book of abstracts operator semigroups in …book of abstracts operator semigroups in analysis:...

Book of Abstracts

Operator Semigroups in Analysis Modern Developments

Organisers Ralph Chill TU Dresden and Yuri Tomilov IM PAN Warsaw

April 24-28 2017 Będlewo

PROGRAM

Monday 24 April 2017

900-940 Yoshikazu Giga (Tokyo)Analyticity of semigroups in end point spaces

945-1025 Sylvie Monniaux (Marseille)First order approach to Lp estimates for the Stokes operator on Lipschitz domains

1030-1100 Coffee break

1100-1130 Robert Haller-Dintelmann (Darmstadt)The Kato square root property for mixed boundary conditions

1135-1205 Moritz Egert (Paris)Cauchy-Riemann system for non-autonomous parabolic PDEs

1210-1240 Alex Amenta (Delft)A first-order approach to elliptic boundary value problems with complex coefficientsand fractional regularity data

1245 Lunch

1430-1515 Jan Pruumlss (Halle)Critical spaces for quasilinear parabolic evolution equations and applications

1520-1550 Juumlrgen Saal (Duumlsseldorf)Multiplication in anisotropic spaces and applications to quasilinear systems


1620-1650 Robert Denk (Konstanz)Generation of semigroups for linear plate equations in Lp-Sobolev spaces

1655-1725 Abdelaziz Rhandi (Salerno)Unbounded perturbations of the generator domain

1730-1800 Nick Lindemulder (Delft)Maximal regularity with weights for parabolic problems with inhomogeneous boundaryconditions

1830 Dinner

1

Tuesday 25 April 2017

900-940 Jared Wunsch (Northwestern U)Decay of damped waves on noncompact manifolds without geometric control

945-1025 Lutz Weis (Karlsruhe)Stochastic non-linear Schroumldinger equations on manifolds


1100-1130 Łukasz Rzepnicki (Toruń)Polynomial stability and a system of coupled strings

1135-1205 Bernhard Haak (Bordeaux)Exact observation of a wave equation on non-cylindrical domains

1210-1240 Marjeta Kramar Fijavž (Ljubljana)Second order differential operators on Lp([0 1]C) with general boundary conditions

1245 Lunch

1430-1515 Matthias Hieber (Darmstadt)Mild and strong periodic solutions to semilinear evolution equations

1520-1550 Birgit Jacob (Wuppertal)Input to state stability of evolution equations


1620-1650 Mustapha Mokhtar-Kharoubi (Besanccedilon)Compactness properties of perturbed sub-stochastic C0-semigroups on L1-spaces

1655-1725 David Seifert (Oxford)Asymptotic behaviour of periodic systems

1730-1800 Lassi Paunonen (Tampere)Asymptotic behaviour of infinite systems of differential equations

1830 Dinner

2

Wednesday 26 April 2017

900-940 Boguslaw Zegarlinski (Imperial College)Construction and ergodicity of dissipative dynamics in noncommutative spaces

945-1025 Wolfgang Arendt (Ulm)Diffusion with non-local boundary conditions


1100-1130 Diego Pallara (Lecce)Heat semigroup and perimeters the local and the nonlocal cases

1135-1205 Hendrik Vogt (Bremen)Linfin-estimates for the torsion function and semigroups dominated by the free heatsemigroup

1210-1240 Giorgio Metafune (Lecce)Sharp heat kernel bounds for a class of parabolic operators with singular coefficients

1245-1315 Krzysztof Bogdan (Wroclaw)Heat kernel of anisotropic nonlocal operators

1330 Lunch

Free afternoon (Hurra)

1830 Dinner

3

Thursday 27 April 2017

900-940 Charles Batty (Oxford)Holomorphic functions which preserve holomorphic semigroups

945-1025 Christian Le Merdy (Besanccedilon)Subordination on K-convex spaces


1100-1130 Andrea Carbonaro (Genova)Bounded Hinfin-calculus for generators of analytic contraction semigroups on Lp spaces

1135-1205 Markus Haase (Kiel)The square function(-)al calculus

1210-1240 Stephan Fackler (Ulm)A new approach to the Akcoglu-Sucheston dilation theorem for positive contractionson Lp-spaces

1245 Lunch

1430-1500 Hans Zwart (Twente)An ideal of Hinfin with a bounded functional calculus

1505-1535 Felix Schwenninger (Hamburg)Hinfin-calculus and the Weiss conjecture for Linfin


1610-1640 Alexander Gomilko (Toruń)On the approximation of bounded C0-semigroups via completely monotone functions

1645-1715 Adam Bobrowski (Lublin)On Hille-type approximation of degenerate semigroups of operators

1720-1750 Peer Kunstmann (Karlsruhe)Lq-Helmholtz decomposition on periodic domains and applications to Navier-Stokesequations

1755-1825 Marcel Schmidt (Jena)Uniqueness of form extensions and domination of semigroups

1900 Dinner

4

Friday 28 April 2017

900-940 Alessandra Lunardi (Parma)Surface measures in Banach spaces

945-1025 Jan van Neerven (Delft)Weyl calculus with respect to the Gaussian measure and Lp-Lq boundedness of theOrnstein-Uhlenbeck semigroup in complex time


1100-1130 Mark Veraar (Delft)Operator-valued Lp-Lq Fourier multiplier theorems

1135-1205 Jan Rozendaal (Warsaw)Stability theory for semigroups using (Lp Lq) Fourier multipliers

1210-1240 Sebastian Kroacutel (Toruń)The Fourier embedding theorems

1245 Lunch

1430-1510 Tom ter Elst (Auckland)The Dirichlet-to-Neumann operator on exterior domains

1515-1545 Valentin Zagrebnov (Marseille)Construction of dynamical semigroups by a functional regularisation agrave la Kato


1620-1650 Juumlrgen Voigt (Dresden)On holomorphic dependence of forms

1655-1725 Jochen Gluumlck (Ulm)Long term behaviour of positive operator semigroups

1730-1800 Andraacutes Baacutetkai (Feldkirch)Boundary delay problems

1830 Dinner

END

5

ABSTRACTS

A First-order Approach to Elliptic Boundary Value Problems with Com-plex Coefficients and Fractional Regularity DataAlex AmentaDelft University of Technology Netherlands

We consider well-posedness of boundary value problems associated with divergence-formelliptic equations with complex t-independent coefficients on the upper half-space andwith boundary data in BesovndashHardyndashSobolev (BHS) spaces Our work is based on atheory of BHS spaces adapted to bisectorial operators with bounded Hinfin functional cal-culus and which satisfy certain off-diagonal estimates

Within a range of exponents determined by properties of adapted BHS spaces weshow that well-posedness of a boundary value problem is equivalent to an associatedprojection being an isomorphism As an application in the case of real coefficients weextend known well-posedness results for the Regularity problem with data in Hardy andLebesgue spaces to a large range of BHS spacesJoint work with Pascal Auscher

Diffusion with Non-local Boundary ConditionsWolfgang ArendtUniversity of Ulm Germany

Non-local boundary conditions are quite natural and occur for several models A particlereaching the boundary is sent back to the interiour with a probability which depends onthe distribution in the interiour at the given moment Mathematically there are severalchallenges One is the space Hilbert spaces seem not appropriate and spaces of continu-ous functions turn out to give the right framework The most difficult is the holomorphicestimateWe will show that the Laplacian with non-local Dirichlet and non-local Robin bound-ary conditions generates a holomorphic semigroup However these semigroups are neverstrongly continuous at 0 Compactness can be proved using the Feller propertyConcluding as a final result one obtains existence and uniqueness as well as regularityand a precise description of the asymptotic behaviourThis is joint work with Stefan Kunkel and Markus Kunze

[1] W Arendt S Kunkel and M Kunze Diffusion with non-local boundary conditionsJ Funct Anal 270 (2016) 2483-2507

[2] W Arendt S Kunkel and M Kunze Diffusion with non-local Robin boundaryconditions arXiv161006894

6

Boundary Delay ProblemsAndraacutes BaacutetkaiUniversity of Vorarlberg Austria

In many cases partial differential equations can be modelled as abstract boundary valueproblems Based on the theory of infinite dimensional regular systems we study a classof boundary perturbation problems with distributed and boundary delay termsAs an application we consider a flow in a network with unbounded delays We provewell-posedness and present conditions on asymptotic stability of such equationsJoint work with S Hadd M Kramar Fijavž and A Rhandi

[1] A Baacutetkai M Kramar Fijavž and A Rhandi Positive Operator Semigroups fromFinite to Infinite Dimensions Operator Theory Advances and Applications 257Birkhaumluser-Verlag Basel 2017

Holomorphic Functions which Preserve Holomorphic SemigroupsCharles BattyUniversity of Oxford UK

Operator semigroups provide an abstract approach to various types of PDEs partic-ularly diffusion equations involving a time variable and a generator A which is typicallya differential operator in space variables The greatest regularity of the solutions occurswhen the semigroup is holomorphic in the time-variable The generators of such semi-groups are known as sectorial operators There are many situations where one wishesto replace the generator A by f(A) for some holomorphic function f For exampleBochnerrsquos notion of subordination in probability corresponds exactly to this procedurefor the class of Bernstein functions (various other names are used for the same class)Thus it is natural to ask when f(A) is sectorial This talk will discuss versions of thisquestion and provide some answers

On Hille-type Approximation of Degenerate Semigroups of OperatorsAdam BobrowskiLublin University of Technology Poland

The result that goes essentially back to Euler says that for any element a of a unitalBanach algebra A with unit u the limit lim983270rarr0(u+ 983185a)[983270

minus1t] (where [middot] denotes the inte-gral part) exists for all t isin R and equals eta As developed by E Hille in his classical

7

monograph in the case where a is replaced by the generator A of a strongly continuoussemigroup etA t ge 0 in a Banach space X a proper counterpart of this formula isetA = lim983270rarr0(IXminus 983185A)minus[983270minus1t] strongly in X Motivated by an example from mathematicalbiology (related to Rotenbergrsquos model of cell growth) I will present results pertaining toconvergence of a similar approximation in which u (resp IX) is replaced by j isin A (respJ isin L(X)) such that for some ℓ ge 2 jℓ = u (resp J ℓ = IX) As it transpires Euler-Hilleapproximation is much more sensible to such changes than that of Yosida and quicklylooses some of its useful properties

Heat Kernel of Anisotropic Nonlocal OperatorsKrzysztof BogdanWrocław University of Technology Poland

I will report on a joint work with Victoria Knopova (Kiev) and Paweł Sztonyk (Wrocław)We construct and estimate Markovian semigroups generated by highly anisotropic space-inhomogeneous integro-differential operators

Bounded Hinfin-calculus for Generators of Analytic Contraction Semi-groups on Lp SpacesAndrea CarbonaroUniversity of Genova Italy

Suppose that T = (T (t))tgt0 is a contraction semigroup on Lp 1 le p le infin Supposefurther that T extends to an analytic contraction semigroup on L2 In this talk I willdiscuss the functional calculus problem for the negative generator Ap of the semigroupT on Lp 1 lt p lt infin More specifically I will show how to reduce the functional cal-culus problem to the proof of a particular bilinear estimate which is an extension of theLumer-Phillips theorem The main tool here is the analysis of the complex time heatflow associated with a particular Bellman function This technique together with thestudy of the convexity properties of the Bellman function gives the following two sharpresults

i) Suppose that A2 is symmetric Then Ap has bounded Hinfin-calculus in any conez isin C 0 |arg(z)| lt φlowast

p + 983185 983185 gt 0 where φlowastp = arcsin |1minus 2p| is optimal

ii) Suppose that A = L is a nonsymmetric finite or infinite dimensional Ornstein-Uhlenbeck operator If minusL generates an analytic contraction semigroup on L2(γinfin)then L has bounded Hinfin-calculus on Lp(γinfin) in any cone of angle θ gt θlowastp where

8

γinfin is the associated invariant measure and θlowastp is the sectoriality angle of L onLp(γinfin) The angle θlowastp is optimal

The talk is based on joint works with Oliver Dragičević (U Ljubljana)

Generation of Semigroups for Linear Plate Equations in Lp-SobolevSpacesRobert DenkUniversity of Konstanz Germany

We consider the linear thermoelastic plate equation with free boundary conditions inLp-Sobolev spaces It can be shown that this equation in uniform C4-domains is uniquelysolvable with maximal regularity and that the associated C0-semigroup is analytic Theproof is based on careful symbol estimates for the solution operators Similar results canbe obtained for the structurally damped plate equation However if Fourierrsquos law of heatconduction is replaced by Cattaneorsquos law then the operator generates a C0-semigroup inLp-spaces only if p = 2 or if the space dimension equals 1

The talk is based on joint results with Yoshihiro Shibata (Tokyo) Roland Schnaubelt(Karlsruhe) and Felix Hummel (Konstanz)

Cauchy-Riemann Sytem for Non-autonomous Parabolic PDEsMoritz EgertUniversity Paris-Sud Orsay France

We report on some recent results on boundary value problems for non-autonomousparabolic equations (or systems) in divergence form

parttuminus divXA(X t)nablaXu = 0

on the upper parabolic half-space (tX) isin R times Rn+1+ obtained in collaboration with

P Auscher and K Nystroumlm Coefficients will depend merely measurably on time andthe tangential variables and enjoy some natural smoothness in the transversal directionWe associate an accretive form with such equations and construct a semigroup flowtransversal to the boundary that describes all weak solutions to Dirichlet- and Neumannboundary value problems with data in L2-Sobolev spaces prior to knowing any solvabilityresults This semigroup evolution will become apparent not on the level of the secondorder parabolic equation but when reformulating the latter as a first order system of

9

Cauchy-Riemann type In implementing this strategy we solve on the boundary Rn+1

the Kato square root problem for parabolic divergence form operators with coefficientsdepending measurably on all variables

The Dirichlet-to-Neumann Operator on Exterior DomainsTom ter ElstUniversity of Auckland New Zealand

We define two versions of the Dirichlet-to-Neumann operator on exterior domains andstudy convergence properties when the domain is truncated

This is joint work with Wolfgang Arendt

A new Approach to the Akcoglu-Sucheston Dilation Theorem for Posi-tive Contractions on Lp-SpacesStephan FacklerUniversity of Ulm Germany

A celebrated result of Akcoglu and Sucheston with profound applications in ergodictheory and functional calculus shows that every positive contraction on a reflexive Lp-space has a dilation to a positive invertible isometry on some bigger Lp-spaces In thistalk we present recent joint work with J Gluumlck (Ulm University) in which we establisha new operator theoretic toolkit to prove dilation theorems on general reflexive Banachspaces As a particular consequence we obtain a new elementary and conceptually clearproof of the dilation result of Akcoglu and Sucheston

Analyticity of Semigroups in End Point SpacesYoshikazu GigaUniversity of Tokyo Japan

It is by now well known that a wide class of elliptic operators with boundary condi-tions has the property generating an analytic semigroup in spaces of bounded functionsAn original proof estimating resolvent equations goes back to K Masuda (1972) and itwas extended by H B Stewart (1974) and others

10

However it is quite recent that one is able to prove that the Stokes operator generates ananalytic semigroup in the space of bounded solenoidal functions even if the domain fluidoccupies is bounded (K Abe and Y Giga (2013)) The original proof based on a con-tradiction argument by estimating evolution equations directly The proof by extendingthe Masuda-Stewart method has been given by K Abe Y Giga and M Hieber (2015)In this talk we give several methods to prove analyticity In particular we give a way toderive necessary resolvent estimates (which goes back to T Suzuki (2016)) by contradic-tion argument This way enables us to derive analyticity of the semigroup generated bythe bidomain operator in mathematical physiology (Y Giga and N Kajiwara (2016))Moreover in this talk we give several results of analyticity of the heat and the Stokesoperator in spaces of functions of bounded mean oscillation BMO In this topic th ereare several possible choices of BMO defined in a domain This last part related to ana-lyticity in BMO is my joint work by M Bolkart (TU Darmstadt) T Suzuki (U Tokyo)and T Tsuyoshi (Shinshu U)

Long Term Behaviour of Positive Operator SemigroupsJochen GluumlckUniversity of Ulm Germany

Let E be an Lp-space for p isin [1infin) or more generally a Banach lattice with ordercontinuous norm and let (Tt)tisin[0infin) be a positive and bounded C0-semigroup on E Inmany applications one is interested in theorems which ensure convergence of Tt as timetends to infinity

In this talk we consider the case where the semigroup contains in some sense a kerneloperator Various theorems from the literature assert that under appropriate technicalassumptions this already implies strong convergence of Tt as t rarr infin We present ageneralisation and at the same time a unification of those results Motivated amongothers by applications to Markov processes we are particularly interested in droppingthe strong continuity assumption on the mapping t 983347rarr Tt thus being left with a purelyalgebraic semigroup This renders classical tools such as spectral theory and Laplacetransform techniques ineffective and therefore we develop a novel and very algebraicapproach to prove convergence of the semigroup

Our approach yields results not only for one-parameter semigroups (Tt)tisin[0infin) butfor positive representations of quite general semigroups As a consequence we demon-strate that in convergence theorems for one-parameter semigroups (Tt)tisin[0infin) the roleplayed by the time interval [0infin) is constituted by its algebraic rather than its topolog-ical propertiesThis talk is based on joint work with Moritz Gerlach (Institut fuumlr Mathematik Univer-sitaumlt Potsdam Germany)

11

On the Approximation of Bounded C0-Semigroups via Completely Mono-tone FunctionsAlexander GomilkoNicholas Copernicus University Torun Poland

Approximation theory is a classical chapter in the theory C0-semigroups with variousapplications to PDEs and their numerical analysis The article [1] proposed a unifiedapproach to approximation formulas for C0-semigroups on Banach spaces by puttingthem into the framework of functional calculus and Bernstein functions One of the ba-sic observations in [1] is that a number of approximation formulas for C0-semigroups canbe derived from the next approximation property for a scalar exponent

eminusnϕ(zn) minus eminusz rarr 0 n rarr infin Re z ge 0

with ϕ being a Bernstein function such that

ϕ(0) = 0 ϕprime(0) = 1 |ϕprimeprime(0)| lt infin

In this talk based on ideas from [1] we develop an approach to approximation of C0-semigroups using completely monotone functions rather than Bernstein functions as in[1]

One of our main results is as follows

Theorem Let minusA be the generator of a bounded C0-semigroup (eminustA)t9841620 on a Banachspace X and let g be a bounded completely monotone function satisfying

g(0) = 1 gprime(0) = minus1 gprimeprime(0) lt infin

If α isin (0 2] then for all t gt 0 n isin N and x isin dom (Aα)

983348(gn(tAn)minus eminustA)x983348 984176 8M((gprimeprime(0)minus 1)t2n)α2 983348Aαx983348

where M = supt9841620

983348eminustA983348

Better estimates are available if the semigroup (eminustA)t9841620 is analytic and this casewill be considered in details This is a joint work with S Kosowicz and Yu Tomilov

[1] A Gomilko and Yu Tomilov On rates in approximation theory for operator semi-groups J Funct Anal 266 (2014) 3040-3082

12

Exact Observation of a Wave Equation on Non-Cylindrical DomainsBernhard HaakUniversity of Bordeaux France

We discuss a 1D wave equation on a non-cylindrical domain given by a boundary curve sof class C2 satisfying 983348sprime983348infin lt 1 We obtain several results on exact observation in finite(and optimal) time

The Square Function(-)al CalculusMarkus HaaseUniversity of Kiel Germany

We develop further the pioneering work of Kalton and Weis on the relation of squarefunctions to Hinfin-functional calculus and show how by a slight change of perspectivesquare functions associated with a scalar functional calculus can again be incorporatedinto a new calculus for vector-valued functions (Hence the name of the talk) I willdemonstrate the elegance and effectivity of this calculus by focussing on some particularinstances (Joint work with Bernhard Haak)

The Kato Square Root Property for Mixed Boundary ConditionsRobert Haller-DintelmannTU Darmstadt Germany

We consider a second order divergence form operator A = minus div micronabla with mixed boundaryconditions in Lp(Ω) for a class of domains Ω that in particular comprises all Lipschitzdomains It turns out that in a very general geometric framework this operator has theKato square root property ie the domain of its square root in L2(Ω) is equal to W 12

D (Ω)where the index D refers to the boundary condition

From this one can deduce that the square root of A is also an isomorphism betweenW 1p

D (Ω) and Lp(Ω) for all 1 lt p le 2 Thus the adjoint of the square root provides uswith an isomorphism between Lp(Ω) and Wminus1p

D (Ω) for 2 le p lt infin that commutes withA thus opening the possibility to transfer many good properties of A known on Lp(Ω)to the spaces Wminus1p(Ω)

This is joint work with Moritz Egert and Patrick Tolksdorf for p = 2 and with PascalAuscher Nadine Badr and Joachim Rehberg for the extrapolation to p ∕= 2

13

Mild and Strong Periodic Solutions to Semilinear Evolution EquationsMatthias HieberDarmstadt University of Technology Germany

In this talk we discuss various approaches to mild and strong periodic solutions tosemilinear evolution equations and apply it to assorted examples ranging from incom-pressible fluid flow over Ornstein-Uhlenbeck processes to electrophysiology Our firstapproach is based on smoothing properties of the underlying linear equation and inter-polation methods and yields results for small forces A weak-strong uniqueness propertyallows us further to obtain strong periodic solutions even for large forces in special situa-tions Finally we consider the bidomain operator and show how to obtain strong periodicsolutions to the FitzHugh-Nagumo model

This is joint work with M Geissert H Nguyen and G Galdi T Kashiwabara as wellas N Kajiwara K Kress and P Tolksdorf

Input to State Stability of Evolution EquationsBirgit JacobUniversity of Wuppertal

In this talk we study the notions of input to state stability (ISS) and integral inputto state stability (iISS) for boundary control systems which are stronger notions thanexponential stability of the corresponding semigroup and include stability with respectto input functions as well It will be shown that if the semigroup is exponentially stablethen ISS is equivalent to admissibility of the input operator with respect to Linfin Fur-ther under the assumption of exponential stability iISS is just admissibility of the inputoperator with respect to an Orlicz space Further we prove that for parabolic diagonalsystems ISS and iISS are equivalent notions

Joint work with Robert Nabiullin (University of Wuppertal) Jonathan R Partington(University of Leeds) and Felix Schwenninger (University of Hamburg)

14

Second Order Differential Operators on Lp([0 1]Cm) with General Bound-ary ConditionsMarjeta Kramar-FijavžUniversity of Ljubljana Slovenia

We shall consider the Banach space X = Lp([0 1]Cm) for some p ge 1 and defineon it the operator

G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063

where a(middot) isin C([0 1]Mm(C)) are diagonalizable positive definite matrices and Φ0 Φ1

are so-called ldquoboundary functionalsrdquo

Φ0 isin L(C([0 1]Cm) Y0) and Φ1 isin L(C([0 1]Cm) Y1)

mapping to ldquoboundary spacesrdquo Y0 Y1 sube C2m respectively satisfying Y0 oplus Y1 = C2mand B isin L(X) a ldquoboundary operatorrdquo Our aim is to give conditions for the functionalsΦ0 Φ1 implying that G generates a cosine family on X To this end we use the operatortheoretical version of the Staffans-Weiss type of perturbation developed recently by AdlerBombieri and Engel

Finally we will apply the abstract results to show well-posedness of wave- and diffu-sion equations on networks

The Fourier Embedding TheoremsSebastian KroacutelNicholas Copernicus University Torun Poland

I will present a complement to the classical results on Fourier multipliers More pre-cisely I will provide a variant of Hytoumlnenrsquos embedding theorem which allows to extendand unify several sufficient conditions for a function to be a Fourier multiplier on weightedHardy spaces The obtained conditions allow to control simultaneously the supremumnorm of dyadic parts of multiplier functions as well as the linfin- and l1-norm of multi-indices of their partial derivatives

15

Lq-Helmholtz Decomposition on Periodic Domains and Applications toNavier-Stokes EquationsPeer KunstmannKarlsruhe Institute of Technology Germany

We prove the existence of the Helmholtz decomposition for vector fields in Lq(Ω) fordomains Ω sube Rd that are invariant under integer translations ie that satisfy Ω+ z = Ωfor all z isin Zd The range of q depends on the boundary regularity of Ω The proof of theHelmholtz decomposition builds upon recent Bloch multiplier theorems due to B BarthWe give several applications to Stokes operators and Navier-Stokes equations on suchdomains(joint work with Jens Babutzka KIT)

Maximal Regularity with Weights for Parabolic Problems with Inho-mogeneous Boundary ConditionsNick LindemulderDelft University of Technology Netherlands

In this talk we consider weighted Lq-Lp-maximal regularity for linear vector-valuedparabolic initial-boundary value problems with inhomogeneous boundary conditions ofstatic type The weights we consider are power weights in time and in space and yieldflexibility in the optimal regularity of the initial-boundary data and allow to avoid com-patibility conditions at the boundary The novelty of the followed approach is the use ofweighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev Besselpotential Triebel-Lizorkin and Besov type which is the main focus of the talk In parti-cular we discuss trace theory and intersection representations for these function spaces

Subordination on K-convex SpacesChristian Le MerdyUniversity of Franche-Comteacute France

This talk is mostly devoted to discrete semigroups and the recently developed notionof subordination of power bounded operators Let (ck)kisinZ be a nonnegative sequencewith

983134k ck = 1 and let T X rarr X be an invertible operator on some Banach space X

Assume that the sequence Tn n isin Z is bounded This allows to define an operatorS =

983134k ckT

k called lsquosubordinated to T rsquo We give conditions implying that S is a Ritt

16

operator (the discrete analogue of lsquobounded analytic semigroupsrsquo) and admits a boundedHinfin-functional calculus with respect to a Stolz domain Banach space geometry comesinto play and we focus on the case when X is a K-convex Banach space (Joint workwith Florence Lancien)

Surface Measures in Banach SpacesAlessandra LunardiUniversity of Parma Italy

Let X be a Banach space endowed with a probability measure m I will describe dif-ferent approaches for the construction of surfaces measures associated to m and relatedintegration by parts formulae on smooth enough subsets of X

The available literature deals mainly with non-degenerate Gaussian measures in sepa-rable Banach spaces In that case integration by parts formulae are similar (as far aspossible) to the finite dimensional case They may be extended to Sobolev functions sincea trace theory for Sobolev functions on smooth surfaces is available For non Gaussianmeasures the theory is not as well developed and several basic questions remain open

Sharp Heat Kernel Bounds for a Class of Parabolic Operators with Sin-gular CoefficientsGiorgio MetafuneUniversity of Salento Italy

We study parabolic problems associated to the second order elliptic operator in RN

L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x

|x|2 middotnablaminus b|x|minus2

with a gt 0 and b c real coefficientsNote that the second order coefficients are discontinuous when a = 1 and singularitiesappear in the drift and potential terms The choice a = 1 c = 0 yields the Schroumldingeroperator with inverse square potential The condition

D =b

a+

983072N minus 1 + cminus a

2a

9830732

ge 0

is necessary and sufficient for the existence of a realization of L generating a positivesemigroup and reduces to the classical one in the case of Schroumldinger operators

17

The operator L becomes self-adjoint in a suitable weighted L2 -space which we use as atool for construncting the generated semigroup However generation in the unweightedLp -spaces is also characterized Letting

s1 =N minus 1 + cminus a

2aminus

radicD s2 =

N minus 1 + cminus a

2a+

radicD

it turns out that there exists a realization Lpint between the minimal and the maximaloperator that generates a semigroup in Lp(RN ) if and only if s1 lt Np lt s2 + 2

We describe the domain and show that the generated semigroup is bounded analyticof angle π2 and positive for t gt 0 As a consequence the spectrum of Lpint coincideswith the half-line (minusinfin 0]

We prove that the semigroup is represented by a kernel p(t x y) which satisfies thedouble side estimates

p(z x y) asymp CtminusN2

983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp

983072minusc|xminus y|2

t

983073

where γ = (N minus 1 + c)a minus N + 1 and the constants c C may differ in the upper andlower bounds Note that γ = 0 if and only if L is self-adjointIntegrating the above kernel estimates with respect to t we also obtain precise kernelbounds of the Green function

Finally we remark that kernel estimates for |x|αL can be obtained from the resultsabove via a change of variables However this tranformation does not relate the kernelof |x|α∆ to that of the Laplacian but rather to the kernel of a suitable L as above wherediscontinuities necessarily appear

Most of the content of these lecture is based on joint works with Chiara Spina LuigiNegro (University of Salento) and Motohiro Sobajima (Tokyo University of Science)

Compactness Properties of Perturbed Sub-stochastic C0-Semigroups onL1-SpacesMustapha Mokhtar-KharroubiUniversity of Franche-Comteacute France

We deal with positive strongly continuous semigroups (U(t))t9841620 of contractions inL1(ΩA micro) with generator T where (ΩA micro) is an abstract measure space and provide asystematic approach of compactness properties of perturbed C0-semigroups

983054et(ldquoTminusV rdquo)983055

tge0

(or their generators) induced by singular potentials V (Ωmicro) rarr R+ More precise re-sults are given in metric measure spaces (Ω d micro) This new construction is based on

18

several ingredients new a priori estimates peculiar to L1-spaces local weak compactnessassumptions on unperturbed operators ldquoDunford-Pettisrdquo arguments and the assump-tion that the sublevel sets ΩM = xV (x) le M are ldquothin at infinity with respect to(U(t))t9841620rdquo We show also how spectral gaps occur when the sublevel sets are not ldquothinat infinityrdquo This formalism combines intimately the kernel of (U(t))t9841620 and the sublevelsets ΩM

[1] M Mokhtar-Kharroubi Compactness properties of perturbed sub-stochastic C0-semigroups on L1(micro) with applications to discreteness and spectral gaps Meacutemoiresde la socieacuteteacute matheacutematique de France N148 2016

First Order Approach to Lp Estimates for the Stokes Operator on Lip-schitz DomainsSylvie MonniauxAix-Marseille University France

In this talk I will describe a first order approach to developing an Lp theory for theHodge-Laplacian and the Stokes operator with Hodge boundary conditions acting on abounded open subset of Rn In particular conditions on the domain and p under whichthese operators have bounded resolvents generate analytic semigroups have boundedRiesz transforms or have bounded holomorphic functional calculi will be given The firstorder approach of initially investigating the Hodge-Dirac operator provides a frameworkfor strengthening known results and obtaining new ones on general classes of domainsin what we believe is a straightforward manner

This is a joint work with Alan McIntosh

Weyl Calculus with Respect to the Gaussian Measure and Lp-Lq Bound-edness of the Ornstein-Uhlenbeck Semigroup in Complex TimeJan van NeervenDelft University of Technology Netherlands

We introduce a Weyl functional calculus for the Ornstein-Uhlenbeck operator L =minus∆ + x middot nabla and give a simple criterion for Lp-Lq boundedness of operators in thisfunctional calculus It allows us to recover unify and extend old and new results con-cerning the boundedness of exp(minuszL) as an operator from Lp(Rd γα) to Lq(Rd γβ) forsuitable values of z isin C with Re z gt 0 p q isin [1infin) and αβ gt 0 Here γτ denotes thecentred Gaussian measure on Rd with density (2πτ)minusd2 exp(minus|x|22τ)

19

Heat Semigroup and Perimeters The Local and the Nonlocal CasesDiego PallaraUniversity of Salento Lecce Italy

The equality

P (E) = limtrarr0

983168π

t

983144

Ec

T (t)χE = limtrarr0

983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144

EdivΦdx Φ isin [C1

c (Rn)]n 983348Φ983348infin le 1983165

is the perimeter of E sub Rn (T (t))tge0 is the heat semigroup and pn(x y t) is the heatkernel Formula (1) can be suitably extended in different contexts such as Wienerspaces and some Carnot groups It can also be extended in the fractional (ie nonlocal)perimeters case

Pα(E) =

983144

EtimesEc

1

|xminus y|n+αdxdy 0 lt α lt 1

in Rn and in Carnot groups by using the fractional heat semigroups generated by theα-powers of the (sub)-Laplacean operator As a by-product in some particular cases theBourgain-Breacutezis-Mironescu result

P (E) = K limαrarr1minus

Pα(E)

can be recovered and extended

Asymptotic Behaviour of Infinite Systems of Differential EquationsLassi PaunonenTampere University of Technology Finland

In this presentation we study infinite systems of differential equations of the form

xk(t) = A0xk(t) +A1xkminus1(t) xk(0) isin Cm k isin Z t ge 0 (1)

where A0 isin Cmtimesm and A1 isin Cmtimesm are independent of the index k isin Z This class ofsystems in particular includes the so-called robot rendezvous problem and platoon systemsthat are used to approximate the dynamics of very large systems of identical vehicles [1]

Our main interest is in studying the stability properties and rates of convergence ofthe solution x(t) = (xk(t))kisinZ isin ℓp(Cm) of the full coupled system (1) as t rarr infin Thesequestions are particularly interesting due to the fact that many systems of the form (1)are known to lack uniform exponential stability As our main results we introduce

20

general conditions for strong stability of the system and present spaces of initial statesx(0) = (xk(0))kisinZ that lead to solutions converging at rational rates as t rarr infin

The presented results are joint work with David Seifert (University of Oxford UK)

[1] L Paunonen and D Seifert Asymptotics for infinite systems of differential equationsSIAM J Control Optim (to appear) available athttparxivorgabs151105374

Critical Spaces for Quasilinear Parabolic Evolution Equations and Ap-plicationsJan PruumlszligMartin Luther University of Halle-Wittenberg Germany

In the last decades considerable effort in pdersquos has been put into finding the mostgeneral spaces of initial data such that a given nonlinear pde is well-posed So far thishas been achieved with case studies each equation has its own theory Moreover theavailable proofs in the literature are usually arkwardly involved sometimes going eveninto the miscroscopic theory of the relevant spaces in most cases Besov spaces

In my talk I present an abstract approach in the framework of parabolic evolutionequations relying on maximal Lp-regularity in time weighted Lp-spaces This approacheasily recovers many known results in applications for example for the Navier-Stokesequation the Keller-Segal equations quasi-geostrophic equations and many others butalso leads in a direct smooth way to new results eg for the vorticity equations

Unbounded Perturbations of the Generator DomainAbdelaziz RhandiUniversity of Salerno Italy

Let XU and Z be Banach spaces such that Z sub X (with continuous and dense em-bedding) L Z rarr X be a closed linear operator and consider closed linear operatorsGM Z rarr U Putting conditions on G and M we show that the operator A = L withdomain D(A) = z isin Z Gz = Mz generates a C0-semigroup on X Moreover we givea variation of constants formula for the solution of the following inhomogeneous problem

983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0

Gz(t) = Mz(t) + g(t) t ge 0

z(0) = z0

21

Several examples will be given in particular a heat equation with distributed un-bounded delay at the boundary condition and a delayed transport problem on a networkOur approach is based on matrices transformations and the theory of closed-loop sys-tems

Stability Theory for Semigroups Using (Lp Lq) Fourier MultipliersJan RozendaalIM PAN Warsaw Poland

This talk concerns a recent development in the stability theory for C0-semigroups per-taining to the link between stability theory and operator-valued Fourier multipliersLet A be the generator of a C0 -semigroup (T (t))tge0 on a Banach space X It is well-known that decay properties of (T (t))tge0 are linked to Fourier multiplier properties ofthe resolvent of A So far this connection has been of limited use for stability theoryThis is due in part to the difficulty of obtaining multiplier estimates for the resolvent ofA using Mikhlinrsquos TheoremIn this talk I will suggest a new approach to the connection between stability theoryvector-valued harmonic analysis and Banach space geometry using a recently developedtheory of operator-valued (Lp Lq) Fourier multipliers for p = q This theory can be usedto explain known results on exponential stability as consequences of (Lp Lq) multipliertheorems and also yields new results on exponential stabilityMoreover I will explain a novel connection between (Lp Lq) multiplier properties of theresolvent of A and polynomial stability of (T (t))tge0 The latter occurs eg in dampedwave equations where the spectrum of A approaches the imaginary axis at infinity or incase of a polynomial singularity of the resolvent at zero For such equations the theoryof (Lp Lp) Fourier multipliers does not suffice Using (Lp Lq) multipliers one can estab-lish concrete results which take into account the geometry of the underlying space in aquantitative mannerThis is joint work with Mark Veraar (Delft University of Technology) The theory ofoperator-valued (Lp Lq) Fourier multipliers will be presented in his talk

Polynomial Stability and a System of Coupled StringsŁukasz RzepnickiNicholas Copernicus University Torun Poland

We study an energy decay problem in a system of two connected vibrating strings It isknown that the rate of the decay depends on coupling conditions and the ratio of wave

22

speeds (see [123] ) Moreover there are some cases for which the energy converges tozero but not exponentially The natural question is if the decay could be polynomialWe use the C0-semigroup approach and Rothrsquos theorem to show that if the ratio of wavespeeds is irrational and algebraic then the answer for the above question is positive

ndashjoint work with Roland Schnaubelt

[1] G Chen M Coleman and HH West Pointwise stabilization in the middle of thespan for second order systems nonuniform and uniform exponential decay of solu-tions SIAM J Appl Math 47 (1987) 751-780

[2] BZ Guo and WD Zhu On the energy decay of two coupled strings through a jointdamper Journal of Sound and Vibration 203 (1997) 447-455

[3] K-S Liu Energy decay problems in the design of a point stabilizer for coupled stringvibrating systems SIAM J Control Optim 26 (1988) 1348-1356

Multiplication in Anisotropic Spaces and Applications to QuasilinearSystemsJuumlrgen SaalUniversity of Duumlsseldorf Germany

Quasilinear mixed order systems arise in countless applications in natural sciences andtechnology Important representatives of this class of PDE are free boundary problemsin fluid dynamics Relying on the maximal regularity approach not seldom intricatenonlinearities of quasilinear mixed order systems have to be estimated in anisotropic (inspace and time) function spaces By the lack of results on multiplication in anisotropicspaces in previous approaches this is more or less done by hand This can take pagesof technical estimates that are not even optimal In my talk I would like to presentrecently derived results on multiplication and analytic Nemytskii operators on scales ofanisotropic function spaces By these results the estimation of nonlinear terms is es-sentially reduced to veryfying an elementary condition for the corresponding anisotropicSobolev indices I also intent to discuss applications and improvements for quasilinearproblems such as the Stefan problem and free boundary problems in fluid dynamics

23

Uniqueness of Form Extensions and Domination of SemigroupsMarcel SchmidtUniversity of Jena Germany

It is an important problem in operator theory and mathematical physics to determinewhen the minimal and maximal quadratic form associated with a differential expressioncoincide In particular this question has been extensively studied for Schroumldinger oper-ators with electric and magnetic potential on Euclidean domains manifolds and graphsWe study this question from an abstract point of view using the concept of dominationof semigroups which is an abstract formulation of Katorsquos inequality As a main resultwe show that roughly speaking uniqueness of form extensions passes from the domi-nating form to the dominated form We conclude by giving applications to magneticSchroumldinger operators (based on joint work with Daniel Lenz and Melchior Wirth)

Hinfin-calculus and the Weiss Conjecture for Linfin

Felix SchwenningerUniversity of Hamburg Germany

Due to a fundamental result by Le Merdy the Weiss conjecture for L2-admissibilityis strongly linked to the Hinfin-calculus for analytic semigroups Recently there has beengrowing interest in the study of Linfin-admissible control operators and in a related oldquestion by George Weiss on the continuity of mild solutions In this talk we will par-tially answer the later and again draw the connection to the Hinfin-calculus Furthermorewe will discuss how this relates to admissibility with respect to Orlicz spacesThis is joint work with Birgit Jacob and Hans Zwart

Asymptotic Behaviour of Periodic SystemsDavid SeifertUniversity of Oxford UK

Consider the non-autonomous Cauchy problem983094

z(t) = A(t)z(t) t ge 0

z(0) = x

where x isin X for some Hilbert space X and where the family A(t) t ge 0 is assumedto have an associated evolution family U(t s) t ge s ge 0 which is uniformly bounded

24

If the system is periodic in the sense that there exists τ gt 0 such that A(t + τ) = A(t)for all t ge 0 then the asymptotic behaviour as t rarr infin of the solution

z(t) = U(t 0)x t ge 0

is determined by the monodromy operator T = U(τ 0) For instance the ABLV theoremshows that 983348z(t)983348 rarr 0 as t rarr infin for all x isin X provided the boundary spectrum σ(T )capTis at most countably infinite and contains no eigenvalues In this talk I shall presenta quantified version of this result involving rates of convergence for a special class ofperiodic families A(t) t ge 0 The crucial property of these families is that theassociated monodromy operator T turns out to be a so-called Ritt operator As anapplication we investigate rates of energy decay for solutions of the one-dimensionaldamped wave equation in the case where the damping is periodic in time The talk isbased on joint work with Lassi Paunonen (Tampere Finland)

Operator-valued Lp-Lq Fourier Multiplier TheoremsMark VeraarDelft University of Technology Netherlands

Fourier multiplier theorems play an important role in mathematical analysis For ex-ample they can be applied in the regularity theory for PDEs and evolution equations Inmany situations Lp rarr Lq with p = q is the most relevant and conditions on the multiplierand its derivatives can be used to obtain boundedness results Motivated by applicationsto stability analysis in the theory of evolution equations we have developed a theory ofoperator-valued Fourier multipliers for p ∕= q Previously the scalar case was treatedby Houmlrmander and Lizorkin and many others Unlike in the situation p = q one canavoid conditions on the derivatives of the multipliers In the vector-valued setting we use(Fourier) type and cotype of the underlying Banach space to obtain sufficient conditionsfor Lp-Lq-boundedness of Fourier multipliers Moreover several converse statements andexamples will be given to prove the necessity of the conditionsThe talk is based on joint work with Jan Rozendaal Applications to stability will bepresented in his talk

25

Linfin-estimates for the Torsion Function and Semigroups Dominated bythe Free Heat SemigroupHendrik VogtUniversity of Bremen Germany

The torsion function uD of an open set D sube Rd can be defined as follows uD(x) is the ex-pected time for the Brownian motion starting at x to leave the set D Let ∆D denote theDirichlet Laplacian acting in L2(D) with ground state energy E0(minus∆D) = inf σ(minus∆D)If E0(minus∆D) gt 0 then the torsion function uD is the unique solution of minus∆Du = 1

We show that1 le E0(minus∆D) middot 983348uD983348infin le d

8+ 061

radicd+ 1 (1)

The constant in the right-hand side is quite sharp if Bd is the unit ball in Rd then

d

8le E0(minus∆Bd

) middot 983348uBd983348infin le d

8+ Cd13

with some absolute constant C gt 0 The upper bound in (1) is derived from a suitableLinfin-estimate for the semigroup generated by ∆D

On Holomorphic Dependence of FormsJuumlrgen VoigtTechnische Universitaumlt Dresden Germany

Let H be a Hilbert space and let z 983347rarr az be a holomorphic family of sectorial forms(a holomorphic family of type (a) in the sense of Kato [2 VII sect4]) Then the functionz 983347rarr Az where Az is the operator associated with az is holomorphic We present a proofof this result making use of the LaxndashMilgram lemma We recall a striking applicationof this result (see [1]) and discuss a question concerning the holomorphic dependence ofsectorial formsThe talk is a report on joint work with H Vogt

[1] T Kato Trotterrsquos product formula for an arbitrary pair of self-adjoint contractionsemigroups Topics in functional analysis (essays dedicated to MG Kreın on theoccasion of his 70th birthday) pp 185ndash195 Adv in Math Suppl Stud vol 3Academic Press New York 1978

[2] T Kato Perturbation Theory for Linear Operators Corrected printing of the secondedition Springer-Verlag Berlin 1980

26

Stochastic Non-linear Schroumldinger Equations on ManifoldsLutz WeisKarlsruhe Institute of Technology Germany

We prove existence and uniqueness of stochastic non-linear Schroumldinger equations onmanifolds The case of the cubic equation on three dimensional compact manifolds isparticularly challenging here we use spectrally localized Strichartz estimates which aredefined in terms of a Littlewood-Paley decomposition of the Laplace-Beltrami operator

Decay of Damped Waves on Noncompact Manifolds Without Geomet-ric ControlJared WunschNorthwestern University USA

I will review recent results on estimates for the decay rate of solutions to the dampedwave equation on noncompact manifolds subject to various hypotheses on the structureof the damping near infinity

Construction of Dynamical Semigroups by a Functional Regularisationagrave la KatoValentin A ZagrebnovAix-Marseille University France

A functional version of the Kato one-parametric regularisation for the construction ofa dynamical semigroup generator of a relative bound one perturbation is introduced Itdoes not require that the minus generator of the unperturbed semigroup is a positivitypreserving operator The regularisation is illustrated by an example of a boson-numbercut-off regularisation of unbounded Kossakowski-Lindblad-Davies generator correspond-ing to evolution of an open system

Based on a joint paper with AFM ter Elst [arXiv170103506v1]

27

Construction and Ergodicity of Dissipative Dynamics in Noncommuta-tive SpacesBoguslaw ZegarlinskiImperial College London UK

I will review some results and problems concerning Dissipative Dynamics for large inter-acting systems

An Ideal of Hinfin with a Bounded Functional CalculusHans ZwartUniversity of Twente Holland

Let minusA be the infinitesimal generator of an exponentially stable semigroup (T (t))tge0

on the Hilbert space X In Haase and Rozendaal [2] it is shown that there exists aconstant mA such that for all f isin Hinfin = Hinfin(C+) the following holds

983348f(A)T (t)983348 le mA| log(t)|983348f983348infin t isin (0 12) (1)

We show that this type of estimates imply that if for f isin Hinfin there exists a c1 ge 0and a δ gt 1 such that

|f(iω)| le c1(log(|ω|+ e))δ

for ae ω isin R (1)

then f(A) is a bounded operatorConcerning this result we make the following observations and remarks

i) Let S denote the set of all Hinfin-functions satisfying (1) for some c1 ge 0 and δ gt 1Clearly this class forms an ideal in Hinfin

We point out that S does not trivially compare with the ideals eminusmiddotτHinfin τ gt 0emerging from estimates of the form (1) In fact it is not hard to see that neitherS sub eminusmiddotτHinfin nor S sup eminusmiddotτHinfin holds

Consider g(z) = (iπ + log(z))minusδ Then g isin S but eτ middotg isin Hinfin(C+) as |eτzg(z)|becomes unbounded for z = rei

π3 and r rarr infin Hence g isin eminusmiddotτHinfin for any τ gt 0

Conversely let h isin L1(0infin) and consider g = L(h lowast δτ ) isin eminusmiddotτHinfin If g was anelement of S then the Fourier transform |F(hlowastδτ )|(ω) = |F(h)|(ω) is O(logminus1(ω))as ω rarr infin This however is not true for general h isin L1(0infin)

ii) The logarithm functions in (1) and (1) are strongly linked This means for instancethat if for a given A (1) holds with | log(t)| replace by

983166| log(t)| then f(A) is a

bounded linear operator for all f satisfying (1) for a δ gt 12

28

iii) The limiting case δ = 1 is open and is strongly related to the question if a resultby Nollau [3] (see also Section 35 of Haase [1]) holds for non-analytic semigroups

[1] M Haase The Functional Calculus for Sectorial Operators Operator Theory Ad-vances and Applications 169 Birkhaumluser Verlag Basel 2006

[2] M Haase and J Rozendaal Functional calculus for semigroup generators via trans-ference Journal of Funct Anal 265 (2013) 3345-3368

[3] N Nollau Uumlber den Logarithmus abgeschlossener Operatoren in Banachschen Raumlu-men (German) Acta Sci Math (Szeged) 30 (1969) 161-174

29

PROGRAM

Monday 24 April 2017

900-940 Yoshikazu Giga (Tokyo)Analyticity of semigroups in end point spaces

945-1025 Sylvie Monniaux (Marseille)First order approach to Lp estimates for the Stokes operator on Lipschitz domains


1100-1130 Robert Haller-Dintelmann (Darmstadt)The Kato square root property for mixed boundary conditions

1135-1205 Moritz Egert (Paris)Cauchy-Riemann system for non-autonomous parabolic PDEs

1210-1240 Alex Amenta (Delft)A first-order approach to elliptic boundary value problems with complex coefficientsand fractional regularity data

1245 Lunch

1430-1515 Jan Pruumlss (Halle)Critical spaces for quasilinear parabolic evolution equations and applications

1520-1550 Juumlrgen Saal (Duumlsseldorf)Multiplication in anisotropic spaces and applications to quasilinear systems


1620-1650 Robert Denk (Konstanz)Generation of semigroups for linear plate equations in Lp-Sobolev spaces

1655-1725 Abdelaziz Rhandi (Salerno)Unbounded perturbations of the generator domain

1730-1800 Nick Lindemulder (Delft)Maximal regularity with weights for parabolic problems with inhomogeneous boundaryconditions

1830 Dinner

1








1245 Lunch







1830 Dinner

2









1330 Lunch


1830 Dinner

3








1245 Lunch








1900 Dinner

4








1245 Lunch







1830 Dinner

END

5

ABSTRACTS








6









7









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29








1245 Lunch







1830 Dinner

2









1330 Lunch


1830 Dinner

3








1245 Lunch








1900 Dinner

4








1245 Lunch







1830 Dinner

END

5

ABSTRACTS








6









7









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29









1330 Lunch


1830 Dinner

3








1245 Lunch








1900 Dinner

4








1245 Lunch







1830 Dinner

END

5

ABSTRACTS








6









7









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29








1245 Lunch








1900 Dinner

4








1245 Lunch







1830 Dinner

END

5

ABSTRACTS








6









7









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29








1245 Lunch







1830 Dinner

END

5

ABSTRACTS








6









7









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29

ABSTRACTS








6









7









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29









7









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29









8











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29











9










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29










10






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29






11
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29
















12












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29












13







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29







14



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29



G = a(middot) d2

ds2 D(G) =

983062f isin W 2p([0 1]Cm) Φ0f = 0 Φ1f

prime + Φ1Bf = 0983063








15









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29









983134k ckT


16







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29







L = ∆+ (aminus 1)

N983142

ij=1

xixj|x|2 Dij + c

x



D =b

a+


2a

9830732

ge 0


17



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29



2aminus

radicD s2 =


2a+

radicD





983072|x||y|

983073minus γ2983074983072

|x|t12

and 1

983073983072|y|t12

and 1

983073983075minusN2+1+

radicD

exp


t

983073







tge0


18








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29








19


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29


The equality

P (E) = limtrarr0

983168π

t

983144

Ec


983168π

t

983144

EtimesEc

pn(x y t) dydx (1)

holds where

P (E) = sup983164983144


c (Rn)]n 983348Φ983348infin le 1983165


Pα(E) =

983144

EtimesEc

1




Pα(E)







20









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29









983110983116983114

983116983112

z(t) = Lz(t) + f(t) t ge 0


z(0) = z0

21






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29






22








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29








23









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29









z(0) = x


24


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29


z(t) = U(t 0)x t ge 0




25




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29




8+ 061

radicd+ 1 (1)


d

8le E0(minus∆Bd


8+ Cd13






26








27



















28





29








27



















28





29



















28





29





29

book of abstracts operator semigroups in …book of abstracts operator semigroups in analysis:...

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