such assumptions, we have

g∗(v∗) = maxi∈1,...,N

g∗i (v∗) (12.38)

Now we present the duality principle.

Theorem 12.5.4. For (G ∇) : U → R defined as above, that is,

G(∇u) =

∫

S

mini∈1,...,N

gi(∇u)dx,

where here gi : Rn → R is convex, for all i ∈ 1, ..., N and F : U → R, defined as

F (u) = 〈u, f〉L2(S),

we haveminu∈U

G∗∗(∇u)− F (u) = supv∗∈C∗

−G∗(v∗) + 〈u0, v∗.n〉L2(Γ).

Here

G∗(v∗) =

∫

S

maxi∈1,...,N

g∗i (v∗)dx

andC∗ = v∗ ∈ Y ∗| div(v∗) + f(x) = 0, a.e. in S.


Proof: We haveG∗(v∗) = G∗∗∗(v∗) = sup

v∈Y〈v, v∗〉Y −G∗∗(v)

that is,

G∗(v∗) ≥ 〈∇u, v∗〉Y −G∗∗(∇u)

= 〈u,−div(v∗)〉L2(S) + 〈u0, v∗.n〉L2(Γ) −G∗∗(∇u), ∀u ∈ U, v∗ ∈ Y ∗.(12.39)

Thus, for v∗ ∈ C∗ we can write:

G∗(v∗) ≥ 〈u0, v∗.n〉L2(Γ) + 〈u, f〉L2(S) −G∗∗(∇u), ∀u ∈ U,

orinfu∈U

G∗∗(∇u)− 〈u, f〉L2(S) ≥ supv∗∈C∗

−G∗(v∗) + 〈u0, v∗.n〉L2(Γ). ¤

The equality in the last line follows from hypothesis (12.36) and Theorem 7.2.5.

Observe that the dual formulation is convex but non-smooth. In the next lines we willsee, through the dual formulation, how the micro-structure is formed, particularly when theoriginal primal formulation has no minimizers in the classical sense.

12.5.2 A Scalar Multi-Well Formulation

To start this section, we present duality for the solution of a scalar multi-well problem (inFiroozye and Kohn [16] you may find a similar vectorial formulation). Consider the openbounded set S ⊂ R3 with a regular boundary Γ and the function (W ∇) defined as

W (∇u) = mini∈1,...,N

1

2|∇u− ai|2

whereU = u ∈ W 1,2(S) | u = u0 on Γ

ai are known matrices, for all i ∈ 1, ..., N. The energy of the system is modeled byJ : U → R, where

J(u) =

∫

S

W (∇u)dx− 〈u, f〉L2(S)

or

J(u) =1

2

∫

S

mini∈1,...,N

|∇u− ai|2

dx− 〈u, f〉L2(S).

From Theorem 12.5.4 we have

infu∈U

J(u) = supv∗∈C∗

−

∫

S

maxi∈1,...,N

1

2|v∗|2 + v∗T ai

dx + 〈u0, v

∗.n〉L2(Γ)


or

infu∈U

J(u) = supv∗∈C∗

−

∫

S

maxλ∈B

1

2|v∗|2 +

N∑i=1

λiv∗T ai

dx + 〈u0, v

∗.n〉L2(Γ)

where

B = (λ1(x), ..., λN(x)), | λk(x) ≥ 0,∀k ∈ 1, ..., N and

N∑

k=1

λk(x) = 1, in S,

andC∗ = v∗ ∈ Y ∗ | div(v∗) + f = 0, a.e. in S

The solution of the dual problem seems not to be difficult. However, it is important toemphasize that, in general, this kind of problem does not present minimizers in the classicalsense. The solution of the dual problem (which is well-posed and convex), reflects the averagebehavior of minimizing sequences, as weak cluster points of such sequences.

12.5.3 An Example - A Two-dimensional Two-Well Problem

Consider the same hypothesis as above for S ⊂ R2 and J : U → R defined as:

J(u) =

∫

S

ming1(∇u), g2(∇u)dx− 〈u, f〉L2(S),

whereU = u ∈ W 1,2(S) | u = u0 on Γ,

g1(∇u) =1

2

(∂u

∂x− 1

)2

+1

2

(∂u

∂y+ 1

)2

and

g2(∇u) =1

2

(∂u

∂x+ 1

)2

+1

2

(∂u

∂y− 1

)2

.

From Theorem 12.5.4 we can write

infu∈U


−

∫

S

maxg∗1(v∗), g∗2(v∗)dx + 〈u0, v∗.n〉L2(Γ)

whereA∗ = v∗ ∈ Y ∗ | div(v∗) + f = 0, a.e. in S,

g∗1(v∗) =

1

2(v∗1)

2 + v∗1 +1

2(v∗2)

2 − v∗2

and

g∗2(v∗) =

1

2(v∗1)

2 − v∗1 +1

2(v∗2)

2 + v∗2.


We solve the dual problem. The corresponding Euler-Lagrange equations are given by:

δ

−〈u, div(v∗) + f〉U +

∫

S

−g∗1(v∗) + λ(g∗1(v

∗)− g∗2(v∗))dx + 〈u0, v

∗.n〉L2(Γ)

= θ

or more explicitly

δ

〈∂u

∂x, v∗1〉Y + 〈∂u

∂y, v∗2〉Y −

(1

2

∫

S

(v∗1)2dx +

∫

S

v∗1dx +1

2

∫

S

(v∗2)2dx−

∫

S

v∗2dx

)

+

∫

S

2λ(v∗1 − v∗2)dx

= θ. (12.40)

Remark 12.5.5. Suppose that |f(x, y)| is almost everywhere small enough so that the op-timum for the dual formulation occurs at a point in which g∗1(v

∗) = g∗2(v∗) what justify the

Lagrange multiplier λ.

Thus, we obtain the following system

∂u

∂x− v∗1 − 1 + 2λ = 0,

∂u

∂y− v∗2 + 1− 2λ = 0

andv∗1 = v∗2

so that

λ =1

2+

1

4

(∂u

∂y− ∂u

∂x

),

v∗1 = v∗2 =1

2

(∂u

∂x+

∂u

∂y

)

and therefore, from div(v∗) + f = 0, the variable u must satisfy the equations

1

2

∂2u

∂x2+

∂2u

∂x∂y+

1

2

∂2u

∂y2+ f(x, y) = 0, a.e. in S, u = u0 on Γ. ¤

Finally, the duality principle could be summarized in the general case as

infu∈U


−

∫

S

maxλ∈[0,1]

(1− λ)g∗1(v∗) + λg2(v

∗)dx + 〈u0, v∗.n〉L2(Γ)

which is in fact a quadratic constrained optimization problem in λ amenable to numericalcomputation. Furthermore, the variable λ(x) gives the proportion of mixture between thephases.


12.6 Duality Suitable for Vectorial Variational Prob-

lems

We just recall the very simple result, namely Theorem earlier labeled as 7.4.7 and respectiveRemark 7.4.8

Theorem 12.6.1. Consider (G Λ) : U → R (not necessarily convex) such that J : U → Rdefined as

J(u) = G(Λu)− 〈u, f〉U ,∀u ∈ U,


infu∈U


−(G Λ)∗(Λ∗v∗)

whereA∗ = v∗ ∈ Y ∗ | Λ∗v∗ − f = 0.

12.6.1 The Multi-Well Formulation Applied to Phase Transitions

Consider an open bounded connected set S ⊂ R3 with a regular boundary Γ and the fieldof displacements u = (u1, u2, u3) of a solid that can present the phases e1, e2, ..., eN, hereek ∈ R3×3,∀k ∈ 1, ..., N. The elastic energy of the system is given by J : U → R where

J(u) =

∫

S

mink∈1,...,N

gk(∇u)dx− 〈u, f〉L2(S;R3),

∇u =

∂ui

∂xj

,

gk(∇u) =1

2

(∂ui

∂xj

− ekij

)Ck

ijlm

(∂ul

∂xm

− eklm

), (12.41)

and,U = u ∈ W 1,2(S;R3) | u = θ on Γ = W 1,2

0 (S;R3).

Here Ckijlm are positive definite matrices for each k ∈ 1, ..., N, which guarantees coer-

civity for the primal formulation and f ∈ L2(S;R3) is a external load. We apply Theorem12.6.1 and obtain

infu∈U


−(G Λ)∗(Λ∗v∗) (12.42)

whereΛu = ∇u


and(G Λ)∗(Λ∗v∗) = sup

u∈U〈∇u, v∗〉L2(S;R9) −G(∇u).

Here

G(Λu) =

∫

S

mink∈1,...,N

gk(∇u)dx

andA∗ = v∗ ∈ Y ∗ | v∗ij,j + fi = 0, a.e. in in S. (12.43)

From (12.41), we have

(G Λ)∗(Λ∗v∗) = supu∈U

〈∇u, v∗〉L2(S;R9) −

∫

S

mink∈1,...,N

gk(∇u)dx

(12.44)

or


〈∇u, v∗〉L2(S;R9) − inf

λ∈B

∫

S

N∑

k=1

λkgk(∇u)dx

(12.45)

where

B =

(λ1(x), ..., λN(x)) | λk(x) ≥ 0, ∀k ∈ 1, ..., N and

N∑

k=1

λk(x) = 1, a.e. in S

.

Hence from (12.45) we can write


supλ∈B

∫

S

∇u.v∗ −

N∑

k=1

λkgk(∇u)

dx

.

That is,

(G Λ)∗(Λ∗v∗) = supλ∈B

supu∈U

∫

S

∇u.v∗ −

N∑

k=1

λkgk(∇u)

dx

,

and thus from Theorem 7.2.5

(G Λ)∗(Λ∗v∗) = supλ∈B

infσ∈C∗

∫

S

(v∗ij + σij)(Dijlm)(v∗lm + σlm)dx

+

∫

S

(σij + v∗ij)DijlmλkCkmlope

kopdx

− 1

2

∫

S



lm)dx

,

whereηij = λkC

kijlmek

lmDijlm = λkC

kijlm−1,


andC∗ = σ ∈ Y ∗ | σij,j = 0, a.e. in S. (12.46)

So to summarize, the duality principle for the multi-well problem may be written as

infu∈U


infλ∈B

supσ∈C∗

−

∫

S

(v∗ij + σij)(Dijlm)(v∗lm + σlm)dx

−∫

S

(σij + v∗ij)DijlmλkCkmlope

kopdx

+1

2

∫

S



lm)dx

where A∗ is indicated in (12.43) and C∗ in (12.46). Even though we have not performed yetnumerical results the dual formulation seems to be simple to compute.

A numerical example including algorithm to compute the solution of dual formulation isplanned for future work.

12.6.2 A More Complex Phase Transition Problem

As in the last section, consider an open bounded connected set S ⊂ R3 with a regularboundary Γ and the field of displacements u = (u1, u2, u3) of a solid that can present thephases e1, e2, ..., eN, here ek ∈ R3×3,∀k ∈ 1, ..., N. Now we are concerned with theminimization of J : U → R where for α > 0,

J(u) =

∫

S

mink∈1,...,N

gk(∇u)dx +α

2〈ui, ui〉L2(S) − 〈u, f〉L2(S;R3),

∇u = ∂ui

∂xj

,

gk(∇u) =1

2

(∂ui

∂xj

− ekij

)Ck

ijlm

(∂ul

∂xm

− eklm

),

and,U = u ∈ W 1,2(S;R3) | u = θ on Γ.

As above Ckijlm are positive definite matrices for each k ∈ 1, ..., N and f ∈ L2(S;R3) is

a external load.

Similarly to last section, considering again Theorem 12.6.1, we may obtain the following


duality principle

infu∈U


infλ∈B

supσ∈C∗

−

∫

S

(v∗1ij + σ1ij)(Dijlm)(v∗1lm + σ1lm)dx

−∫

S

(σ1ij + v∗1ij)DijlmλkCklmope

kopdx− 1

2α〈v∗2i + σ2i , v∗2i + σ2i〉L2(S)

+1

2

∫

S



lm)dx

,

whereηij = DijlmλkC

klmope

kop,

Dijlm = λkCkijlm−1,

C∗ = σ ∈ Y ∗ | σ1ij,j − σ2i = 0, a.e. in S, ∀i ∈ 1, 2, 3,and

A∗ = v∗ ∈ Y ∗ | v∗1ij,j − v∗2i + fi = 0, a.e. in S, ∀i ∈ 1, 2, 3.Remark 12.6.2. It is worth noting that dual formulation is concave in v∗ and the dualproblem has a solution, which is related to the average behavior of minimizing sequenceseven as the primal formulation has no solution in the classical sense.

12.7 Another Multi-Well Problem

In this section we consider duality for another class of multi-well problems similar as thosefound in [7]. The format of our problem is more general, not restricted to two-well formula-tions.

Now we describe the primal formulation. For an open bounded set S ⊂ R3 with a sufficientlyregular boundary denoted by ∂S, consider the functional J : U → R where

J(u) =

∫

S

mink∈1,...,N

gk(ε(u)) + βkdx− 〈u, f〉L2(S;R3),

εij(u) =1

2(ui,j + uj,i),

gk(ε(u)) =1

2(εij(u)− ek

ij)Ckijlm(εlm(u)− ek

lm).

As above Ckijlm are positive definite matrices for each k ∈ 1, ..., N and f ∈ L2(S;R3) is

a external load. Here again ek ∈ R3×3, for k ∈ 1, ..., N represent the phases presented


by a solid with field of displacements (u1, u2, u3) ∈ W 1,2(S;R3) due to a external loadf ∈ L2(S;R3). Also

U = u ∈ W 1,2(S;R3) | u = (0, 0, 0) ≡ θ on ∂S ≡ W 1,20 (S;R3).

We may writeJ(u) = G(u)− F (ε(u))

where

G(u) = J(u) +K

2

∫

S

(εij(u))Hijlm(εlm(u))dx,

F (ε(u)) =K

2

∫

S

(εij(u))Hijlm(εlm(u))dx,

and Hijlm is a positive definite matrix. Observe that ε : U → Y is given by

ε(u) = εij(u),

so that from Toland [35], we have

infu∈U

G∗∗(u)− F (ε(u)) = infz∗∈Y ∗

F ∗(z∗)−G∗(ε∗(z∗)) (12.47)

where

F ∗(z∗) = supz∈Y

〈zij, z∗ij〉L2(S) − F (z) =

1

2K

∫

S

z∗ijHijlmz∗lmdx,

andHijlm = Hijlm−1.

Also,G∗(ε∗(z∗)) = sup

u∈U〈εij(u), z∗ij〉L2(S) −G(u)

and thusG∗(ε∗(z∗)) = inf

v∗∈A∗G∗(z∗, v∗),

whereA∗ = v∗ ∈ Y ∗ | v∗ij,j + fi = 0, a.e. in S,

G(v, z∗) = −〈vij, z∗ij〉L2(S) +

1

2

∫

S

mink∈1,...,N

(vij − ekij)C

kijlm(vlm − ek

ij) + βkdx

+K

2

∫

S

vijHijlmvlmdx, (12.48)

andG∗(z∗, v∗) = sup

v∈Y〈vij, v

∗ij〉L2(S) − G(v, z).


Therefore

G∗(v, z∗) =

∫

S

maxk∈1,...,N

1

2(v∗ij + z∗ij)D

kijlm(v∗lm + z∗lm) + (v∗ij + z∗ij)C

kijlmek

lm − βk

dx,

whereDk

ijlm = Ckijlm + KHijlm−1,

andCk

ijlm = Ckijlm + KHijlm−1Ck

ijlm.Hence, the duality principle given by (12.47) may be expressed as

infu∈U

G∗∗(u)− F (ε(u)) = infz∗∈Y ∗

1

2K

∫

S

z∗ijHijlmz∗lmdx

+ supv∗∈A∗

∫

S

mink∈1,...,N

−1

2(v∗ij + z∗ij)D

kijlm(v∗lm + z∗lm)

− (v∗ij + z∗ij)Ckijlmek

lm + βk

dx

Interchanging the infimum and supremum in the right side of last equality, we obtain

infu∈U

G∗∗(u)− F (ε(u)) ≥ supv∗∈A∗

inf

z∗∈Y ∗

1

2K

∫

S

z∗ijHijlmz∗lmdx

+

∫

S

mink∈1,...,N

−1

2(v∗ij + z∗ij)D

kijlm(v∗lm + z∗lm)

− (v∗ij + z∗ij)Ckijlmek

lm + βk

dx

so that

infu∈U

G∗∗(u)− F (ε(u)) ≥ supv∗∈A∗

inft∈B

inf

z∗∈Y ∗

1

2K

∫

S

z∗ijHijlmz∗lmdx

−∫

S

tk2

(v∗ij + z∗ij)Dkijlm(v∗lm + z∗lm)dx

−∫

S

(v∗ij + z∗ij)tkCkijlmek

lmdx +

∫

S

tkβkdx

where

B = (t1, ..., tN) measurable | tk(x) ∈ [0, 1] ∀k ∈ 1, ..., N,N∑

k=1

tk(x) = 1, a.e. in S.

Observe that the infimum in z∗ is attained for functions satisfying

1

KHijlmz∗lm −

N∑

k=1

tkDkijlm(v∗lm + z∗lm) −

N∑

k=1

tkCkijlmek

lm = 0. (12.49)


The final format of concerned duality principle is given by

infu∈U

G∗∗(u)− F (ε(u)) ≥ supv∗∈A∗

inft∈B

1

2K

∫

S

z∗ij(v∗, t)Hijlmz∗lm(v∗, t)dx

−∫

S

tk2

(v∗ij + z∗ij(v∗, t))Dk

ijlm(v∗lm + z∗lm(v∗, t))dx

−∫

S

(v∗ij + z∗ij(v∗, t))tkCk

ijlmeklmdx +

∫

S

tkβkdx

where,A∗ = v∗ ∈ Y ∗ | v∗ij,j + fi = 0, a.e. in S,

B = (t1, ..., tN) measurable | tk(x) ∈ [0, 1] ∀k ∈ 1, ..., N,N∑

k=1

tk(x) = 1, a.e. in S.

Finally, z∗(v∗, t) is obtained through equation (12.49).

Remark 12.7.1. The final dual formulation is concave in v∗ (as the infimum of concavefunctionals) and, if K is big enough (we may choose H as the identity matrix), so that for aminimizing sequence un we have G∗∗(un) = G(un) (as n → +∞), the duality gap is zero(we have not proved it in the present work) and the last inequality is in fact an equality.

12.8 A Numerical Example

In this section we present numerical results for the one-dimensional example (originally dueto Bolza, see P.Pedregal [25]).

Consider J : U → R expressed as

J(u) =1

2

∫ 1

0

((u,x)2 − 1)2dx +

1

2

∫ 1

0

(u− f)2dx

or, defining S = [0, 1],

G(Λu) =1

2

∫ 1

0

((u,x)2 − 1)2dx

and

F (u) =1

2

∫ 1

0

(u− f)2dx

we may writeJ(u) = G(Λu) + F (u)

where, for convenience we define, Λ : U → Y ≡ L4(S)× L2(S) as

Λu = u,x, 0.


Furthermore, we have

U = u ∈ W 1,4(S) | u(0) = 0 and u(1) = 0.5

For Y = Y ∗ = L4(S)× L2(S), defining

G(Λu + p) =1

2

∫

S

((u,x + p1)2 − 1.0 + p0)

2dx

for v∗0 > 0 we obtain

G(Λu) + F (u) ≥ infp∈Y−〈p0, v

∗0〉L2(S) − 〈p1, v

∗1〉L2(S) + G(Λu + p) + F (u)

or

G(Λu)+F (u) ≥ infp∈Y−〈q0, v

∗0〉L2(S)−〈q1, v

∗1〉L2(S) +G(q)+ 〈0, v∗0〉L2(S) + 〈u′, v∗1〉L2(S) +F (u).

Here q = Λu + p so that

G(Λu) + F (u) ≥ −G∗(v∗) + 〈0, v∗0〉L2(S) + 〈u,x, v∗1〉L2(S) + F (u).

That is

G(Λu) + F (u) ≥ −G∗(v∗) + infu∈U

〈0, v∗0〉L2(S) + 〈u,x, v∗1〉L2(S) + F (u),

orinfu∈U

G(Λu) + F (u) ≥ supv∗∈A∗

−G∗(v∗)− F ∗(−Λ∗v∗)

where

G∗(v∗) =1

2

∫

S

(v∗1)2

v∗0dx +

1

2

∫

S

(v∗0)2dx,

if v∗0 > 0, a.e. in S. Also

F ∗(−Λ∗v∗) =1

2

∫

S

[(v∗1),x]2dx + 〈f, (v∗1),x〉L2(S) − v∗1(1)u(1)

andA∗ = v∗ ∈ Y ∗ | v∗0 > 0, a.e. in S.

Remark 12.8.1. Through the extremal condition v∗0 = ((u,x)2−1) and Weierstrass condition

(u,x)2 − 1.0 ≥ 0 we can see that the dual formulation is convex for v∗0 > 0, however it is

possible that the primal formulation has no minimizers, and we could expect a microstructureformation through v∗0 = 0 (that is, u,x = ±1, depending on f(x)). To allow v∗0 = 0 we willredefine the primal functional as below indicated.


Define G1 : U → R and F1 : U → R by

G1(u) = G(Λu) + F (u) +K

2

∫

S

(u,x)2dx

and

F1(u) =K

2

∫

S

(u,x)2dx.

Also defining G(Λu) = G(Λu) + K2

∫S(u,x)

2dx, from Theorem 12.3.1 we can write

infu∈U

J(u) ≤ infz∗∈Y ∗


F ∗1 (z∗)− G∗(v∗0, v

∗2)− F ∗(v∗1) (12.50)

where

F ∗1 (z∗) =

1

2K

∫

S

(z∗)2dx,

G∗(v∗0, v∗2) =

1

2

∫

S

(v∗2)2

v∗0 + Kdx +

1

2

∫

S

(v∗0)2dx,

F ∗(v∗1) =1

2

∫

S

(v∗1)2dx + 〈f, v∗1〉L2(S) − v∗2(1)u(1)

andB∗(z∗) = v∗ ∈ Y ∗ | − (v∗2),x + v∗1 − z∗ = 0 and v∗0 ≥ 0 a.e. in S.

We developed an algorithm based on the dual formulation indicated in (12.50). It is relevantto emphasize that such a dual formulation is convex for v∗0 ≥ 0 (this results follows from thetraditional Weierstrass condition, so that there is no duality gap between the primal anddual formulations and the inequality indicated in (12.50) is in fact an equality).

We present numerical results for f(x) = 0 (figure 12.1), f(x) = 0.3 ∗ Sin(π ∗ x) (figure 12.2)and f(x) = 0.3 ∗ Cos(π ∗ x) (figure 12.3). The solutions indicated as optima through thedual formulations (denoted by u0), are in fact weak cluster points of minimizing sequencesfor the primal formulations.

12.9 Conclusion

In this chapter we developed dual variational formulations for the Ginzburg-Landau equa-tions. Also we present a study about the Multi-Well problem, introducing duality as anefficient tool to tackle the problem. It is fascinating how the standard results of ConvexAnalysis can be used to clarify the understanding of mixture of the phases, as illustrated inSection 12.5.3.

In our view, the importance of duality for theoretical and numerical analysis of the Multi-Well and related phase transition problems seems to have been clarified.


0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 12.1: Vertical axis: u0(x)-weak limit of minimizing sequences for f(x)=0

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 12.2: Vertical axis: u0(x)-weak limit of minimizing sequences for f(x) = 0.3 ∗ Sin(π ∗ x)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 12.3: Vertical axis: u0(x)-weak limit of minimizing sequences for f(x) = 0.3 ∗ Cos(π ∗ x)

Chapter 13

Duality Applied to Conductivity inComposites

13.1 Introduction

For the primal formulation we repeat the statements found in reference [15] (U.Fidalgo,P.Pedregal). Consider a material confined into a bounded domain Ω ⊂ RN , N > 1. Themedium is obtained by mixing two constituents with different electric permitivity and con-ductivity. Let Q0 and Q1 denote the two N ×N symmetric matrices of electric permitivitycorresponding to each phase. For each phase, we also denote by Lj, j = 0, 1, the anisotropicN×N symmetric matrix of conductivity. Let 0 ≤ t1 ≤ 1 be the proportion of the constituent1 into the mixture. Constituent 1 occupies a space in the physical domain Ω which we de-note by E ⊂ Ω. Regarding the set E as our design variable, we introduce the characteristicfunction χ : Ω → 0, 1:

χ(x) =

1, if x ∈ E,0, otherwise,

(13.1)

Thus, ∫

E

dx =

∫

Ω

χ(x)dx = t1

∫

Ω

dx = t1|Ω|. (13.2)

The matrix of conductivity corresponding to the material as a whole is L = χL1 +(1−χ)L0.

Finally, the electrostatic potential, denoted by u : Ω → R is supposed to satisfy the equation

div[χL1∇u + (1− χ)L0∇u] = P (x), in Ω, (13.3)

with the boundary conditionsu = u0, on ∂Ω (13.4)

where P : Ω → R is a given source or sink of current (we assume P ∈ L2(Ω)).

159

CHAPTER 13. DUALITY APPLIED TO CONDUCTIVITY IN COMPOSITES 160

13.2 The Primal Formulation

Now and on we assume N = 3. Consider the problem of minimizing the cost functional,

I(χ, u) =

∫

Ω

χ

2(∇u)T Q1∇u +

(1− χ)

2(∇u)T Q0∇udx (13.5)

subject todiv[χL1∇u + (1− χ)L0∇u] = P (x) (13.6)

where u ∈ U, here U = u ∈ W 1,2(Ω) | u = u0 on ∂Ω.We will rewrite this problem as the minimization of J : U × Y → R, where Y = L2(S;R3),

J(u, f) = inft∈B

∫

Ω

( t

2(∇u)T Q1∇u +

(1− t)

2(∇u)T Q0∇u) + Ind1(u, f)dx + Ind2(u, f),

Ind1(∇u, f) =

0, if (tL1 + (1− t)L0)∇u− f = 0,+∞, otherwise,

and

Ind2(u, f) =

0, if div(f) = P a.e. in Ω,+∞, otherwise.

Here

B = t measurable | t(x) ∈ 0, 1, a.e. in Ω,

∫

Ω

t(x)dx = t1|Ω|.

13.3 The Duality Principle

Observe that we may write

J(u, f) = inft∈BG(Λu, f, t) + F (u, f),

where Λ : U → Y is given byΛu = ∇u,

G(Λu, f, t) =

∫

Ω

( t

2(∇u)T Q1∇u +

(1− t)

2(∇u)T Q0∇u) + Ind1(Λu, f)dx,

andF (u, f) = Ind2(u, f).

Also, we have that

inf(u,f)∈U×Y

J(u, f) = inft∈B

inf(u,f)∈U×Y

G(Λu, f, t) + F (u, f).


However, from Theorem 7.2.5 we obtain

inf(u,f)∈U×Y

G(u, f, t) + F (u, f) = sup(v∗,f∗)∈Y ∗×Y ∗

−G∗(v∗, f ∗, t)− F ∗(−Λ∗v∗,−f ∗).

Thus,G∗(v∗, f ∗, t) = sup

(v,f)∈Y×Y

〈v, v∗〉L2(Ω;R2) + 〈f, f ∗〉L2(Ω;R2) −G(v, f, t)

or

G∗(v∗, f ∗, t) = sup(v,f)∈Y×Y

〈v, v∗〉L2(Ω;R2) + 〈f, f ∗〉L2(Ω;R2)

−∫

Ω

( t

2(v)T Q1v +

(1− t)

2(v)T Q0v) + Ind1(v, f)dx (13.7)

so that

G∗(v∗, f ∗, t) = sup(v,f)∈Y×Y

〈v, v∗〉L2(Ω;R2) + 〈(tL1 + (1− t)L0)v, f ∗〉L2(Ω;R2)

−∫

Ω

(t12(v)T Q1v + (1− t)

1

2(v)T Q0v)dx (13.8)

orG∗(v∗, f ∗, t) =

=1

2

∫

Ω

(v∗ + (t(L1) + (1− t)L0)T f ∗)(tQ1 + (1− t)Q0)

−1(v∗ + (t(L1) + (1− t)L0)T f ∗)dx.

On the other hand

F ∗(−Λ∗v∗,−f ∗) = sup(u,f)∈U×Y

−〈∇u, v∗〉L2(Ω;R2) − 〈f, f ∗〉L2(Ω;R2) − F (u, f),

orF ∗(−Λ∗v∗,−f ∗) = sup

(u,f)∈U×Y

−〈∇u, v∗〉L2(Ω;R2) − 〈f, f ∗〉L2(Ω;R2) − Ind2(u, f).

That is

F ∗(−Λ∗v∗,−f ∗) = sup(u,f)∈U×Y

−〈∇u, v∗〉L2(Ω;R2) − 〈f, f ∗〉L2(Ω;R2) − 〈λ, div(f)− P 〉L2(Ω),

where λ is an appropriate Lagrange Multiplier. Hence we have

F ∗(−Λ∗v∗,−f ∗) =

−〈λ, P 〉L2(Ω) + 〈u0, v∗.n〉L2(∂Ω), if (v∗, f ∗) ∈ B∗,

+∞, otherwise,

where

B∗ = (v∗, f ∗) ∈ Y ∗ × Y ∗ | div(v∗) = 0, f ∗ = ∇λ, a.e. in Ω, λ = 0 on ∂Ω.


Therefore, we may summarize the last results by the following duality principle,

inf(u,f)∈U×Y

J(u, f) =

inft∈B sup

(v∗,λ)∈C∗−1

2

∫

Ω

(v∗ + (t(L1) + (1− t)L0)T∇λ)Q(t)(v∗ + (t(L1) + (1− t)L0)

T∇λ)dx+

〈λ, P 〉L2(Ω) − 〈u0, v∗.n〉L2(∂Ω),

whereQ(t) = (tQ1 + (1− t)Q0)

−1,

C∗ = (v∗, λ) ∈ Y ∗ × U | div(v∗) = 0, a.e. in Ω, λ = 0 on ∂Ω,and

B = t measurable | t(x) ∈ 0, 1, a.e. in Ω,

∫

Ω

t(x)dx = t1|Ω|. ¤

13.4 Conclusion

In this chapter we developed duality for a two-phase non-convex variational problem inconductivity. As we may not have minimizers for this kind of problem, a possible solutionof the dual variational formulation reflects the average behavior of minimizing sequences, asa weak cluster point of such (minimizing) sequences. Finally, it seems that the solution ofdual problem is not difficult to compute.

Chapter 14

Duality Applied to the OptimalDesign in Elasticity

14.1 Optimal Design of a Plate

The first objective of the present chapter is the establishment of a dual variational formu-lation for the optimal design, concerning the minimization of internal work, of a plate ofvariable thickness. Such a thickness is denoted by h(x) and allowed to assume the valuesbetween a minimum h0 and maximum h1. The total plate volume, assume fixed, is a designconstraint denoted by V .

Consider a plate which the middle surface is denoted by S ⊂ R2, where S is an open boundedconnected set with a sufficiently regular boundary denoted by Γ. The plate thickness isassumed to be the design variable and, as mentioned above, is denoted by h(x), wherex = (x1, x2) ∈ S ⊂ R2 and h0 ≤ h(x) ≤ h1. The field of normal displacements to S, due toa external load P ∈ L2(S), is denoted by w : S → R.

The optimization problem is the minimization of J : U → R, where

J(w) = inft∈C

∫

S

Hαβλµ(t)

2w,αβw,λµ

dx, (14.1)

subject to(Hαβλµ(t)w,λµ),αβ = P, in S (14.2)

and ∫

S

(th1 + (1− t)h0)dS = t1h1|S| = V , (14.3)

whereC = t measurable t(x) ∈ [0, 1], a.e. in S,

163

CHAPTER 14. DUALITY APPLIED TO THE OPTIMAL DESIGN IN ELASTICITY164

0 < t1 < 1 and |S| denotes the Lebesgue measure of S and

U = W 2,20 (S) = w ∈ W 2,2(S) | w =

∂w

∂n= 0 on Γ. (14.4)

Finally,Hαβλµ(t) = (th1 + (1− t)h0)

3Aαβλµ (14.5)

where h(x) = t(x)h1 + (1− t(x))h0 represents the plate thickness and Aαβλµ is a positivedefinite matrix related to Hooke’s Law. Observe that 0 ≤ t(x) ≤ 1, a.e. in S.

14.1.1 The First Duality Principle

Now we rewrite the primal formulation, so that we express J : U × Y → R, as

J(w, f) = inft∈B

∫

S

Hαβλµ(t)

2w,αβw,λµ + Ind1(Λw, f)

dx

+ Ind2(w, f), (14.6)

where,

Ind1(Λw, f) =

0, if fαβ = Hαβλµ(t)w,λµ,+∞, otherwise,

(14.7)

Ind2(w, f) =

0, if fαβ,αβ = P, a.e in S,+∞, otherwise,

(14.8)

Λ : U → Y is given byΛw = w,αβ,

and

B = t measurable | t(x) ∈ [0, 1] a.e. in S,

∫

S

(th1 + (1− t)h0)dS = t1h1|S| = V , (14.9)

and also Y = L2(S;R4). Observe that we may write

inf(w,f)∈U×Y

J(w, f) = inft∈B inf

(w,f)∈U×YG(Λw, f, t) + F (w, f), (14.10)

where

G(Λw, f, t) =

∫

S

Hαβλµ(t)

2w,αβw,λµ + Ind1(Λw, f)dx,

andF (w, f) = Ind2(w, f).

From Theorem 7.2.5, we may write

inf(w,f)∈U×Y

G(Λw, f, t)+F (w, f) = sup(v∗,f∗)∈Y ∗×Y ∗

−G∗(v∗, f ∗, t)−F ∗(−Λ∗v∗,−f ∗), (14.11)


whereG∗(v∗, f ∗, t) = sup

(v,f)∈Y×Y

〈vαβ, v∗αβ〉L2(S) + 〈fαβ, f ∗αβ〉L2(S) −G(v, f, t),

or

G∗(v∗, f ∗, t) = sup(v,f)∈Y×Y

〈vαβ, v∗αβ〉L2(S) + 〈fαβ, f ∗αβ〉L2(S)

−∫

S

Hαβλµ(t)

2vαβvλµ + Ind1(v, f)dx. (14.12)

Thus,

G∗(v∗, f ∗, t) = sup(v,f)∈Y×Y

〈vαβ, v∗αβ〉L2(S) + 〈Hαβλµ(t)vλµ, f

∗αβ〉L2(S) −

∫

S

Hαβλµ(t)

2vαβvλµdx

,

and we may write

G∗(v∗, f ∗, t) =1

2

∫

S

Hαβλµ(t)f ∗αβf ∗λµdS + 〈v∗αβ, f ∗αβ〉L2(S) +1

2

∫

S

Hαβλµ(t)v∗αβv∗λµdS.

whereHαβλµ(t) = Hαβλµ(t)−1.

On the other hand

F ∗(−Λ∗v∗,−f ∗) = sup(w,f)∈U×Y

−〈w,αβ, v∗αβ〉L2(S) − 〈fαβ, f ∗αβ〉L2(S) − F (w, f),

orF ∗(−Λ∗v∗,−f ∗) = sup

(w,f)∈U×Y

−〈w,αβ, v∗αβ〉L2(S) − 〈fαβ, f ∗αβ〉L2(S) − Ind2(w, f).

That is,

F ∗(−Λ∗v∗,−f ∗) = sup(w,f)∈U×Y

−〈w,αβ, v∗αβ〉L2(S) − 〈fαβ, f ∗αβ〉L2(S) + 〈w, fαβ,αβ − P 〉L2(S),

where w is an appropriate Lagrange Multiplier. Thus

F ∗(−Λ∗v∗,−f ∗) =

−〈w, P 〉L2(S), if (v∗, f ∗) ∈ B∗,+∞, otherwise,

(14.13)

whereB∗ = (v∗, f ∗) ∈ Y ∗ × Y ∗ | f ∗αβ = wαβ, v∗αβ,αβ = 0, a.e. in S.

Hence, the duality principle indicated in (14.11), may be expressed as

inf(w,f)∈U×Y

G(Λw, f, t) + F (w, f) =


sup(v∗,w)∈B∗

−1

2

∫

S

Hαβλµ(t)wαβwλµdS−〈v∗αβ, wαβ〉L2(S)− 1

2

∫

S

Hαβλµ(t)v∗αβv∗λµdS + 〈w, P 〉L2(S),

We may evaluate the last supremum and obtain v∗ = θ. Therefore,

inf(w,f)inU×Y

G(Λw, f, t) + F (w, f) = supw∈U

−1

2

∫

S

Hαβλµ(t)wαβwλµdS + 〈w, P 〉L2(S)

,

However, from Theorem 7.2.5, we may conclude that

supw∈U

−1

2

∫

S

Hαβλµ(t)wαβwλµdS + 〈w, P 〉L2(S)

= inf

Mαβ∈D∗

1

2

∫

S

Hαβλµ(t)MαβMλµdS

,

whereD∗ = Mαβ ∈ Y ∗ | Mαβ,αβ + P = 0, a.e. in S.

And thus, the final format of the duality principle would be

inf(w,f)∈U×Y

J(w, f) = inf(t,Mαβ)∈B×D∗

1

2

∫

S

Hαβλµ(t)MαβMλµdS

. ¤

14.2 Optimal Design in Three-Dimensional Elasticity

In this section we develop duality for a two phase problem in elasticity. Consider V ⊂ R3,and open connected bounded set with a sufficiently regular boundary denoted by ∂V . HereV stands for the volume of a elastic solid under the action of a load P ∈ L2(V,R3). Thefield of displacements is denoted by u = (u1, u2, u3) ∈ U where

U = u ∈ W 1,2(V ;R3) | u = (0, 0, 0) on ∂V = W 1,20 (V ;R3). (14.14)

The strain tensor, denoted by e = eij, is defined as

eij(u) =1

2(ui,j + uj,i). (14.15)

The solid V is composed by mixing two constituents, namely 1 and 0, with elasticity matricesrelated to Hooke’s Law denoted by H1

ijkl and H0ijkl, respectively. The part occupied by

constituent 1 is denoted by E and represented by the characteristic function χ : V → 0, 1where

χ(x) =

1, if x ∈ E,0, otherwise,

Now we define the optimization problem of minimizing J(u, χ) where

J(u, χ) =1

2

∫

V

(χH1ijkleijekl + (1− χ)H0

ijkleijekl)dV, (14.16)


subject to(χH1

ijklekl + (1− χ)H0ijklekl),j + Pi = 0, in V, (14.17)

u ∈ U and ∫

V

XdV ≤ t1|V |, (14.18)

where 0 < t1 < 1 and |V | denotes the Lebesgue measure of V .

We rewrite the primal formulation, now denoting it by J : U × Y → R as

J(u, f) = inft∈B

∫

V

Hijkl(t)

2eijekl + Ind1(eij(u), f)

dV + Ind2(u, f), (14.19)

whereHijkl(t) = tH1

ijkl + (1− t)H0ijkl,

Ind1(eij(u), f) =

0, if fij = tH1

ijklekl + (1− t)H0ijklekl,

+∞, otherwise,

Ind2(u, f) =

0, if fij,j + Pi = 0, a.e in S,+∞, otherwise,

B = t measurable | t(x) ∈ 0, 1, a.e. in V,

∫

V

t(x)dx ≤ t1|V |,

and also Y = L2(V ;R9).

By analogy to last section, we may obtain

inf(u,f)∈U×Y

J(u, f) = inf(t,σ)∈B×B∗

1

2

∫

V

Hijkl(t)σijσkldx

,

whereHijkl(t) = Hijkl(t)−1,

B∗ = σ ∈ Y ∗ | σij,j + Pi = 0, a.e. in V ,and

B = t measurable | t(x) ∈ 0, 1, a.e. in V,

∫

V

t(x)dx ≤ t1|V |.

14.3 A Numerical Example

Consider a plate which the middle surface is represented by S = [0, 1] × [0, 1], over whichis applied the distributed vertical external load P = 2000. The plate is supposed to be acomposite of materials 1 and 2, with stiffness coefficients E1 = 20000, E2 = 5000, respectively


05

1015

20

0

5

10

15

200

0.2

0.4

0.6

0.8

1

Figure 14.1: Vertical axis: solution t(x, y) for the dual problem with∫S tdS ≤ 0.68

(unities related to international system). We present numerical results for the problemof obtaining the optimal mixture between these two constituents , in order to minimizeJ : U → R, the internal work produced by the displacement field w : S → R, where

J(w) = inft∈B

1

2

∫

S

(tE1 + (1− t)E2)(∇2w)2dS

, (14.20)

subject to∇2((tE1 + (1− t)E2)∇2w) = P (14.21)

and ∫

S

tdS ≤ 0.68, (14.22)

whereU = w ∈ W 2,2(S) | w = 0 on ∂S. (14.23)

We compute the optimal composite through the dual problem, as above, given by,

inft∈B

supw∈U

−1

2

∫

S

(tE1 + (1− t)E2)(∇2w)2dS + 〈w, P 〉L2(S)

, (14.24)

where

B = t measurable | t(x) ∈ [0, 1] a.e. in S,

∫

S

tdS ≤ 0.68. (14.25)

See figure 14.1 for the results for t(x, y), which expresses the proportion of constituent ofstiffness E1. The field of displacements, denoted by w0 is indicated in figure 14.2.

For∫

StdS ≤ 0.60, for the same problem see figure 14.3.


05

1015

20

0

5

10

15

20−5

−4

−3

−2

−1

0

1

x 10−4

Figure 14.2: Vertical axis: Field of displacements w0(x, y) (in m) for the dual problem, with∫S tdS ≤ 0.68

05

1015

20

0

5

10

15

200

0.2

0.4

0.6

0.8

1

Figure 14.3: Vertical axis: solution t(x, y) for the dual problem with∫S tdS ≤ 0.60


14.4 Conclusion

In this chapter we developed dual variational formulations for the optimal design of thevariable thickness of a plate and for a two-phase problem in elasticity. The infima in t indi-cated in the dual formulations represent the structure search for stiffness in the optimizationprocess, which implies the minimization of the internal work. In some cases, the primalproblem may not have solutions, so that the solution of dual problem is a weak cluster pointof minimizing sequences for the primal formulation. Finally, about the numerical results,we may see a clear preference of material with greater stiffness to concentrate in the centralregion of the plate (see figures 14.1 and 14.3), where we have the greatest deformations andmoments (see figure 14.2).

Chapter 15

Duality Applied to Micro-Magnetism

15.1 Introduction

In this chapter we develop dual variational formulations for models in micro-magnetism. Forthe primal formulation we refer to P.Pedregal and B.Yan [26] for details.

Let Ω ⊂ R3 be an open bounded set with a finite Lebesgue measure and a regular boundarydenoted by ∂Ω. Consider the model of micro-magnetism in which the magnetization m :Ω → R3, is given by the minimization of the functional

J(m, f) =α

2

∫

Ω

|∇m|2dx +

∫

Ω

ϕ(m(x))dx−∫

Ω

H(x).mdx +1

2

∫

R3

|f(z)|2dz, (15.1)

m ∈ W 1,2(Ω;R3) ≡ Y1, |m(x)| = 1, a.e. in Ω (15.2)

and f ∈ L2(R3;R3) ≡ Y2 is the unique field determined by the simplified Maxwell’s equations

Curl(f) = 0, div(−f + mχΩ) = 0, a.e. in R3. (15.3)

Here H ∈ L2(Ω;R3) is a known external field and χΩ is a function defined by

χΩ(x) =

1, if x ∈ Ω,0, otherwise.

(15.4)

The termα

2

∫

Ω

|∇m|2dx

is called the exchange energy. Finally, ϕ(m) represents the anisotropic contribution and isgiven by a multi-well functional whose minima establish the prefered directions of magneti-zation.

171

CHAPTER 15. DUALITY APPLIED TO MICRO-MAGNETISM 172

15.2 The Primal formulations and the Duality Princi-

ples

15.2.1 Summary of Results for the Hard Uniaxial Case

We examine first the case of uniaxial material with no exchange energy. That is, α = 0 andϕ(x) = β(1− |m.e|).Observe that

ϕ(m) = minβ(1 + m.e), β(1−m.e)where β > 0 and e ∈ R3 is a unit vector. Thus we can express the the functional J : V →R = R ∪ +∞, (here V ≡ Y1 × Y2), as

J(m, f) = G(m, f) + F (m)

where

G(m, f) =

∫

Ω

ming1(m), g2(m)dx +1

2

∫

R3

|f(z)|2dz + Ind0(m) + Ind1(f) + Ind2(m, f),

and

F (m) = −∫

Ω

H(x)mdx.

Here,g1(m) = β(1 + m.e),

g2(m) = β(1−m.e),

Ind0(m) =

0, if |m(x)| = 1 a.e. in Ω,+∞, otherwise,

Ind1(m, f) =

0, if div(−f + mχΩ) = 0 a.e. in R3,+∞, otherwise,

and

Ind2(f) =

0, if Curl(f) = 0, a.e. in R3,+∞, otherwise.

The dual functional for such a variational formulation can be expressed by the followingduality principle:

inf(m,f)∈Y1×Y2

J(m, f) = sup(λ1,λ2)∈Y ∗

inft∈B

−

∫

Ω

(3∑

k=1

(∂λ2

∂xi

+ Hi + β(1− 2t)ei)2)1/2 dx

−1

2

∫

R3

|Curl∗λ1 +∇λ2|2 dx

+

∫

Ω

β dx (15.5)


whereB = t measurable | t(x) ∈ [0, 1], a.e. in Ω.

andY ∗ = (λ1, λ2) ∈ W 1,2(R3;R3)×W 1,2(R3) | λ2 = 0 on ∂Ω.

15.2.2 The Results for the Full Semi-linear Case

Now we present the duality principle for the full semi-linear case, that is, for α > 0. Firstwe define G : Y1 × Y2 → R and F : Y1 × Y2 → R as

G(m, f) =α

2

∫

Ω

|∇m|2dx +

∫

Ω


2

∫

R3

|f(z)|2dz

+Ind0(m) + Ind1(m) + Ind2(m), (15.6)

and

F (m, f) = −∫

Ω

H.mdx.

Also,g1(m) = β(1 + m.e),

g2(m) = β(1−m.e),

Ind0(m) =


Ind1(m, f) =


and

Ind2(f) =


For J(m, f) = G(m, f) + F (m, f), the dual variational formulation is given by the followingduality principle

inf(m,f)∈Y1×Y2

J(m, f) = sup(λ1,λ2,y∗)∈Y ∗×Y ∗0

inft∈B

− 1

2α

∫

Ω

|y∗|2dx

−∫

Ω

(3∑

i=1

(div(y∗i ) + Hi + (1− 2t)βei +∂λ2

∂xi

)2)1/2 dx

−1

2

∫

R3

|∇λ2 + Curl∗λ1|2dz

+

∫

Ω

βdx, (15.7)


whereB = t measurable | t(x) ∈ [0, 1], a.e. in Ω,

Y ∗0 = y∗ ∈ W 1,2(Ω;R3×3) | y∗i .n = 0 on ∂Ω, ∀i ∈ 1, 2, 3,

andY ∗ = (λ1, λ2) ∈ W 1,2(R3;R3)×W 1,2(R3) | λ2 = 0 on ∂Ω. ¤

Remark 15.2.1. It is important to emphasize that in both cases the dual formulations areconcave. Thus the dual problems always have solutions, even when for the hard uniaxialcase the minimizer in the primal problem is not attained. In this latter case the solution ofdual problem reflects the average behavior of minimizing sequences, as a weak limit of suchsequences.

15.3 A Preliminary Result

Now we recall a simple but very useful result, through which we establish our duality prin-ciples.

Theorem 15.3.1. Consider (G Λ) : V → R (not necessarily convex) such that J : V → Rdefined as

J(m) = G(Λm)− 〈m, f〉V , ∀m ∈ V,


infm∈V

J(m) = supy∗∈A∗

−(G Λ)∗(Λ∗y∗)

whereA∗ = y∗ ∈ Y ∗ | Λ∗y∗ − f = 0.

Remark 15.3.2. What seems to be relevant is that, when computing (G Λ)∗(Λ∗y∗), weobtain a duality which is perfect concerning the convex envelop of the primal formulation.

15.4 The Duality Principle for the Hard Case

We recall the primal formulation for the hard uniaxial case, expressed by J(m, f) where

J(m, f) = G(m, f) + F (m),

G(m, f) =

∫

Ω


2

∫

R3

|f(z)|2dz + Ind0(m) + Ind1(f) + Ind2(m, f),


and

F (m, f) = −∫

Ω

H(x).mdx.

Also,g1(m) = β(1 + m.e),

g2(m) = β(1−m.e),

Ind0(m) =


Ind1(m, f) =


and

Ind2(f) =


From Theorem 15.3.1, we may write

inf(m,f)∈Y1×Y2

J(m, f) = sup(m∗,f∗)∈Y ∗1 ×Y ∗2

−G∗(m∗, f ∗)− F ∗(−m∗,−f ∗)). (15.8)

We now calculate the dual functionals. First we have that

G∗(m∗, f ∗) = sup(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) −G(m, f),

or

G∗(m∗, f ∗) = sup(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) −

∫

Ω

ming1(m), g2(m)dx

−∫

R3

|f(z)|2dz − Ind0(m)− Ind1(f)− Ind2(m, f)

. (15.9)

That is,

G∗(m∗, f ∗) = sup(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3)

− inft∈B

∫

Ω

(tg1(m) + (1− t)tg2(m))dx−∫

R3

|f(z)|2dz

−Ind0(m)− Ind1(f)− Ind2(m, f), (15.10)



Thus,

G∗(m∗, f ∗) = sup(m,f,t)∈Y1×Y2×B

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3)

−∫

Ω

(tg1(m) + (1− t)g2(m))dx−∫

R3

|f(z)|2dz

−Ind0(m)− Ind1(f)− Ind2(m, f), (15.11)

or,

G∗(m∗, f ∗) = supt∈B sup

(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3)

−∫

Ω

(tg1(m) + (1− t)g2(m))dx−∫

R3

|f(z)|2dz

−Ind0(m)− Ind1(f)− Ind2(m, f). (15.12)

Hence


(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3)

−∫

Ω

(tg1(m) + (1− t)g2(m))dx−∫

R3

|f(z)|2dz

−∫

Ω

λ

2(

3∑i=1

m2i − 1)dx− 〈Curl(f), λ1〉L2(R3,R3)

−〈div(−f + mχΩ), λ2〉L2(R3),where λ, λ1 and λ2 are appropriate Lagrange Multipliers concerning the respective con-straints.

Therefore,


(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3)

−∫

Ω

β(t(1 + m.e) + (1− t)(1−m.e))dx−∫

R3

|f(z)|2dz

−∫

Ω

λ

2(

3∑i=1

m2i − 1)dx− 〈Curl(f), λ1〉L2(R3;R3)

−〈div(−f + mχΩ), λ2〉L2(R3).

The last indicated supremum is attained for functions satisfying the equations

m∗i + β(1− 2t)ei − λmi +

∂λ2

∂xi

= 0


or

mi =m∗

i + β(1− 2t)ei + ∂λ2

∂xi

λ= 0

and thus from the constraint3∑

i=1

m2i − 1 = 0

we obtain

λ = (3∑

i=1

(m∗i + β(1− 2t)ei +

∂λ2

∂xi

)2)1/2.

Also, the supremum in f is achieved for functions satisfying

f ∗ − f − Curl∗λ1 −∇λ2 = 0.

Observe that we need the condition λ2 = 0 on ∂Ω to have a finite supremum, so that

G∗(m∗, f ∗) = supt∈B

inf(λ1,λ2)∈Y

∫

Ω

(3∑

i=1

(m∗i + β(1− 2t)ei +

∂λ2

∂xi

)2)1/2dx

−1

2

∫

R3

|f ∗ − Curl∗λ1 −∇λ2|2dx

−

∫

Ω

βdx. (15.13)

if λ2 = 0 on ∂Ω, +∞ otherwise.

Furthermore

F ∗(−m∗,−f ∗) = sup(m,f)∈Y1×Y2

〈m,−m∗〉L2(Ω;R3) + 〈f,−f ∗〉L2(R3;R3) − F (m, f),

F ∗(−m∗,−f ∗) = sup(m,f)∈Y1×Y2

〈m,−m∗〉L2(Ω;R3) + 〈f,−f ∗〉L2(R3;R3) −∫

Ω

H(x)mdx,

so that

F ∗(−m∗,−f ∗) =

0, if (m∗, f ∗) ∈ B∗,+∞, otherwise,

(15.14)

where

B∗(m∗, f ∗) ∈ Y ∗1 × Y ∗

2 | m∗ = H a.e. in Ω, f ∗ = θ, a.e. in R3.Therefore we may summarize the duality principle indicated in (15.8) as

inf(m,f)∈Y1×Y2

J(m, f) = inft∈B

sup(λ1,λ2)∈Y

−

∫

Ω

(3∑

i=1

(Hi + β(1− 2t)ei +∂λ2

∂xi

)2)1/2dx

−1

2

∫

R3

|Curl∗λ1 +∇λ2|2 dx +

∫

Ω

βdx

, (15.15)




15.5 The Full Semi-linear Case

Now we present a study concerning duality for the full semi-linear case, that is, for α > 0.First we recall the definition of G : Y1 × Y2 → R and F : Y1 × Y2 → R, that is,

G(m, f) =α

2

∫

Ω

|∇m|2dx +

∫

Ω


2

∫

R3

|f(z)|2dz

+Ind0(m) + Ind1(m) + Ind2(m),

and

F (m, f) = −∫

Ω

H.mdx.

Also,g1(m) = β(1 + m.e),

g2(m) = β(1−m.e),

Ind0(m) =


Ind1(m, f) =


and

Ind2(f) =


From Theorem 15.3.1, we have

inf(m,f)∈Y1×Y2

G(m, f) + F (m, f) = sup(m∗,f∗)∈Y ∗1 ×Y ∗2

−G∗(m∗, f ∗)− F ∗(−m∗,−f ∗), (15.16)

whereG∗(m∗, f ∗) = sup

(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) −G(m, f),

and

F ∗(−m∗,−f ∗) = sup(m,f)∈Y1×Y2

〈m,−m∗〉L2(Ω;R3) + 〈f,−f ∗〉L2(R3;R3) − F (m, f).


Thus, from above definitions we may write

G∗(m∗, f ∗) = sup(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) − α

2

∫

Ω

|∇m|2dx

−∫

Ω

ming1(m), g2(m)dx− 1

2

∫

R3

|f(z)|2dz − Ind0(m)− Ind1(m)− Ind2(m)

.(15.17)

Or

G∗(m∗, f ∗) = sup(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) − α

2

∫

Ω

|∇m|2dx

− inft∈B

∫

Ω

(tg1(m) + (1− t)g2(m))dx− 1

2

∫

R3

|f(z)|2 dz

−Ind0(m)− Ind1(m)− Ind2(m), (15.18)


Hence

G∗(m∗, f ∗) = sup(m,f,t)∈Y1×Y2×B

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) − α

2

∫

Ω

|∇m|2dx

−∫

Ω

(tg1(m) + (1− t)g2(m))dx− 1

2

∫

R3

|f(z)|2 dz

−Ind0(m)− Ind1(m)− Ind2(m), (15.19)

or


sup(m,f)∈Y1×Y2

〈m, m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) − α

2

∫

Ω

|∇m|2dx

−∫

Ω

(tg1(m) + (1− t)g2(m))dx− 1

2

∫

R3

|f(z)|2 dz

−Ind0(m)− Ind1(m)− Ind2(m). (15.20)

Thus, as the second supremum is a convex optimization problem, there exist (λ, λ1, λ2) ∈L2(S)× L2(S;R3)× L2(S), such that


sup(m,f)∈Y1×Y2

〈m,m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(R3;R3) − α

2

∫

Ω

|∇m|2dx

−∫

Ω

(tg1(m) + (1− t)g2(m))dx− 1

2

∫

R3

|f(z)|2dz

−∫

Ω

λ

2(

3∑i=1

m2i − 1)dx− 〈Curl(f), λ1〉L2(R3;R3) − 〈div(−f + mχΩ), λ2〉L2(R3).


Thus we may write,


m∈Y1

−G(Λm)− F (m, t)+ F ∗1 (f ∗),

where

F ∗1 (f ∗) =

1

2

∫

R3

|f ∗ + Curl∗λ1 +∇λ2|2dx

G(Λm) =α

2

∫

Ω

|∇m|2dx,

Λm = ∇m,

and

F (m, t) = −〈m,m∗〉L2(Ω;R3) +

∫

Ω

tg1(m) + (1− t)g2(m)dx

+

∫

Ω

λ

2(

3∑i=1

m2i − 1)dx + 〈div(mχΩ), λ2〉L2(R3). (15.21)

Therefore, from Theorem 7.2.5,

sup(m,f)∈Y1×Y2

−G(Λm)− F (m, t) = infy∗∈Y ∗0

G∗(y∗)− F ∗(−Λ∗y∗, t),

where

G∗(y∗) = supy∈L2(Ω,R3×3)

〈y, y∗〉L2(Ω;R3×3) − α

2

∫

Ω

|y|2dx =1

2α

∫

Ω

|y∗|2dx

andF ∗(−Λ∗y∗, t) = sup

m∈Y1

−〈∇mi, y∗i 〉L2(Ω;R3) − F (m, t).

Thus we may write

F ∗(−Λ∗y∗, t) = supm∈Y1

−〈∇mi, y∗i 〉L2(Ω,R3) + 〈m,m∗〉L2(Ω;R3)

−∫

Ω

(t(1 + m.e) + (1− t)(1−m.e))βdx

−∫

Ω

λ

2(

3∑i=1

m2i − 1)dx− 〈div(mχΩ), λ2〉L2(R3). (15.22)

The last supremum is attained for functions satisfying y∗i .n + λ2ni = 0 on ∂Ω, for all i ∈1, 2, 3, where n denotes the outer normal to ∂Ω (such a condition is necessary to guaranteea finite supremum). Furthermore

div(y∗i ) + m∗i + (1− 2t)βei − λmi +

∂λ2

∂xi

= 0,


or

mi =div(y∗i ) + m∗

i + (1− 2t)βei + ∂λ2

∂xi

λ.

From the constraint∑3

i=1 m2i − 1 = 0, we obtain

λ =

(3∑

i=1

(div(y∗i ) + m∗

i + (1− 2t)βei +∂λ2

∂xi

)2)1/2

,

so that

F ∗(−Λ∗y∗, t) =

∫

Ω

(3∑

i=1

(div(y∗i ) + m∗

i + (1− 2t)βei +∂λ2

∂xi

)2)1/2

dx−∫

Ω

βdx.

Therefore, summarizing the last results, we may write


infy∗∈Y ∗0

1

2α

∫

Ω

|y∗|2dx +

∫

Ω

(3∑

i=1

(div(y∗i ) + m∗i + (1− 2t)βei +

∂λ2

∂xi

)2)1/2dx

+1

2

∫

R3

| − f ∗ + Curl∗λ1 +∇λ2|2dz −∫

Ω

βdx, (15.23)

where, Y ∗0 = y∗ ∈ W 1,2(Ω;R3×3) | y∗i .n + λ2ni = 0 on ∂Ω, ∀i ∈ 1, 2, 3 . On the other

hand

F ∗(−m∗,−f ∗) = sup(m,f)∈Y1×Y2

〈m,−m∗〉L2(Ω;R3) + 〈f,−f ∗〉L2(Ω;R3) +

∫

Ω

H.mdx

so that

F ∗(−m∗,−f ∗) =

0, if (m∗, f ∗) ∈ B∗,+∞, otherwise,

where

B∗ = (m∗, f ∗) ∈ Y ∗1 × Y ∗

2 | m∗ = H, a.e. in Ω, f ∗ = θ a.e. in R3.

Therefore, we could summarize the duality principle indicated in (15.16) as

inf(m,f)∈Y1×Y2

J(m, f) = inft∈B

sup

(λ1,λ2)∈Y ∗sup

y∗∈Y ∗0

− 1

2α

∫

Ω

|y∗|2dx

−∫

Ω

(3∑

i=1

(div(y∗i ) + Hi + (1− 2t)βei +∂λ2

∂xi

)2)1/2 dx

−1

2

∫

R3

|∇λ2 + Curl∗λ1|2 dz

+

∫

Ω

β dx, (15.24)



Y ∗0 = y∗ ∈ W 1,2(Ω;R3×3) | y∗i .n + λ2ni = 0 on ∂Ω,∀i ∈ 1, 2, 3,

andY ∗ = W 1,2(R3;R3)×W 1,2(R3). ¤

15.6 Final Results, Convex Dual Formulations

Consider again the functional given by J : Y1 × Y2 → R, where

J(m, f) =

∫

Ω

ming1(m), g2(m)dx + Ind0(m)−∫

Ω

H(x).mdx

+1

2

∫

Ω

|f(x)|2dx + Ind1(f) + Ind2(m, f). (15.25)

Considering the expression of Ind0(m) given in the last section, may write J(m, f) =G(m, f) + F (m, f) where

G(m, f) =

∫

Ω

ming1(m), g2(m)dx + Ind0(m)−∫

Ω

H(x).mdx +K

2〈mi,mi〉L2(Ω) − K

2

and

F (m, f) =1

2

∫

Ω

|f(x)|2dx +K

2〈mi,mi〉L2(Ω) − K

2+ Ind1(f) + Ind2(m, f) + Ind0(m).

It is known that

inf(m,f)∈Y1×Y2

J(m, f) ≥ supz∗∈Y ∗

−G∗(z∗)− F ∗(−z∗), (15.26)

whereG∗(z∗) = sup

(m,f)∈Y1×Y2

〈f, z∗1〉L2(R3;R3) + 〈m, z∗2〉L2(Ω;R3) −G(m, f),

andF ∗(−z∗) = sup

(m,f)∈Y1×Y2

−〈f, z∗1〉L2(R3;R3) − 〈m, z∗2〉L2(Ω;R3) − F (m, f).

Thus,

G∗(z∗) = sup(m,f)∈Y1×Y2

〈f, z∗1〉L2(R3;R3) + 〈m, z∗2〉L2(Ω;R3)

−∫

Ω

ming1(m), g2(m)dx− Ind0(m)

+

∫

Ω

H(x).mdx− K

2〈mi,mi〉L2(Ω) +

K

2, (15.27)


so that

G∗(z∗) =

supt∈B

∫Ω(∑3

i=1(z∗2i + Hi + β(1− 2t)ei)

2)1/2dx − β∫Ω

dx, if z∗ ∈ B∗,+∞, otherwise,

(15.28)where

B = t measurable | t(x) ∈ [0, 1], a.e. in Ω,and

B∗ = z∗ ∈ L2(R3;R3)× L2(R3) | z∗1 = θ, a.e. in R3.Also,

F ∗(−z∗) = sup(m,f)∈Y1×Y2

〈f,−z∗1〉L2(R3;R3) + 〈m,−z∗2〉L2(Ω;R3) − 1

2

∫

Ω

|f(x)|2dx

−K


K

2− Ind1(m)− Ind2(m, f)− Ind0(m). (15.29)

The calculation of F ∗(−z∗) is a standard quadratic optimization problem. Therefore the lastsupremum indicated is attained through appropriate Lagrange multipliers λ0, λ1, λ2, that is,

F ∗(−z∗) = sup(m,f)∈Y1×Y2

〈f,−z∗1〉L2(R3;R3) + 〈m,−z∗2〉L2(Ω;R3) − 1

2

∫

Ω

|f(x)|2dx

−K


K

2−

∫

R3

λ1.Curl(f)dx

−∫

Ω

λ2(div(−f + mχΩ))dx−∫

Ω

λ0

2(

3∑i=1

m2i − 1)dx. (15.30)

Evaluating such a supremum, we obtain

F ∗(−z∗) = inf(λ1,λ2)∈Y ∗

∫

Ω

(3∑

i=1

(z∗2i − ∂λ2

∂xi

)2)1/2dx +1

2

∫

R3

|z∗1 + Curl∗λ1 +∇λ2|2dx

whereY ∗ = (λ1, λ2) ∈ W 1,2(R3;R3)×W 1,2(R3) | λ2 = 0 on ∂Ω.

Observe that if K is big enough so that for a minimizing sequence (mn, fn) we have G(mn, fn) =G∗∗(mn, fn) for all sufficiently big n, then there is no duality gap between the primal anddual formulations. In this case we may replace G by G∗∗ and the duality is perfect.


We may summarize the last results by the following duality principle

inf(m,f)∈Y1×Y2

J(m, f) = sup(z∗2 ,λ1,λ2)∈L2(Ω)×Y ∗

inft∈B−

∫

Ω

(3∑

i=1

(z∗2i + Hi + β(1− 2t)ei)2)1/2dx

−∫

Ω

(3∑

i=1

(z∗2i − ∂λ2

∂xi

)2)1/2dx− 1

2

∫

R3

|Curl∗λ1 +∇λ2|2dx

+

∫

Ω

βdx, (15.31)


From (15.15) we have

inf(m,f)∈Y1×Y2

J(m, f) ≥ sup(λ1,λ2)∈Y ∗

inft∈B

−

∫

Ω

(3∑

i=1

(∂λ2

∂xi

+ Hi + β(1− 2t)ei)2)1/2 dx

−1

2

∫

R3


+

∫

Ω

β dx. (15.32)

Hence, from (15.31) and (15.32) we finally obtain

inf(m,f)∈Y1×Y2

J(m, f) = sup(λ1,λ2)∈Y ∗

inft∈B

−

∫

Ω

(3∑

i=1

(∂λ2

∂xi

+ Hi + β(1− 2t)ei)2)1/2 dx

−1

2

∫

R3


+

∫

Ω

β dx (15.33)

whereB = t measurable | t(x) ∈ [0, 1] a.e. in Ω.

andY ∗ = (λ1, λ2) ∈ W 1,2(R3;R3)×W 1,2(R3) | λ2 = 0 on ∂Ω.

Similar results may be obtained for the semi-linear case. The final format of the concernedduality principle is given by

inf(m,f)∈Y1×Y2

J(m, f) = sup(λ1,λ2,y∗)∈Y ∗×Y ∗0

inft∈B

− 1

2α

∫

Ω

|y∗|2 dx

−∫

Ω

(3∑

i=1

(div(y∗i ) + Hi + (1− 2t)βei +∂λ2

∂xi

)2)1/2 dx

−1

2

∫

R3

|∇λ2 + Curl∗λ1|2 dz

+

∫

Ω

β dx, (15.34)



Y ∗0 = y∗ ∈ W 1,2(Ω;R3×3) | y∗i .n = 0 on ∂Ω, ∀i ∈ 1, 2, 3,


15.7 The Cubic Case in Micro-magnetism

In this section we present the result for the cubic case. For the primal formulation we referto references [22, 26] for details.

Let Ω ⊂ R3 be an open bounded set with a finite Lebesgue measure and a regular boundarydenoted by ∂Ω and, consider the model of micro-magnetism in which the magnetizationm : Ω → R3, is given by the minimization of the functional J(m, f), where

J(m, f) =α

2

∫

Ω

|∇m|2dx +

∫

Ω

ϕ(m(x))dx−∫

Ω

H.mdx +1

2

∫

R3

|f(z)|2dz,

m ∈ W 1,2(Ω;R3) ≡ Y1, |m(x)| = 1, a.e. in Ω

and f ∈ L2(R3;R3) ≡ Y2 is the unique field determined by the simplified Maxwell’s equations

Curl(f) = 0, div(−f + mχΩ) = 0, in R3.

Here the function ϕ(m), for the cubic anisotropy is given by

ϕ(m) = K0 + K1

∑

i 6=j

m2i m

2j + K2m

21m

22m

23,

where K1, K2 > 0. Also H ∈ L2(Ω;R3) is a known external field and χΩ is a function definedas

χΩ(x) =

1, if x ∈ Ω,0, otherwise.

The termα

2

∫

Ω

|∇m|2dx

stands for the exchange energy. Finally, ϕ(m) represents the anisotropic contribution and isgiven by a multi-well functional whose minima establish the preferred directions of magne-tization.

Remark 15.7.1. It is worth noting that ϕ(m) has six points of minimum, namely r1 =(1, 0, 0), r2 = (0, 1, 0) and r3 = (0, 0, 1), r4 = (−1, 0, 0), r5 = (0,−1, 0) and r6 = (0, 0,−1)which define the preferred directions of magnetization.


15.7.1 The Primal Formulation

For α = 0, we define the primal formulation as J : Y1 × Y2 → R = R ∪+∞, where

J(m, f) = G(m, f) + F (m, f).

Here G : Y1 × Y2 → R = R ∪ +∞ is defined as

G(m, f) =

∫

Ω

ϕ(m)dx +K

2

∫

Ω

(3∑

k=1

m2k − 1)dx (15.35)

and we shall rewrite ϕ as the approximation

ϕ(m) = ming1(m), g2(m), g3(m), g4(m), g5(m), g6(m),

where

gk(m) = ϕ(rk) +3∑

i=1

∂ϕ(rk)

∂mi(mi − rki) +

1

2

3∑i=1

3∑j=1

∂2ϕ(rk)

∂mi∂mj

(mi − rki)(mj − rkj).

As above mentioned, r1 = (1, 0, 0), r2 = (0, 1, 0), r3 = (0, 0, 1), r4 = (−1, 0, 0), r5 = (0,−1, 0)and r6 = (0, 0,−1) are the points through which the possible microstructure is formed forthe cubic case, what justify such expansions. Also, we define F : V → R, as

F (m, f) =1

2

∫

R3

|f(z)|2dz −∫

Ω

H.mdx + Ind1(m) + Ind2(m, f) + Ind3(f), (15.36)

where

Ind1(m) =


Ind2(m, f) =


and

Ind3(f) =


15.7.2 The Duality Principles


inf(m,f)∈Y1×Y2

G∗∗(m, f) + F (m, f) = sup(m∗,f∗)∈Y ∗1 ×Y ∗2

−G∗(m∗, f ∗)− F ∗(−m∗,−f ∗), (15.37)


whereG∗(m∗, f ∗) = sup

(m,f)∈Y1×Y2

〈m, m∗〉L2(Ω;R3) + 〈f, f ∗〉L2(Ω;R3) −G(m, f),

so that, defining gk(m) = gk(m) + K2(∑3

k=1 m2k), we have

G∗(m, f ∗) =

∫Ω

maxk∈1,...,6g∗k(m)dx + K2|Ω|, if (m∗, f ∗) ∈ B∗,

+∞, otherwise,

whereB∗ = (m∗, f ∗) ∈ Y ∗

1 × Y ∗2 | f ∗ = (0, 0, 0) ≡ θ, a.e. in R3.

Also,F ∗(−m∗, f ∗) = sup

(m,f)∈Y1×Y2

−〈m,m∗〉L2(Ω;R3) − 〈f, f ∗〉L2(Ω;R3) − F (m, f),or

F ∗(−m∗,−f ∗) = sup(m,f)∈Y1×Y2

−〈m,m∗〉L2(Ω;R3) − 〈f, f ∗〉L2(Ω;R3) − 1

2

∫

R3

|f(z)|2dz

+

∫

Ω

H.mdx− Ind1(m)− Ind2(m, f)− Ind3(f), (15.38)

that is

F ∗(−m∗,−f ∗) = sup(m,f)∈Y1×Y2

−〈m,m∗〉L2(Ω;R3) − 〈f, f ∗〉L2(Ω;R3) − 1

2

∫

R3

|f(z)|2dz

+

∫

Ω

H.mdx−∫

Ω

λ

2(

3∑

k=1

m2k − 1)dx

−〈Curl(f), λ1〉L2(R3;R3) − 〈(div(−f + mχΩ), λ2〉L2(R3), (15.39)

so that

F ∗(−m∗,−f ∗) =

∫

Ω

(3∑

i=1

(∂λ2

∂xi

+ m∗i + Hi)

2)1/2dx− 1

2

∫

R3

|f ∗ + Curl∗λ1 +∇λ2|2dx.

Hence the duality principle indicated in (15.37) may be expressed as

inf(m,f)∈Y1×Y2

G∗∗(m, f) + F (m, f) = sup(m∗,λ1,λ2)∈Y ∗

−

∫

Ω

maxk∈1,...,6

g∗k(m∗) dx

−∫

Ω

(3∑

i=1

(∂λ2

∂xi

+ m∗i + Hi)

2)1/2 dx− 1

2

∫

R3

|Curl∗λ1 +∇λ2|2dz

− K

2|Ω|,

where

Y ∗ = (m∗, λ1, λ2) ∈ H−1(Ω;R3)×W 1,2(R3;R3)×W 1,2(R3)) | λ2 = 0 on ∂Ω. ¤


Remark 15.7.2. If K is big enough so that for a minimizing sequence (mn, fn) (of theoriginal primal approach) we have G∗∗(mn, fn) = G(mn, fn) for any n sufficiently big, thenwe may replace G∗∗ by G in the last duality principle, with no duality gap between the primaland dual formulations. Also, observe that the minimizing sequences for the primal problemdoes not depend on K.

By analogy, we may obtain the results for the semi-linear cubic case (for α > 0), namely

inf(m,f)∈Y1×Y2

G∗∗(m, f) + F (m, f) = sup(m∗,λ1,λ2,y∗)∈Y ∗×Y ∗0

−

∫

Ω

maxk∈1,...,6

g∗k(m∗)dx

− 1

2α

∫

Ω

|y∗|2dx−∫

Ω

(3∑

i=1

(div(y∗i ) +∂λ2

∂xi

+ m∗i + Hi)

2)1/2 dx

−1

2

∫

R3


− K

2|Ω|,

whereY ∗

0 = y∗ ∈ W 1,2(Ω;R3×3) | y∗i .n = 0 on ∂Ω,and

Y ∗ = (m∗, λ1, λ2) ∈ H−1(Ω;R3)×W 1,2(R3;R3)×W 1,2(R3)) | λ2 = 0 on ∂Ω. ¤

15.8 Conclusion

In this chapter we develop duality principles for models in ferromagnetism met in references[22, 26], for example . The last dual variational formulations here presented are convex(in fact concave) either for the hard and full (semi-linear) uniaxial cases or for the cubiccases. The results are obtained through standard tools of convex analysis. It is important toemphasize that in some situations (specially the hard cases), the minima may not be attainedthrough the primal approaches, so that the minimizers of the dual formulations reflect theaverage behavior of minimizing sequences for the primal problems, as weak cluster points ofsuch sequences.

Chapter 16

Duality Applied to Fluid Mechanics


In this chapter we develop dual variational formulations for the incompressible two-dimensionalsteady Navier-Stokes system. We establish as a primal formulation the sum of L2 norm ofeach of equations, and obtain the dual formulation through the Legendre Transform concept.Now we present the primal formulation.

Consider S ⊂ R2 an open, bounded and connected set, whose the internal boundary isdenoted by Γ0 and, the external boundary is denoted by Γ1. Denoting by u : S → R thefield of velocity in direction x of the Cartesian system (x, y), by v : S → R, the velocity fieldin the direction y, by p : S → R, the pressure field, so that P = p/ρ, where ρ is the constantfluid density and ν is the viscosity coefficient, the Navier-Stokes system is expressed by

ν∇2u− u∂xu− v∂yu− ∂xP = 0, a.e. in S, (16.1)

ν∇2v − u∂xv − v∂yv − ∂yP = 0, a.e. in S, (16.2)

∂xu + ∂yv = 0, a.e. in S, (16.3)

u = v = 0, on Γ0 (16.4)

andu = u∞, v = 0, P = P∞ on Γ1. (16.5)

The primal variational formulation, denoted by J : U → R, is expressed as:

J(u) =1

2(‖L1(u)‖2

L2(S) + ‖L2(u)‖2L2(S) + ‖L3(u)‖2

L2(S)) (16.6)

where u = (u, v, P ) ∈ U , and

U = u ∈ H2(S)×H2(S)×H1(S) | u = v = 0, Γ0, and, u = u∞, v = 0, P = P∞ on Γ1.(16.7)

189

CHAPTER 16. DUALITY APPLIED TO FLUID MECHANICS 190

AlsoL1(u) = ν∇2u− u∂xu− v∂yu− ∂xP, (16.8)

L2(u) = ν∇2v − u∂xv − v∂yv − ∂yP (16.9)

andL3(u) = ∂xu + ∂yv. (16.10)

Clearly we can write

J(u) =

∫

S

g(Λu)dS (16.11)

where Λu = Λiu, for i ∈ 1, ..., 14 or more explicitly

Λ1u = ν∇2u, Λ2u = u, Λ3u = −∂xu,

Λ4u = v, Λ5u = −∂yu, Λ6u = −∂xP,

Λ7u = ν∇2v, Λ8u = u, Λ9u = −∂xv,

Λ10u = v, Λ11u = −∂yv, Λ12u = −∂yP,

Λ13u = ∂xu, Λ14u = ∂yv. (16.12)

Hereg(y) = g1(y) + g2(y) + g3(y) (16.13)

where

g1(y) =1

2(y1 + y2y3 + y4y5 + y6)

2, (16.14)

g2(y) =1

2(y7 + y8y9 + y10y11 + y12)

2, (16.15)

g3(y) =1

2(y13 + y14)

2. (16.16)


Applying the definition of Legendre transform to g(y) = g1(y) + g2(y) + g3(y) we obtaing∗L(y∗) = g∗1L(y∗) + g∗2L(y∗) + g∗3L(y∗), where

g∗1L(y∗) =y∗2y

∗3

y∗1+

y∗4y∗5

y∗1+

(y∗1)2

2, (16.17)

g∗2L(y∗) =y∗8y

∗9

y∗7+

y∗10y∗11

y∗7+

(y∗7)2

2(16.18)

and

g∗3L =1

2(y∗13)

2. (16.19)


Remark 16.2.1. Observe that any solution system

δ(−GL(v∗) + 〈u, Λ∗v∗〉U) = θ (16.20)

yields a solution of the Navier-Stokes system (the Euler-Lagrange equations of such a systemis equivalent to the Navier-Stokes system).

Here, Λ∗v∗ = θ denotes:

ν∇2v∗1 + v∗2 + ∂xv∗3 + ∂yv

∗5 − ∂xv

∗13 = 0, a.e. in S, (16.21)

ν∇2v∗7 + v∗10 + ∂xv∗9 + ∂yv

∗11 − ∂yv

∗13 = 0, a.e. in S, (16.22)

and∂xv

∗1 + ∂yv

∗7 = 0, a.e. in S. (16.23)

and

G∗L(v∗) =

∫

S

g∗1L(v∗)dS +

∫

S

g∗2L(v∗)dS +

∫

S

g∗3L(v∗)dS (16.24)

or, more explicitly:

G∗L(v∗) =

∫

S

v∗2v∗3

v∗1dS +

∫

S

v∗4v∗5

v∗1dS +

1

2

∫

S

(v∗1)2dS

+

∫

S

v∗8v∗9

v∗7dS +

∫

S

v∗10v∗11

v∗7dS +

1

2

∫

S

(v∗7)2dS. (16.25)

16.3 The Dual Variational Formulation

Firstly we define (G Λ) : U → R as

G(Λu) =

∫

S

g1(Λu)dS +K

2

∫

S

u2dS +K

2

∫

S

(∂xu)2dSK

2

∫

S

v2dS +K

2

∫

S

(∂yu)2dS

+

∫

S

g2(Λu)dS +K

2

∫

S

u2dS +K

2

∫

S

(∂xv)2dSK

2

∫

S

v2dS

+K

2

∫

S

(∂yv)2dS +

∫

S

g3(Λu)dS, (16.26)

so that clearly we can write

G(Λu) =

∫

S

g1(Λu)dS +

∫

S

g2(Λu)dS +

∫

S

g3(Λu)dS (16.27)

where

g1(y) = g1(y) +K

2y2

2 +K

2y2

3 +K

2y2

4 +K

2y2

5, (16.28)


g2(y) = g2(y) +K

2y2

8 +K

2y2

9 +K

2y2

10 +K

2y2

11, (16.29)

g3(y) = g3(y) (16.30)

and hence, through the definition of Legendre transform we have

G∗L(v∗) =

∫

S

g∗1L(v∗)dS +

∫

S

g∗2L(v∗)dS +

∫

S

g∗3L(v∗)dS (16.31)

where

g∗1L(y∗) =1

2∆1

(K(y∗2)2−2y∗1y

∗2y∗3+K(y∗3)

2)+1

2∆1

(K(y∗4)2−2y∗1y

∗4y∗5+K(y∗5)

2)+(y∗1)

2

2, (16.32)

g∗2L(y∗) =1

2∆7

(K(y∗8)2 − 2y∗7y

∗8y∗9 + K(y∗9)

2) +1

2∆7

(K(y∗10)2 − 2y∗7y

∗10y

∗11 + K(y∗11)

2) +(y∗7)

2

2,

(16.33)where ∆1 = K2 − (y∗1)

2, ∆7 = K2 − (y∗7)2 and

g∗3L(y∗) =(y∗13)

2

2. (16.34)

The dual variational formulation is indicated in the next theorem.

Theorem 16.3.1. For J : U → R defined as

J(u) = G(Λu)− F (Λu), (16.35)

where G(Λu) is indicated in (16.26) and F (Λu) is defined as

F (Λu) =K

2

∫

S

u2dS +K

2

∫

S

(∂xu)2dSK

2

∫

S

v2dS +K

2

∫

S

(∂yu)2dS

+K

2

∫

S

u2dS +K

2

∫

S

(∂xv)2dSK

2

∫

S

v2dS +K

2

∫

S

(∂yv)2dS, (16.36)

whereΛu = u, ∂xu, ∂yu, v, v, ∂xv, ∂yv, u, (16.37)

we may writeinfu∈U

J(u) ≤ infu∗∈E∗

supv∗∈C∗(u∗)

F ∗(u∗)− G∗(v∗) (16.38)

where v∗ ∈ C∗(u∗) if and only if v∗ ∈ Y ∗ and

ν∇2v∗1 + v∗2 + ∂xv∗3 + ∂yv

∗5 − ∂xv

∗13 − u∗1 + ∂xu

∗2 − u∗4 + ∂yu

∗3 = 0, a.e. in S, (16.39)


ν∇2v∗7 + v∗10 + ∂xv∗9 + ∂yv

∗11 − ∂yv

∗13 − u∗8 + ∂xu

∗6 − u∗5 + ∂yu

∗7 = 0, a.e. in S, (16.40)

and∂xv

∗1 + ∂yv

∗7 = 0, a.e. in S. (16.41)

Also,

F ∗(u∗) =1

2K

∫

S

(u∗1)2dS +

1

2K

∫

S

(u∗2)2dS +

1

2K

∫

S

(u∗3)2dS +

1

2K

∫

S

(u∗4)2dS

+1

2K

∫

S

(u∗5)2dS +

1

2K

∫

S

(u∗6)2dS +

1

2K

∫

S

(u∗7)2dS +

1

2K

∫

S

(u∗8)2dS,(16.42)

andG∗(v∗) = G∗

L(v∗), if v∗ ∈ B∗, (16.43)

whereB∗ = v∗ ∈ Y ∗ | ∆1 ≥ 0 and ∆7 ≥ 0. (16.44)

Finally,inf

v∗∈C∗(u∗)G∗(v∗) ≡ G∗(u∗) (16.45)

andE∗ = u∗ ∈ U∗ | (G Λ)∗(Λ∗u∗) = G∗(u∗). (16.46)

Proof: We have that infu∈UJ(u) = 0 so that

−F (Λu) ≥ −G(Λu), ∀u ∈ U (16.47)

and hence we may write

F ∗(u∗) ≥ supu∈U

〈Λu, u∗〉 − F (Λu) ≥ supu∈U

〈Λu, u∗〉 − G(Λu) (16.48)

However, from the Theorem 7.2.5

(G Λ)∗(Λ∗u∗) = supu∈U

〈Λu, u∗〉 − G(Λu) ≤ infv∗∈C∗(u∗)

G∗(v∗) ≡ G∗(u∗) (16.49)

and thus, recalling that

E∗ = u∗ ∈ U∗ | (G Λ)∗(Λ∗u∗) = G∗(u∗) (16.50)

we haveF ∗(u∗) + sup

v∗∈C∗(u∗)G∗(v∗) ≥ 0 = inf

u∈UJ(u), (16.51)

∀u∗ ∈ E∗.

Finally, we have to show that G∗(v∗) = G∗L(v∗) on B∗. In fact, it is sufficient to show that:

g∗L(y∗) ≥ 〈y, y∗〉R14 − g(y),∀y ∈ R14, y∗ ∈ B∗, (16.52)


or

〈y, y∗〉R14 − 1

2(y1 + y2y3 + y4y5 + y6)

2 − K

2y2

2 −K

2y2

3 −K

2y2

4 −K

2y2

5

− 1

2(y7 + y8y9 + y10y11 + y12)

2 − K

2y2

8 −K

2y2

9 +K

2y2

10 +K

2y2

11

− 1

2(y13 + y14)

2

≤ g∗L(y∗). (16.53)

Through the transformations:

y1 = y1 + y2y3 + y4y5 + y6 (16.54)

andy7 = y7 + y8y9 + y10y11 + y12 (16.55)

this is equivalent to

y∗1 y1 − y∗1y2y3 − y∗1y4y5 − y∗1y6 + y∗7 y7 − y∗7y8y9 − y∗7y10y11 − y∗7y12

+ y∗2y2 + y∗3y3 + y∗4y4 + y∗5y5 + y∗6y6 + y∗8y8

+ y∗9y9 + y∗10y10 + y∗11y11 + y∗12y12 + y∗13y13 + y∗14y14

− 1

2y2

1 −1

2y2

7 −1

2(y13 + y14)

2 − K

2y2

2 −K

2y2

3 −K

2y2

4 −K

2y2

5

− K

2y2

8 −K

2y2

9 −K

2y2

10 −K

2y2

11 − g∗L(y∗)

≤ 0. (16.56)

On the other hand, if ∆1 ≥ 0 it is easy to see that

sup(y2,y3)∈R2

−y∗1y2y3 + y∗2y2 + y∗3y3− K

2y2

2 −K

2y2

3 =1

2∆1

(K(y∗2)2− 2y∗1y

∗2y∗3 +K(y∗3)

2) (16.57)

and also,

sup(y4,y5)∈R2

−y∗1y4y5 +y∗4y4 +y∗5y5−K

2y2

4−K

2y2

5 =1

2∆1

(K(y∗4)2−2y∗1y

∗4y∗5 +K(y∗5)

2). (16.58)


If ∆7 ≥ 0 we have

sup(y8,y9)∈R2

−y∗7y8y9 + y∗8y8 + y∗9y9− K

2y2

8 −K

2y2

9 =1

2∆7

(K(y∗8)2− 2y∗7y

∗8y∗9 +K(y∗9)

2) (16.59)

and

sup(y10,y11)∈R2

−y∗7y10y11 +y∗10y10 +y∗11y11−K

2y2

10−K

2y2

11 =1

2∆7

(K(y∗10)2−2y∗7y

∗10y

∗11 +K(y∗11)

2)

(16.60)so that considering that y∗1 = y∗6, y∗7 = y∗12, y∗13 = y∗14 we obtain

supy∈R6

y∗1 y1 − y∗1y6 + y∗7 y7 + y∗6y6 − y∗7y12 + y∗12y12 + y∗13y13 + y∗14y14

− 1

2(y1)

2 − 1

2(y7)

2 − 1

2(y13 + y14)

2

=1

2(y∗1)

2 +1

2(y∗7)

2 +1

2(y∗13)

2 (16.61)

where y = (y1, y7, y6, y12, y13, y14) so that considering the expression of g∗L(y∗) in (16.31), wecan conclude that (16.56) holds. ¤

16.4 Conclusion

In this chapter we obtain a dual variational formulations for the two-dimensional incom-pressible Navier-Stokes system via Legendre Transform. The extension of results to R3,compressible and time dependent cases is not difficult, but postponed for a future work.

Chapter 17

Duality Applied to a Beam Model

17.1 Introduction and Statement of Primal Formula-

tion

In this chapter we present an existence result and duality theory concerning the non-linearbeam model proposed by Gao in [19].

The boundary value form of Gao’s beam model is represented by the equation

EIw,xxxx − a(w,x)2w,xx + λw,xx = f, in [0, l] (17.1)

subject to the conditions:

w(0) = w(l) = w,x(0) = w,x(l) = 0, (17.2)

where w : [0, l] → R denotes the field of vertical displacements.

The corresponding primal variational formulation for such a model, is expressed by thefunctional J : U → R, where:

J(w) =

∫ l

0

1

2(EI(w,xx)

2 +a

6(w,x)

4 − λ(w,x)2)dx−

∫ l

0

fwdx (17.3)

Here E denotes the Young Modulus related to a specific material, I = bh3

12for a beam with

rectangular cross section (rectangle basis b and height h), a is a constant related to the crosssection area. Furthermore, l denotes the beam length (in fact the beam is represented bythe set [0, l] = x ∈ R | 0 ≤ x ≤ l, λ denotes an axial compressive load applied to x = land finally, f(x) denotes the distributed vertical load.

Also, we define:

U = w ∈ W 2,2([0, l]) ∩W 1,4([0, l]) | w(0) = w(l) = 0 = w,x(0) = w,x(l). (17.4)

196

CHAPTER 17. DUALITY APPLIED TO A BEAM MODEL 197

Remark 17.1.1. The boundary conditions refer to a clamped beam at x=0 and x=l.

We consider two different problems.

Problem P1 : To determine w0 ∈ U , such that J(w0) = infw∈U

J(w) (17.5)

Problem P2 : To determine w0 ∈ U+, such that J(w0) = infw∈U+

J(w) (17.6)

where U+ = w ∈ U , such that w(x) ≥ 0,∀x ∈ [0, l]Equation (17.1) stands for the necessary conditions for Problem P1. The Necessary condi-tions for Problem P2 may be similarly obtained, however we postpone their presentation forthe next sections.

Remark 17.1.2. From the Sobolev Imbedding Theorem (Adams [1], page 85) we have thefollowing result (case A for mp > n):

W j+m,p(Ω) → W j,q(Ω), (17.7)

for p ≤ q < ∞. For the present case we have m = n = 1, p = 2, j = 1 and q = 4, whichmeans:

W 2,2([0, l]) ⊂ W 1,4([0, l]) (17.8)

so thatW 2,2([0, l]) ∩W 1,4([0, l]) = W 2,2([0, l]). (17.9)

17.2 Existence and Regularity Results for Problem P1

In this section we show the existence of a minimizer for the unconstrained problem P1. Morespecifically, we establish the following result:

Theorem 17.2.1. Given b, h, a, l, E, λ ∈ R+ and f ∈ L2([0, l]) there exists at least onew0 ∈ U such that

J(w0) = infw∈U

J(w), (17.10)

whereU = w ∈ W 2,2([0, l]) | w(0) = w(l) = 0 = w,x(0) = w,x(l) (17.11)

and

J(w) =

∫ l

0

1

2(EI(w,xx)

2 +a

6(w,x)

4 − λ(w,x)2)dx−

∫ l

0

fwdx, ∀ w ∈ U . (17.12)


Proof : From Poincare Inequality it is clear that J is coercive, that is:

lim‖w‖U→+∞

J(w) = +∞ (17.13)

where‖w‖U = ‖w‖W 2,2([0,l]),∀w ∈ U . (17.14)

Therefore since J is strongly continuous, there exists α ∈ R such that

α = infw∈U

J(w). (17.15)

Thus, if wnn∈N is a minimizing sequence (in the sense that limn→+∞ J(wn) = α), then‖wn‖W 2,2

0 ([0,l]) and ‖wn‖W 1,4([0,l]) are bounded sequences in reflexive Banach spaces (see

Remark 17.1.2).

Hence, there exists w0 ∈ W 2,20 ([0, L]) and a subsequence wnj ⊂ wn such that

wnj → w0 as j → +∞, weakly in W 2,20 ([0, l]). (17.16)

From the Rellich Kondrachov theorem, up to a subsequence, which we also denote by wnjwe have

wnj,x → w0,x as j → +∞, strongly in L2([0, l]), (17.17)

Furthermore we have

J(w) = J1(w)− λ

2

∫ l

0

(w,x)2 dx (17.18)

where

J1(w) =

∫ l

0

1

2(EI(w,xx)

2 +a

6(w,x)

4) dx−∫ l

0

fw dx. (17.19)

As J1 is convex and continuous, it is also weakly lower semi-continuous, so that

lim infk→+∞

J1(wnjk) ≥ J1(w0) (17.20)

From this and equation (17.17), as wnjk is also a minimizing sequence, we can conclude

that:α = inf

w∈UJ(w) = lim inf

k→+∞J(wnjk

) ≥ J(w0) (17.21)

which impliesJ(w0) = α = inf

w∈UJ(w). ¤ (17.22)


Remark 17.2.2. We recall that from the Rellich-Kondrachov Theorem, Part III (see refer-ence [1], page 168), for mp > n , we have the following compact imbedding

W j+m,p(Ω) → Cj(Ω0). (17.23)

In our case consider n = m = 1, p = 2 and j = 1, that is, as w0 ∈ W 2,2((0, l)) (hereΩ = Ω0 = (0, 1)) we can conclude that w0 ∈ C1([0, l]), which means that w0 has continuousderivative in [0, l] (no corners). In fact such a regularity result refers to the space W 2,2([0, l])as a whole, not only to solution w0. To obtain deeper results concerning regularity, we wouldneed to evaluate the effect of necessary conditions on the solution w0.

17.3 A Convex Dual Formulation for the Beam Model

Now, similarly to above, consider the primal variational formulation expressed by J : U → R,where

J(w) =

∫ l

0

EI

2(w,xx)

2dx +

∫ l

0

α

2(w2

,x

2− β)2dx−

∫ l

0

fwdx, (17.24)

where U = W 2,20 ([0, l]) and α, β are positive real constants. We may also write

J(w) = G(Λw)− F (Λ1w), (17.25)

where

G(Λw) =

∫ l

0

EI

2(w,xx)

2dx +

∫ l

0

α

2(w2

,x

2− β)2dx +

K

2

∫ l

0

w2,xdx−

∫ l

0

fwdx, (17.26)

F (Λ1w) =K

2

∫ l

0

w2,xdx, (17.27)

whereΛw = Λ1w, Λ2w, w, (17.28)

andΛ1w = w,x, Λ2w = w,xx. (17.29)


infw∈U

J(w) = infz∗∈Y ∗

supv∗∈A∗

F ∗(z∗)−G∗(v∗), (17.30)

where

F ∗(z∗) =1

2K

∫ l

0

(z∗)2dx, (17.31)


and

G∗(v∗) =1

2EI

∫ l

0

(v∗2)2dx +

1

2

∫ l

0

(v∗1)2

v∗0 + Kdx +

1

2α

∫ l

0

(v∗0)2dx + β

∫ l

0

v∗0dx, (17.32)

andA∗ = v∗ ∈ Y ∗ | Λ∗v∗ − Λ∗1z

∗ − f = 0, (17.33)

orA∗ = (v∗, z∗) ∈ L2([0, l],R4) | v∗2,xx − v1,x + z∗,x = f, in [0, l]. (17.34)

Observe thatG∗(v∗) ≥ 〈Λw, v∗〉Y −G(Λw), ∀w ∈ U, v∗ ∈ Y ∗. (17.35)

Thus−F ∗(z∗) + G∗(v∗) ≥ −F ∗(z∗) + 〈Λ1w, z∗〉+ 〈w, f〉U −G(Λw). (17.36)

We can make z∗ an independent variable though A∗, that is, for v∗2(z, v∗1) given by

v∗2(z∗, v∗1) = (v∗2)

′(0)x + v∗2(0) +

∫ x

0

v∗1(t) dt−∫ x

0

z∗(t) dt +

∫ x

0

∫ t1

0

f dt dt1. (17.37)

From (17.36), we may write

supz∗∈L2([0,l])

−F ∗(z∗) + G∗(v∗2(v∗1, z

∗), v∗1, v∗0)

≥ supz∗∈L2([0,1])

−F ∗(z∗) + 〈Λ1w, z∗〉+ 〈w, f〉U −G(Λw), (17.38)

so that we may infer that

sup(v∗1 ,v∗0)∈L2([0,l];R2)

infz∗∈L2([0,1])

1

2K

∫ 1

0

(z∗)2dx − 1

2EI

∫ 1

0

(v∗2(z∗, v∗1))

2dx

−1

2

∫ 1

0

(v∗1)2

v∗0 + Kdx− 1

2α

∫ 1

0

(v∗0)2dx− β

∫ 1

0

v∗0dx

≤ infw∈U

J(w). (17.39)

Observe that the infimum for the dual formulation indicated in (17.39) is attained, forK < EI/K0 (here K0 denotes the constant concerning Poincare Inequality), through therelation

v∗2 =EIz,x

K, z∗(0) = z∗(l) = 0 (17.40)

so that the final format of our duality principle is given by


infu∈U

J(w) = sup(z∗,v∗1 ,v∗0)∈B∗

− EI

2K2

∫ 1

0

(z∗,x)2 dx +

1

2K

∫ 1

0

(z∗)2 dx

−1

2

∫ 1

0

(v∗1)2

v∗0 + Kdx− 1

2α

∫ 1

0

(v∗0)2dx− β

∫ 1

0

v∗0 dx

. (17.41)

Defining Y ∗0 = W 1,2

0 [0, l]× L2([0, l],R2) we have

B∗ = (z∗, v∗1, v∗0) ∈ Y ∗0 | EI

Kz∗,xxx−v∗1,x +z∗,x = f, and v∗0 +K > 0, a.e. in [0, l]. (17.42)

Remark 17.3.1. It is important to emphasize that the equality indicated in (17.41) holdsonly if there exists a critical point for the dual formulation such that v∗0 +K > 0 a.e. in [0, l]and K < EI/K0, where, as above mentioned, K0 is the constant concerning the Poincareinequality. In such a case, the dual formulation is convex.

17.4 A Necessary Condition for Problem P2

We recall Problem P2

To determine w0 ∈ U+, such that J(w0) = infw∈U+

J(w) (17.43)

whereU+ = w ∈ U , such that w(x) ≥ 0,∀x ∈ [0, l].

Given w ∈ U+, let us extend it to z ∈ U by:

w = |z|, (17.44)

also, define J1 : U → R, by:J1(z) = J(|z|) (17.45)

Hence we can establish Problem P3 (which is equivalent to problem P2):

To determine z0 ∈ U , such that J1(z0) = infz∈UJ(|z|). (17.46)

The Euler-Lagrange Equation for Problem P3 is expressed through the boundary valueproblem (which we call the Modular Necessary Condition):

EIz,xxxx − a(z,x)2z,xx + λz,xx − f

z

|z| = 0, ∀x ∈ (0, l) such that z(x) 6= 0, (17.47)

z(0) = z(l) = z,x(0) = z,x(l) = 0 (17.48)


Remark 17.4.1. Observe that in equation (17.47) the term f z|z| is not defined if z(x) =

0. We can solve this problem and obtain consistent numerical results, if we regularize thefunction |z| =

√z2 through the relation |z| ' √

z2 + ε, for ε > 0 such that O(ε) ' 0.

17.5 A Similar Two-dimensional Model

Finally, we analyze a two-dimensional model which is similar to the presented beam model.Consider S ⊂ R2 open, bounded, connected and with a regular boundary denoted by ∂S,and, J : U → R defined as:

J(u) =ε

2

∫

S

(∇2u)2dS +1

2

∫

S

(|∇u|2 − 1)2dS − 〈u, f〉 (17.49)

where U = u ∈ W 1,2(S) | u = u0 on ∂S.Of great interest in the literature is the system behavior as ε → 0. Anyway, the problem forwhich we obtain numerical results is a little different, and is defined by the functional, alsodenoted by J : U → R, as indicated below,

J(u) =1

2

∫

S

(|∇u|2 − 1)2dS +1

2

∫

S

u2dS − 〈u, f〉 (17.50)

Theorem 17.5.1. Consider J : U → R defined as above. Thus we can write

infu∈U

J(u) ≤

infz∗∈Y ∗

∫

S

1

2K|z∗|2dS + sup

v∗∈A∗

−1

2

∫

S

|v∗1|2v∗0 + K

dS − 1

2

∫

S

(v∗0)2dS − 1

2

∫

S

(v∗2)2dS

(17.51)

where A∗ = v∗ ∈ Y ∗ | div(z∗) + div(v∗1)− v∗2 + f = 0, a.e. in S.

For a proof, we use again Theorem 9.6.1. The idea is to redefine J : U →, as

J(u) = G(Λu)− F (Λ1u) (17.52)

where

G(Λu) =1

2

∫

S

(|∇u|2 − 1)2dS +1

2

∫

S

u2dS +K

2

∫

S

|∇u|2dS (17.53)

and

F (Λ1u) =K

2

∫

S

|∇u|2dS − 〈u, f〉L2(S). (17.54)

Perhaps, it is not easy to see that the dual formulation is convex for v∗0 ≥ 0, a.e. in S.


0

50

100

150

0

50

1000

0.005

0.01

0.015

0.02

0.025

0.03

Figure 17.1: Vertical axis: solution u0(x, y) for the dual problem

0

50

100

150

0

50

100−0.01

−0.005

0

0.005

0.01

Figure 17.2: Vertical axis: solution u0(x, y) for the dual problem

However, through the extremal conditions we have v∗0 = (|∇u|2 − 1) and it is clear that forv∗0 < 0 the Weierstrass necessary condition is not satisfied, that for such points we haveg5(x) 6= g∗∗5 (x), where g5(x) = 1

2(x2 − 1)2. Therefore, the inequality indicated in (17.51) is

in fact an equality, and there is no duality gap, if we proceed the maximization indicated in(17.51) restricted to v∗0 ≥ 0.

We developed numerical results for two examples. We define S = [0, 1] × [0, 1] and for theexample f(x, y) = 0.3 ∗ sin(πx), which the graph is indicated in figure 17.1.

For the second example, f(x, y) = 0.3 ∗ cos(πx), and the respective graph is indicated infigure 17.2. The computation was done first evaluating the supremum in v∗ in (17.51). Thenfor v∗ fixed, the value of z∗ is up-dated through the calculation of the infimum (in z∗). Thenwe calculate the supremum in v∗ again and go on until convergence is achieved.


17.6 Conclusion

In this chapter we present existence theory following the direct method of calculus of vari-ations and a duality principle for the non-convex variational formulation concerning Gao’sbeam model. Also, we introduce the concept of modular necessary condition. Finally, wepresent duality, in fact a convex dual variational formulation, for a bi-dimensional phasetransitional problem closed related to Gao’s beam model. The numerical results were ob-tained through the solution of the dual problem. It is important to emphasize that thesolution of the dual problem is not a minimizer for the primal one, but it is a cluster pointof a weakly convergent minimizing sequence.

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