variational analysis and...

V A R I A T I O N A L ANALYSIS AND A P P L I C A T I O N S

Nonconvex Optimization and Its Applications

V O L U M E 79

Managing Editor:

Panos Pardalos University of Florida, U.S.A.

Advisory Board."

J. R. Birge University of Michigan, U.S.A.

Ding-Zhu Du University of Minnesota, U.S.A.

C. A. Floudas Princeton University, U.S.A.

J. Mockus Lithuanian Academy of Sciences, Lithuania

H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A.

G. Stavroulakis Technical University Braunschweig, Germany

H. Tuy National Centre for Natural Science and Technology, Vietnam

VARIATIONAL ANALYSIS AND APPLICATIONS

Edited by

FRANCO GIANNESSI University of Pisa, Italy

ANTONINO MAUGERI University of Catania, Italy

~1 Springer

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Contents

Preface xi

PART 1

The Work of G. Stampacchia in Variational Inequalities J.-L. Lions

In Memory of Guido Stampacchia M.G. Garroni

31

The Collaboration between Guido Stampacchia and Jacques-Louis Lions On Variational Inequalities 33

E. Magenes

In Memory of Guido Stampacchia O. G. Mancino

39

Guido Stampacchia 47 S. Mazzone

Memories of Guido Stampacchia L. Nirenberg

79

In Memory of Guido Stampacchia C. Sbordone

81

vi

Guido Stampacchia, My Father G. Stampacchia

Variational Analysis and Applications

83

PART 2

Convergence and Stability of a Regularization Method for Maximal Monotone Inclusions and its Applications to Convex Optimization

Ya. L Alber, D. Butnariu and G. Kassay

Partitionable Mixed Variational Inequalities E. AllevL A. Gnudi, 1. V. Konnov and E.O. Mazurkevich

Irreducibility of the Transition Semigroup Associated with the Two Phase Stefan Problem

V. Barbu and G. Da Prato

On Some Boundary Value Problems for Flows with Shear Dependent Viscosity

H. Beir6o da Veiga

Homogenization of Systems of Partial Differential Equations A. Bensoussan

About the Duality Gap in Vector Optimization G. Bigi and M. Pappalardo

Separation of Convex Cones and Extremal Problems V.Boltyanski

Infinitely Many Solutions for the Dirichlet Problem via a Variational Principle of Ricceri

F. Cammaroto, A. Chinni, and B. Di Bella

A Density Result on the Space VMQ, A.O. Caruso and M.S. Fanciullo

Linear Complementarity since 1978 R. W. Cottle

Variational Inequalities in Vector Optimization G.P. CrespL 1. Ginchev and M. Rocca

89

133

147

161

173

195

205

215

231

239

259

Variational Analysis and Applications vii

Variational Inequalities for General Evolutionary Financial Equilibrium

P. Daniele 279

Variational Control Problems with Constraints via Exact Penalization 301 V.F. Demyanov, F. Giannessi and G.Sh. Tamasyan

Continuous Sets and Non-Attaining Functionals in Reflexive Banach Spaces

E. Ernst and M. Th~ra 343

Existence and Multiplicity Results for a Nonlinear Hammerstein Integral Equation 359

F. Faraci

Differentiability of Weak Solutions of Nonlinear Second Order Parabolic Systems with Quadratic Growth and Non Linearity q >_ 2 373

L. Fattorusso

An Optimization Problem with Equilibrium Constraint in Urban Transport

P. Ferrari 393

Sharp Estimates for Green's Functions: Singular Cases M.G. Garroni

409

First-Order Conditions for C °'~ Constrained Vector Optimization L Ginchev, A. Guerraggio and M. Rocca

427

Global Regularity for Solutions to Dirichlet Problem for Elliptic Systems with Nonlinearity q >_ 2 and with Natural Growth 451

S. Giuffr~ and G. Idone

Optimality Conditions for Generalized Complementarity Problems S. Giuffrb, G. 1done and A. Maugeri

465

Variational Inequalities for Time Dependent Financial Equilibrium with Price Constraints 477

S. Giuffrb andS. Pia

Remarks about Diffusion Mediated Transport: Thinking about Motion in Small Systems 497

S. Hastings and D. Kinderlehrer

viii Variational Analysis and Applications

Augmented Lagrangian and Nonlinear Semidefinite Programs X.X. Huang, X.Q. Yang and K.L. Teo

513

Optimality Alternative: a Non-Variational Approach to Necessary Conditions

A.D. Ioffe 531

A Variational Inequality Scheme for Determining an Economic Equilibrium of Classical or Estended Type

A. Jofre, R.T. Rockafellar and R.J.-B. Wets 553

On Time Dependent Vector Equilibrium Problems A. Khan and F. Raciti

579

On Some Nonstandard Dynamic Programming Problems of Control Theory

A.B. Kurzhanski and P. Varaiya 589

Properties of Gap Function for Vector Variational Inequality S.J. Li and G.Y. Chen

605

Zero Gravity Capillary Surfaces and Integral Estimates G.M. Lieberman

633

Asymptotically Critical Points and Multiple Solutions in the Elastic Bounce Problem

A. Marino and C. Saccon

651

A Branch-and-Cut to the Point-to-Point Connection Problem on Multicast Networks 665

C.N. Meneses, C.A.S. Oliveira and P.M. Pardalos

Variational Inequality and Evolutionary Market Disequilibria: The Case of Quantity Formulation 681

M. Milasi and C. Vitanza

Numerical Approximation of Free Boundary Problem by Variational Inequalities. Application to Semiconductor Devices

M. Morandi Cecchi and R. Russo 697

Sensitivity Analysis for Variational Systems B.S. Mordukhovich

723

Variational Analysis and Applications ix

Stable Critical Points for the Ginzburg Landau Functional on Some Plane Domains 745

M.K. Venkatesha Murthy

The Distance Function to the Boundary and Singular Set of Viscosity solutions of Hamilton-Jacobi Equation

L. Nirenberg 765

L p -Regularity for Poincar6 Problem and Applications D.K. Palagachev

773

Minimal Fractions of Compact Convex Sets D. Pallaschke and R. Urbahski

791

On Generalized Variational Inequalities B. Panicucei and M. Pappalardo

813

Bounded (Hausdorff) Convergence: Basic Facts and Applications J.-P. Penot and C. ZMinescu

827

Control Processes with Distributed Parameters in Unbounded Sets. Approximate Controllability with Variable Initial Locus

G. PulvirentL G. Santagati and A. Villani 855

Well Posedness and Optimization Problems L. Pusillo

889

Semismooth Newton Methods for Shape-Preserving Interpolation, Option Price and Semi-Infinite Programs 905

L. Qi

HOlder Regularity Results for Solutions of Parabolic Equations M.A. Ragusa

921

Survey on the Fenchel Problem of Level Sets T. Rapcsdk

935

Integral Functionals on Sobolev Spaces Having Multiple Local Minima

B. Ricceri 953

Aspects of the Projector on Prox-Regular Sets Stephen M. Robinson

963

x Variational Analysis and Applications

Application of Optimal Control Theory to Dynamic Soaring of Seabirds

G. Sachs and P. Bussotti 975

On The Convergence oftheMatricesAssociated to the A~ugate Jacobians

C. Sbordone 995

Quasi-Variational Inequalities Applied to Retarded Equilibria in Time- Dependent Traffic Problems 1007

L. Scrimali

Higher Order Approximation Equations for the Primitive Equations of the Ocean 1025

E. Simonnet, T. Tachim Medjo and R. Temam

Hahn-Banach Theorems and Maximal Monotonicity S. Simons

1049

Concrete Problems and the General Theory of Extremum V.M. Tikhomirov

1085

Numerical Solution for Pseudomonotone Variational Inequality Problems by Extragradient Methods 1101

F. Tinti

Regularity and Existence Results for Degenerate Elliptic Operators C. Vitanza and P. Zamboni

1129

Vector Variational Inequalities and Dynamic Traffic Equilibria X.Q. Yang and H. Yu

1141

A New Proof of the Maximal Monotonicity of the Sum using the Fitzpatrick Function 1159

C. ZMinescu

Contributors 1173

Preface

This Volume contains the (refereed) papers presented at the 38th Conference of the School of Mathematics "G.Stampacchia" of the "E.Majorana" Centre for Scientific Culture of Erice (Sicily), held in Memory ofG. Stampacchia and J.-L. Lions in the period June 20 - July 1, 2003.

The presence of 130 participants from 15 Countries has greatly contributed to the success of the meeting.

The School of Mathematics was dedicated to Stampacchia, not only for his great mathematical achievements, but also because He founded it.

The core of the Conference has been the various features of the Variational Analysis and their motivations and applications to concrete problems. Variational Analysis encompasses a large area of modem Mathematics, such as the classical Calculus of Variations, the theories of perturbation, approximation, subgradient, subderivates, set convergence and Variational Inequalities, and all these topics have been deeply and intensely dealt during the Conference. In particular, Variational Inequalities, which have been initiated by Stampacchia, inspired by Signorini Problem and the related work of G. Fichera, have offered a very great possibility of applications to several fundamental problems of Mathematical Physics, Engineering, Statistics and Economics.

The pioneer work of Stampacchia and Lions can be considered as the basic kernel around which Variational Analysis is going to be outlined and constructed.

The Conference has dealt with both finite and infinite dimensional analysis, showing that to carry on these two aspects disjointly is unsuitable for both.

xii Variational Analysis and Applications

The book is divided into two parts. The former contains the reproduction under kind permission of J.Wiley - of a paper presented in 1978 at

"E.Majorana" Centre by J.-L.Lions on the work of Stampacchia just after His death, and - in alphabetic order - reminiscences and comments on the mathematical achievements of Stampacchia. The latter contains in alphabetic order - the other papers presented at the Conference.

We want to express our deep gratitude to all those who took part in the Conference. Special mention should once more be made of the "E. Majorana" Centre, which offered its facilities and stimulating environment for the meeting. We are all indebted to the "E.Majorana" Centre, the Municipality of Erice, the Italian National Group for Mathematical Analysis, Probability and Applications (GNAMPA), the University of Catania, the Faculty of Sciences and the Dept.of Mathematics and Computer Science of University of Catania, the University of Messina, the University of Pisa, the Dept.of Mathematics of University of Pisa, the University of Reggio Calabria (DIMET), for their financial support. We are grateful to Dr.J.Martindale of Kluwer Publ.Co. and to Professor P.M.Pardalos for having proposed to publish this book. We want also to thank L. Lucarelli Co. for the typing.

F.Giannessi A.Maugeri

PART 1

THE WORK OF G. STAMPACCHIA IN VARIATIONAL INEQUALITIES'

J.-L. Lions

. I N T R O D U C T I O N

An introductory survey on variational inequalities should have been made here by G. Stampacchia.

All those of you who knew him, who had the pleasure to share with him long and stimulating discussions, who knew his warm personality, will share my emotion and my sorrow.

In what follows, I will try to present some of his main ideas and his main contributions in the field of variational inequalities, the main topic of the meeting mentioned in the preface and where he was looking forward to participating and lecturing.

Therefore, I will not speak of his previous contributions; a general report with a complete bibliography will be presented by E. Magenes in the Bollettino dell'Unione Matematica Italiana.

In the field of partial differential equations and functional analysis, in 1958 he published a survey with E. Magenes (Annali Scuola Normale Superiore Pisa, 12 (1958), 247-357), which had a very deep influence on the teaching of partial differential equations (PDE), and he made very important

° Re-printed from "Variational Inequalities and Complementarity Problems. Theory and Applications", Edited by R. W. Cottle, F. Giannessi, J.-L. Lions, J. Wiley, 1980, pp. 1-24.

z Vol. 15-A, No. 3, 1978, pp. 715-756.

4 Variational Analysis and Appls.

contributions to the study of second-order elliptic operators, in particular those without any smoothness hypothesis on the coefficients. It was while he was working on deep questions of regularity of solutions when coefficents are only assumed to be bounded and measurable, and on problems of potential theory, that he was led, at the beginning of the 60's, to variational inequalities.

. VARIATIONAL INEQUALITIES

The following result is now classical [ 1 ]: let Vbe a Hilbert space on IR ; let a(u,v) be a continuous bilinear form on V, which is not necessarily symmetric, and which is V-elliptic, i.e. which satisfies

a(v,v) > a Ilvll 2, a>0, Vv Z (1)

(l[ II denotes the norm in V.) Let K be a closed convex subset of V, K ~ O, and let v ~ ( f ,v) be a continuous linear form on V; then, there exists a unique element u E K such that

a ( u , v - u ) > ( f , v - u ) VvEK (2)

This (2) is what is called a variational inequality (in short VI). Let us remark that:

(i) if K = V, (2) is equivalent to

a(u,v)=(f ,v) , VvEV (3)

and the above result gives the Lax-Milgram lemma; (ii) if a is symmetric (i.e. a(u,v)=a(v,u)Vu, vE V) then (2)is equivalent to

1 2a(u ,u ) - ( f ,u ) = min[Za(v,K L 2 v , v ) - ( f , v ) l (4)

The idea of the original proof of Stampacchia is as follows: (I) the result is immediate, according to (ii) above, i f a is symmetric; (II) if (2) is proven for a(u,v), it will also be proven for a(u,v)+ p(u,v)

where p(u,v) is a not too large perturbation of a(u,v); (III) with this in mind, one introduces

The work o f G. Stampacchia in Variational Inequalities 5

a(v,u)l =

(5)

= ½ E , ( , , u ) -

and, for 0_<0_<1,

a o ( u , v ) = a ( u , v ) + Ofl(u,v ) (6)

By virtue of (I), the result is true for 0 = 0; using (II) one checks that the result is true for a o (u,v), 0 < 0 < 0 o, where 0 0 is a constant depending only on a, and one proceeds in this way.

The main application that Stampacchia had in mind at the beginning of this theory was to potential theory; he was at that time giving a series of lectures in Leray's seminar [ 2, 3 ]. Let us give one example extracted from one of his works (see [ 2 ]).

. A P P L I C A T I O N T O P O T E N T I A L T H E O R Y

Let ~ be a bounded open set of IR" ; we consider the classical Sobolev spaces

Ov i5, e L ( n ) /_/2 (f~) = vV, ox,...,Ox,

H~ ( n ) = {v v e H ' ( n ) , v = 0 on r}

Let ag (x) be a family of functions such that

a o. ~ L ~ (~), (7)

n

i,j=l i= l

and let us define, Vu,v ~ H I ( ~ ) :


Ou Ov dx (9) a(u 'v l : .~' ~au(X) axj j

We do not assume symmetry (a U v aij in general) and we do not assume

regularity on the a0.

Let us define (in a vague manner for the moment) the set K as follows: let E be a closed subset of f2 and let us set

K : { v v ~ H ~ ( n ) , v > l onE} (10)

The precise meaning of 'v >_ 1 on E' is as follows: we say that v > 1 on E in the sense of H~ (f~) if there exists a sequence of smooth functions u,, in H~ (f~) such that:

(i) /'/m -'~ V in H~ (U2)

(ii) u , .>l onE

I f K is not empty, then there exists a unique element u e K satisfying

a(a,v-u)>O V v e K (11)

3.1 In t e rp r e t a t i on o f (11)

Stampacchia shows that

a(u,v)= ~vd,u, VveH~(~)~C°( f2) (12)

where

dkt is a positive measure, with support in the (13)

boundary aE of E

The fact that one has (12) with a positive measure is very simple: let ~o be any smooth function with, say, compact support in ~ , and such that ~o > 0;

The work of G. Stampacchia in Variational Inequalities 7

then if u is the solution o f (11), it is clear that v = u + (9 belongs to K, so that by using this choice o f v in (11) we obtain

a(u,~o)>O fo reve ry ~p>0 (14)

The result follows using a theorem of Schwartz (every distribution which is greater than or equal to 0 is a (positive) measure).

The main point consists o f showing that/ . t has its support in OE. One shows first that

u = 1 o n e (15)

(in the sense of H~ ( ~ ) ) . In order to do that, Stampacchia used a simple technique, but a very powerful one, which is now one o f the classical tools o f the theory o f partial differential equations. Let us define

w = inf {u, 1} (16)

One checks that w ~ K and that

a(w,w-u)=O (17)

Indeed, either u > 1 and then w = 1, or u < 1 and then w - u = 0, so that

(x) O ( w - u ) (x) = 0 aij oxj Ox,

in either case. We can take v = w in (11), and from (11) and (17) we deduce that

a(w-u,w-u)<O

Since 2

a ( w - u , w - u ) > - a l l w - u l l 2

it follows that

W = U


hence (15) follows. Let us take now (o as a smooth function in H~ (f)) , with support in CE;

then v = u + cp ~ K , and taking v = u + fo in (1 I) gives

a(u,~o) = 0, Vfp with compact support in CE ( 1 8 )

Then (13) follows from (15) and (18). The measure /t is called the capacitary measure of E with respect to

a(u,v) and to ~ , and/~(1) in the corresponding capacity of E. In reference [ 2 ](see also reference [ 3 ]), Stampacchia proceeds to study

the properties of this capacity. He introduces, among other things, the notion of regular points with respect to A and shows that this notion is in fact independent of A (in the class of elliptic operators), so that it is equivalent with the Wiener condition (relative to A = -A ). (A is the second-order elliptic operator associated with a).

The techniques and the ideas of Stampacchia gave rise to several interesting contributions in potential theory 3.

. A N U M B E R O F V A R I A N T S A N D E X T E N S I O N S

Let us return now to (8). A number of theoretical questions immediately present themselves.

A first natural question is connected with (10); if one considers, instead of ½ a(v, v) - ( f , v), a general convex function J(v) defined in a Banach space,

one is lead to a V I of the form ( i f J is differentiable).

(J ' (u) ,V-u)>O, V v ~ K

u ~ K

It is then natural to replace the operator J ' by a monotonic operator. This led to the paper of Hartman and Stampacchia [ 4 ] where they study

VI in reflexive Banach spaces, for non-linear partial differential operators of the types of those introduced (in increasing order of generality) by Minty 4,

3 R.M. Herv6. Ann. Inst. Fourier, 14 (1964),493-508; M. Herv6, R. M. Herv6. Ann. Inst. Fourier 22 (1972), 131-145; A. Ancona. J. Mat. Pures et Appl., 54 (1975), 75-124.

4 G. Minty. Duke math. ,i., 29 (1962), 341-6.


Browder 5, and Leray and Lions 6. The good abstract notion for abstract operators A leading to "well set" elliptic V I

(A(u),v-u)>_o,

u ~ K

V v e K

(19)

was introduced by Brezis 7 with the notion of pseudo-monotonic operators. As always in the work of Stampacchia, there is a motivation for the

"abstract" part of the work[ 4 ]; we shall return to that. Another question, motivated by the so-called "unilateral boundary

conditions" arising in elasticity (or "Signorini's problem"; see Fichera 8) is whether the coerciveness hypothesis (7) can be relaxed. This has been studied by Stampacchia and Lions [ 5, 6 ]. Let us mention here one result: suppose that a(u, v) is given as in section 2 but that it satisfies, instead of (7), the much weaker condition:

a(v,v)>O, V v e V (20)

We assume that (8) allows at least one solution, and we denote by X the set of all solutions; one checks immediately that X is a closed convex set; let b(u, v) be a continuous bilinear form on V such that

b(v,v) Pllvll 2, p>O, Vv V (21)

Let v --~ (g,v) be a continuous linear form on V; for every ~ > 0, there exists (according to the result (1) of section 2) a unique u~ e K such that

a ( u c , v - u , ) + c b ( u , , v - u , ) > ( f +eg, v - u , ) , V v E K (22)

Then, as c --, 0, u c ~ u 0 in V, where u 0 is the solution of

b(u0,v-u0) >- (g,v- u0),

u o E S

V v E X

(23)

5 F. Browder. Bull. Am. Math. Soc., 71 (1965), 780-5. 6 J. Leray and J. L. Lions. Bull Soc. Math. Fr., 93 (1965), 97-107. 7 H. Brezis. Ann. Inst. Fourier, 18 (1968),115-75. 8 G. Fichera. Mem. Accad Naz. Lincei, 8 (1964), 91-140.


This result is used in reference [ 6 ], among other things, to solve the unilateral problem.

Still another natural question is the evolution analogue of (2): find a function t ~ u(t), where t is the time, such that 9

u(t) ~ K (24)

Ou(t),v u ( t ) ) + a ( u ( t ) , v - u ( t ) ) > ( f ( t ) , v - u ( t ) ) - - 5 ; - -

V v ~ K (25)

u(t)l,=o=U(O ) = u ° is given (in K) (26)

When K = V, (25) reduces to

Ou(t) ,vl + a ( u ( t ) , v ) = ( f ( t ) , v ) , V v e V (27) Ot )

It is the variational form of "abstract" parabolic equations. This problem has been introduced in [ 6 ]; it was considerably extended

and deepened in the work of Brezis~°; many examples arising from mechanics have been studied~; this problem is also connected with non- linear semi groups Iz .

One difficulty which arises in connection with (25) is in the definition of what we mean by a solution of a V! and an important remark is now in order: let A be a non-linear operator from a reflexive Banach space V into its dual V', and let us assume that A is monotonic, i.e.

( ( A ( u ) - A ( v ) , U - v ) > O , V u , v ~ Z (28)

Then if u is a solution of the V I

9 We do not define in detail the function spaces where u can be taken. 10 H. Brezis. NATO Summer School, Venice, June 1968; H. Brezis. J. Math. Pures et Appl.,

51 (1972), 1-168. )l G. Duvaut and J. L. Lions. "Les indquations en Mdcanique et en Physique". Dunod, Paris

(1972). 12 H. Brezis. "Op6rateurs maximaux monotones et semi groupes de contractions dans les

espaces de Hilbert". North-Holland, Amsterdam (1973).

The work of G. Stampacchia in Variational Inequalities

(A(u) ,V-U)>(f ,v-u) , VvEK

u~K

one has

11

(29)

(A(v),V-U)>_(f,v-u), Vv~K (30)

u~K

This is obvious, since

( A(v),V-U)=( A(u),V-U)+( A(v)- A(u),v-u)--(

(using (28)); but the reciprocal property is true, provided A is hemi- continuous (i.e. 2.--+ (A(u + 2v) ,w) is continuous Vu,v,w~ V). Indeed, if

is given in K, and if we choose in (30)

v=(1-O)u+O~, 0~]0 ,1]

we obtain, after dividing by t3 :

( A((1-O)u +OfO,~-u)>(f ,~-u ) (31)

By virtue of the hemi-continuity, we can let 0--+ 0 in (31) and we obtain (29) (with ~3 instead of v).

This remark allows one to define weak solutions, or generalized solutions, of VI; it is used in the paper with Lewy [ 7 ] (we shall retum to that) and it can also be used for (25) (to "replace" Ou/Ot by Ov/Ot).

. T H E O B S T A C L E P R O B L E M

In section 4 we indicated very briefly some of the problems in variational inequalities which were under study in the years 1966-68; it was at about this time, may be a little earlier, that Stampacchia started working on a problem which is simple, beautiful and deep - and which led to important discoveries some of them being reported in this book.

This is the so-called "obstacle" problem. Let us consider a(u,v) to be given by (9) and let us define


K = {v v e H~ ( n ) , v > V , g given in f2} (32)

Of course, one has to specify in (32) the class where ~' is given, so that, in particular, K is not empty; the function V represents the obstacle. The corresponding VI (2) has a unique solution and the problem is as follows. (1) How to interpret the V I? (2) What are the regularity properties of the solution u?

In solving (1) and (2) a free-boundary problem will appear and the next question will be the following. (3) What are the regularity properties of the free boundary?

Let us explain the basic idea of the work with Brezis [ 8 ] in a simple particular case. According to (30) the VI can be writte#3:

(Av, v - u ) > ( f , v - u ) , V v ~ K (33)

where K is given by (32). Let us assume that

~ H l ( f 2 ) v < O o n F and (34)

A~/<O

Everything is based on a particular choice o fv in (33). For c > 0 we define u c as the solution in H~ (f2) of

~ A u e +t , t e = U

u , = 0 o n F

in (35)

Let us allow for the moment - this & the crucialpoint - that

u , ~ K (i.e. u,>_V i n ~ ) (36)

Then one can choose v = u, in (33) and after dividing by e it gives

( A u , , A u c ) < - ( f , Au,.)

Hence, it follows that

,3 ax i t, (.Q) (dual space of H o (~) )


IIAu~IIL,,o, -IIflIL~,o, (37)

It is a simple matter to check that u, --~ u in H~0 (~) as e --~ 0, so from (37) one obtains that

Au ~ L 2 ( ~ ) (38)

Therefore, if we set Au = f ( ~ ~ f in general!), one can think of u as being given by the solution of the Dirichlet's boundary value problem

Au=? ( ? ~ e ( ~ ) ) , . = o on r

It follows that if the coefficients of A are smooth enough and if the boundary F of ~ is smooth enough, then

u e H 2 ( ~ ) (39)

that is

02u - - ~ L2 (f2), Vi, j axe%

Let us verify now that (36) holds. We write (35) as

cA(u~ -~')+(u, -~')+ eA~ =u-~" (40)

and we take scalar products with ( u , - ~ ) - (where, in general,

v - = s u p ( - v , 0 ) ) . We obtain, since (u,.-V/)-~H~(f2) and since

a(v,v-)=-a(v-,v-), (v,v-)=-(v,v-):

-~a((u,.- ~,)-,(u,- ~,)-)-[l(u~- ~')-~L2,o, + ~(A~',(u,. - ~')-) = = ( u - ~ , ( u , - ~/)-) (41)

Since A~_<0,then (A~,(u -v / ) - )_<0, and (41) gives:


(u -p ' , (u , - f / ) - )+l (u , -~) - ; (n ,+ea( (u , -~) - , (u , -~ ) - ) <-0

But (u -p ' , ( u , - ~ ) - ) >- 0 and therefore (u, - g ) - = 0 , i.e. u, _> g .

The above analysis can be extended, as we show below. Before doing so, let us apply (38) to the interpretation of the VI. One shows easily that u is characterized by

A u - f > O u -~>O

i n f l

(42)

and of course

u = 0 o n F

Consequently, there are two sets in ~ :

the coincidence set, where u = ~,

the equilibrium set, where Au = f (43)

At least in the two-dimensional case, one can think of this problem as giving the displacement of a membrane subjected to forces f and required to stay above the obstacle ~/.

The membrane touches the obstacle on the coincidence set. The two regions are separated by a "surface" S, which is a free surface; S is not given, and on S one has two "boundary" conditions. If ~ E H 2 (f~), one has

u = p '

and (44)

Ou_Op" Vi onS Ox~ Ox~

A natural and important question is now: under suitable hypotheses o n f and on ~/, is it true that

Au ~ L p ( ~ ) ? (45)


This is, of course, important for the regularity of u in spaces like

W' (f~)= v V, ox ~ , Ox~Ox----~j

for p large. For the study of this problem, a more "abstract" presentation is in order, always following the work of Brezis and Stampacchia.

. A N A B S T R A C T R E G U L A R I T Y T H E O R E M . A P P L I C A T I O N T O T H E O B S T A C L E P R O B L E M

We consider the VI

(Av, v - u ) > ( f , v - u ) ,

u ~ K

V v E K

(46)

where K c V, and V is a Hilbert space. The situation extends to cases where A is non-linear and where V is a reflexive Banach space. Let us consider a space X such that

V c X c V' (47)

with continuous embedding, each space being dense in the following one. Example 1 V=H~(f2) , V ' = H -I(f~) and X = L p(f~), for p large enough. The problem considered by Stampacchia and Brezis is: when can we conclude that

A u ~ X ?

One introduces a duality mapping J from X - - - ~ X ' ( v c x ' c v ' ) , i.e. a (non-linear) mapping from X ~ X' such that

(J (u),u) =llJ(u)ll,,llull, IIJ(u)llx is a strictly increasing function o f llullx and goes to +oo as

Ilull~ - ~ oo.


Example 2 If X = L p ( ~ ) , J ( u ) =[ul p-2 u.

I f X is a Hilbert space that we identify with its dual (X = L 2 (f2) in the example), then J = identity. The crucial hypothesis is now:

One can find a duality mapping J from X --~ X ' such that V~ > 0 and Vu ~ K, there exists u~ such that (48) u, e K , Au, E X and u, +EJAu,=u

One can then take v = u, in (46) and using the properties o f J one obtains that

IIAull -< constant

Hence, it follows that

Au ~ X (49)

Application to the problem. We take J(u) = Jp (u) = lul p-2 u and we consider the equation (48), i.e. since

1 1 J-~ = Jp. and - - + - - = 1

p p '

then

We want to show that u, >_ ~. We use the same technique as in section 5,

i.e. we multiply by (u, - p ')- . We obtain

(51)

But

The work o f G. Stampacchia in Variational Inequalities 17

1 lu,-ul Ep'-I

so that (51) gives (since A g < 0 ):

Therefore, either u , . ( x ) - u ( x ) = O , and hence u,.(x)>_u(x)>_g/(x), or (u~ ( x ) - g ( x ) ) - = 0 , i.e. u,. (x)>_ ~/ (x) .

It will follow that, under reasonable assumptions, the solution u of the obstacle problem satisfies

u ~ W"P(f2) (53)

Simple one-dimensional examples show that one cannot obtain L p estimates for higher-order derivatives. One can study the regularity in Schauder spaces; we refer the reader to the book of Kinderlehrer and Stampacchia j4 and to other chapters of this book. See also the report at the International Congress of Mathematicians, Vancouver, 1974, made by Kinderlehrer.

Remark Due to the physical interpretation of the obstacle problem, it is quite natural to consider the problem of minimal surfaces with obstacle. This has been considered by Nitsche ~5 and by Giusti, 16 Giaquinta and Pepe, IVand for surfaces with mean curvature fixed, it has been considered by Mazzone. is

14 An introduction to variational inequalities and their applications. Academic Press, New York, 1980.

Js J.C. Nitsche. "Vorlesungen uber Minimalflachen". Grundlehr. Math. Wiss., vol. 199, Springer, Berlin (1975).

t6 E. Giusti. "Minimal surfaces with obstacles". CIME course on Geometric Measure theory and Minimal surfaces, Rome, 1973,pp. 119-53.

17 M. Giaquinta and L. Pepe. "Esistenza e regolarit~ per il problema dell'area minima con ostacoli in n variabili". Ann. Scu. Norm. Sup., Pisa, 25 (1971), 481-507.

is S. Mazzone. "Un problema di disequazioni variazionali per superficie di curvatura media assegnata". Boll. Unione Mat. Ital., 7 (1973), 318-29.


7. A N I N E Q U A L I T Y F O R T H E O B S T A C L E P R O B L E M

In the above proof of the regularity for the obstacle problem, the hypothesis 'A~, < 0' is much too restrictive. This can be overcome in several ways. (1) One can introduce more flexibility in the abstract hypothesis of section 4 (see [48]); following Brezis and Stampacchia, one introduces families of operators B c : V ~ X, Cc : V ~ X', which are bounded as g ~ 0 and such that the equation

u, + eJ ( Au,. + B,u , ) = u + EC,.u, (54)

has a solution u~ ~ K such that Au~ ~ X ; one then obtains the conclusion (same proof) that

Au ~ X (55)

and this allows one to obtain regularity results similar to the above results under the assumption:

Ag is a measure on ~ ; sup {Ag/,O}~LP(f)) (56)

(2) One can use penalty arguments. (3) One can use an inequality given in Lewy and Stampacchia [9, 10] that we now explain.

Let u be the solution of the obstacle problem. Then one has

f <Au<max{A~, f} (57)

One does not restrict the generality in taking

f = 0 (58)

(Indeed if co is defined by Aco = f , co = 0 on F , then it suffices to work on u - co instead of u). Then one has to show that

0_< Au_< max{A~,0} (59)

Actually, one can obtain a more precise result [9].Let us introduce O(s)= 1


for s<O,O(s)=O f o r s > 0 . Then there exists a unique function u in W 2'p (f2) such that

Au=max{A~/,O}O(u-~)

u = 0 inF

i n ~ (60)

and u is the solution of the obstacle problem. (Of course (59) follows from (60).)

Proof" The proof of(60) is in two essential steps: Step1 One considers an approximation of (60). Let 0. (s) be a sequence

of Lipschitz continuous functions approximating 0 :

oo(s) ! if s<O

= -ns if 0< s<l /n

if s > l/n (61)

One considers the equation

Au. =max{A~,O}O.(u . -~) , u. =0 on F (62)

One proves that this equation has a solution by a fixed-point argument: given co e H~ (~ ) one defines z~ as the solution of

Ah = max { A~,O}O. ( co-~) (63)

One verifies that co--+h=T(co) maps a suitable ball Y. of H ~ ( ~ ) i n t o

itself and that T is continuous. One has also, if max {Ag,0} ~ L p (f~), and if

the coefficients of A are smooth enough, that fi ~ WZ'~'(f)) One then verifies that the mapping T is compact from E --~ Z, and hence it has a fixed point, which is a solution u. of (62),

One has also obtained in this manner that

u. remains in a bounded set of W 2'~' (~ ) (64)

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