Evolution equations and approximations

Series on Advances in Mathematics for Applied Sciences - Vol. 61

EVOLUTION EQUATIONS AND APPROXIMATIONS a-6

ranz Kappel World Scientific

EVOLUTION EQUATIONS AND APPROXIMATIONS

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Series on Advances in Mathematics for Applied Sciences - V o l . 61

EVOLUTION EQUATIONS AND APPROXIMATIONS Kazufumi Ito North Carolina State University, USA

Franz Kappel University of Graz, Austria

m World Scientific m

New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

EVOLUTION EQUATIONS AND APPROXIMATIONS Series on Advances in Mathematics for Applied Sciences — Vol. 61 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-026-4

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Preface Abstract evolution equations provide a unifying framework in order to investigate well-posedness of dynamical systems of various types describing the time evolution of concrete systems. Moreover, the abstract setting can also be used as a framework in order to develop approximation schemes for such dynamical systems. In case of linear autonomous problems the abstract theory is available in a considerable number of textbooks on strongly continuous semigroups of linear bounded operators. Also the cases of linear time dependent problems and of nonlinear autonomous problems are already well represented by monographs. One of the main difficulties arising in connection with nonlinear problems is the fact that the usual requirements of dissipativity resp. accretiveness on the operators governing the evolution equation result in too severe restrictions in case of the concrete problems one wants to consider. The reason is that dissipativity of an operator is global property. A sufficiently general and flexible way for localizing the concept of dissipativity has been developed by K. Kobayashi, Y. Kobayashi and S. Oharu by requiring dissipativity of the operators only on level sets of lower semi-continuous functionals. The common methodological feature of the proofs for existence resp. well-posedness is to prove convergence of the implicit (or backward) Euler scheme on certain time mesh sequences (i.e., to prove convergence of what is usually called a DS-approximation), which leads to the concept of mild solutions. Uniqueness proofs in many cases are based on the concept of integral solutions introduced by Ph. Benilan. This monograph intends to give a unified presentation of the general approach for well-posedness results, the core being the theory developed by K. and Y. Kobayashi together with S. Oharu contained in Sections 6.1 - 6.5. Modifications of this theory are introduced for quasilinear problems in Section 6.9 and in Chapter 7, where a localized version of the theory developed by M. G. Crandall and A. Pazy is presented. In Section 6.10 we show that this approach can be used for nonlinear parabolic problems, where the operator governing the evolution equation is the subdifferential of a convex and lower semi-continuous functional. In Chapter 8 apply the difference scheme approach to variational formulations of the evolution equation using the concept of Gelfand triples as the basic setting. The only exception where well-posedness is not established by a difference scheme approach is the class of semilinear evolution equations considered in Chapter 11. There we use a Banach fix point argument instead.

VI

Preface

For the sake of completeness of the representation we include in Chapter 2 a short introduction into the theory of Co-semigroups of bounded linear operators restricting ourselves to a discussion of strong and mild solutions for abstract Cauchy problems, to the Hille-Yosida theorem, the Lumer-Phillips theorem and Ball's theorem concerning characterization of well-posedness via weak solutions. Because of their importance we give in Chapter 3 the basic facts on analytic semigroups including the characterization of infinitesimal generators as sectorial operators. As a first step to the general nonlinear and time dependent case we discuss in Chapter 5 strongly continuous semigroups of nonlinear Lipschitzian operators. We present the Crandall-Liggett generation theory and give a detailed discussion of various solution concepts for nonlinear evolution equations including the concept of integral solutions (as introduced by Ph. Benilan). We discuss also carefully the concepts of weak and strong generators as well as their relation to the minimal sections of the operator generating the semigroup and present finally a Hille-Yosida type theorem for nonlinear semigroups. The second central theme for this monograph are abstract approximation results for evolution equations. In Chapter 4 we present different versions of the basic approximation result for Co-semigroups of linear bounded operators known as Trotter-Kato theorem. Section 8.2 contains an approximation result of Lax-Richtmyer type (i.e., stability plus consistency imply convergence) for the evolution equations considered in Chapter 8. Approximation of mild solutions of evolution equations in the framework of Chapter 6 is the topic of Chapter 10. An abstract approximation result of Lax-Richtmyer type is given in Section 10.1, whereas a Chernoff type approximation result is obtained in Section 10.2. As an application of the general result given in Section 10.1 we state and prove a Trotter-Kato theorem for nonlinear semigroups. The general Chernoff type theorem is used to prove some results on operator splitting in Section 10.3. Finally, Chapter 11 contains approximation results for the semilinear evolution equations considered there (see Sections 11.3 and 11.4). In order to demonstrate the feasibility of the existence and uniqueness theory presented in this monograph for various types of evolution equations we include applications to more or less concrete problems. As an application of the theory for nonlinear semigroups presented in Chapter 5 (and the results on subdifferentials of lower semi-continuous convex functionals given in Section 1.6) we consider nonlinear diffusion in Section 5.4. The main applications are those of Chapter 9. In Section 9.1 delay equations of various type are considered. Well-posedness for scalar conservation laws of general type is discussed extensively in Section 9.2. Our approach is based on the viscosity method and applies the general theory developed in Chapter 6 to the viscous problems, whereas the limiting problem is handled with the localized version

Preface

vu

of the Crandall-Pazy theory as presented in Chapter 7. We show that the mild solution obtained via the Crandall-Pazy theory is the unique entropy solution of the problem. Finally, in Section 9.3 we apply the general theory to the Navier-Stokes equations of dimension two and three. The presentation of the material in this book is naturally influenced by many sources. Besides the other papers and books quoted at various parts, we in particular have to mention the books by V. Barbu, I. Miyadera, N. H. Pavel and A. Pazy (see [Bb2], [Bb4], [Miy], [Pav2] and [Pal]). In general we did not try to follow the history of the results presented in this book in detail and to give a complete survey on the relevant literature. This means also that we do not claim authorship for results for which there is no quotation of references. However, we believe that the book contains several components which are not available in the presently existing literature. Specifically we want to mention the following items: the statement and proof of Chernoff's formula for rather general time dependent evolution equations (see Theorem 10.16) in Section 10.2; the treatment of scalar conservation laws in Section 9.2 combining the semigroup approach by M. G. Crandall and the more direct one based on the concept of entropy solutions by S. N. Kruzkov (see [Cr2] and [Kr]); the 'localized' version of the Crandall-Pazy theory for evolution equations in Chapter 7; the results on delay equations of various types presented in Section 9.1 and the treatment of the Navier-Stokes equations in the three dimensional case in Subsection 9.3.2. We tried to give a self-contained presentation of the material as far as possible. In order to serve this purpose we present in the introductory Chapter 1 the results (usually with complete proofs) on dissipative operators, monotone operators and the Minty-Browder theory together with the basic fact on subdifferentials of convex functionals. In the Appendix we collect some of the technical results which are needed in other parts of the book. Whenever we had to quote results from the literature we tried to give precise references. It may be helpful for the reader who wants to give a special course or a seminar on one of the topics of this monograph to have some information on the mutual dependence of the various chapters: a) Chapter 2 can serve as an introduction to the theory of Co-semigroups of bounded linear operators. In order to include the Lumer-Phillips theorem one has to use the results of Section 1.2 specialized to linear operators up to Theorem 1.10. In addition one would need the definition of the duality mapping and the material from Definition 1.2 up to Lemma 1.3. In order to deal with the example of Section 2.5 the definition of Gelfand triples and the LaxMilgram theorem should be taken from Section 3.1. If one adds Chapter 3 to Chapter 2 then one could include the basic fact about analytic semigroups into an introduction to Co-semigroups. If one wants to include also approximation

Vlll

Preface

then Chapter 4 or parts of it (for instance Section 4.1 up to Proposition 4.3 and Section 4.2) can be added. b) Sections 5.1 - 5.3 provide a rather self-contained introduction to the theory of strongly continuous semigroups of nonlinear Lipschitzian operators. Of course one has to include most of the material covered in Sections 1.1 - 1.4. If one wants to include the evolution equation describing nonlinear diffusion of Section 5.4 one needs also material from Sections 1.5 and 1.6. c) If one intends to give an introduction to the Kobayashi-Oharu theory, then Sections 6.1 - 6.5 and 6.7 provide a rather self-contained basis. On could also include Section 6.9. As an application one can add without further requirements the results on delay equations given in Section 9.1. For the material of Sections 6.6 and 6.8 we need in addition results on dissipative and m-dissipative operators. Treatment of the parabolic problem of Section 6.10 requires in addition results from Section 1.6. If one wants to include approximation results one can use the material of Chapter 10 or just the material of Section 10.1. d) In order to present the results of Chapter 7 one only needs in addition the material of Section 1.2 (specifically Theorem 1.10 and Proposition 1.8) together with some basic definitions of Chapter 6 (for instance the definition of DS-approximations given in Definition 6.6). e) For a presentation of the material of Chapter 8 one only needs in addition the results of Section 1.5 and the concept of DS-approximations. f) The material on scalar conservation laws (Section 9.2) and on the NavierStokes equations (Section 9.3) could be used for advanced seminars and require also additional material not presented in this monograph. Both of us have to acknowledge material and immaterial support from various sources during the process of writing this book, which took much more time and efforts than envisaged when we started. Despite all the possibilities offered by modern telecommunication technology it was necessary to meet either in Graz or in Raleigh in order to discuss matters. We gratefully acknowledge travel support offered by the Austrian science foundation FWF and by NSF (USA) in the framework of the Spezialforschungsbereich "Optimierung und Kontrolle" and within the US-Austria Cooperative Science Program. The first author extends his special thanks to Karl Kunisch and his family for the long lasting friendship and the hospitality during his stays in Graz. The second author takes this opportunity to thank H. T. Banks and his wife Sue for many years of friendship and in particular for the hospitality provided to him on the occasion of his stays in Raleigh. Thanks go also to World Scientific for the patience during the period of writing the book and to the staff of World Scientific for handling everything very smoothly and professionally. Finally and last but not least we cannot help to thank our wives for being married to

Preface

ix

more frequently than just occasionally absent minded mathematicians and thus introducing a reasonable amount of normality into their - the mathematicians - life.

Raleigh and Graz, December 2001

Kazufumi Ito Franz Kappel

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Contents Preface

v

Chapter 1. Dissipative and Maximal Monotone Operators 1.1 Duality mapping and directional derivatives of norms 1.2 Dissipative operators 1.3 Properties of m-dissipative operators 1.4 Perturbation results for m-dissipative operators 1.5 Maximal monotone operators 1.6 Convex functionals and subdifferentials

1 1 9 17 25 32 40

Chapter 2. Linear Semigroups 2.1 Examples and basic definitions 2.2 Cauchy problems and mild solutions 2.3 The Hille-Yosida theorem 2.4 The Lumer-Phillips theorem 2.5 A second order equation

57 57 66 74 79 83

Chapter 3. Analytic Semigroups 3.1 Dissipative operators and sesquilinear forms 3.2 Analytic semigroups

89 89 97

Chapter 4. Approximation of Co-Semigroups 4.1 The Trotter-Kato theorem 4.2 Approximation of nonhomogeneous problems 4.3 Variational formulations of the Trotter-Kato theorem 4.4 An approximation result for analytic semigroups

115 115 131 135 141

Chapter 5. Nonlinear Semigroups of Contractions 5.1 Generation of nonlinear semigroups 5.2 Cauchy problems with dissipative operators 5.3 The infinitesimal generator 5.4 Nonlinear diffusion

147 147 153 163 175

Chapter 6. Locally Quasi-Dissipative Evolution Equations 6.1 Locally quasi-dissipative operators 6.2 Assumptions on the operators A(t) 6.3 DS-approximations and fundamental estimates

181 183 186 188

xii

Contents

6.4 6.5 6.6 6.7 6.8 6.9 6.10

Existence of DS-approximations Existence and uniqueness of mild solutions Autonomous problems "Nonhomogeneous" problems Strong solutions Quasi-linear equations A "parabolic" problem

202 208 229 231 238 246 253

Chapter 7. The CrandalHPazy Class 7.1 The conditions 7.2 Existence of an evolution operator

263 263 267

Chapter 8. Variational Formulations and Gelfand Triples 8.1 Cauchy problems and Gelfand triples 8.2 An approximation result

285 285 301

Chapter 9. Applications to Concrete Systems 9.1 Delay-differential equations 9.1.1 Equations of retarded type in the state space C 9.1.2 State dependent delays 9.1.3 Equations of neutral type 9.2 Scalar conservation laws 9.2.1 Basic assumptions and preliminaries 9.2.2 Globally bounded functions 9.2.3 Vanishing viscosity, quasi-dissipativity 9.2.4 L2-considerations 9.2.5 L1- and L°°-estimates 9.2.6 Misestimates 9.2.7 Uniformity with respect to v 9.2.8 The limit v 1 0 9.2.9 The limiting operator 9.2.10 The general case 9.3 The Navier-Stokes equations 9.3.1 The two-dimensional case 9.3.2 The three-dimensional case

305 305 306 313 319 326 326 331 333 338 342 347 354 361 378 385 386 398 403

Chapter 10. Approximation of Solutions for Evolution Equations 10.1 Approximation by approximating evolution problems 10.2 Chernoff's theorem 10.3 Operator splitting

409 409 427 440

Chapter 11. Semilinear Evolution Equations 11.1 Well-posedness

447 447

Contents

11.2 Delay equations with time and state dependent delays 11.3 Approximation theory 11.4 A concrete approximation scheme for delay systems

xiii

458 463 469

Appendix A.l Some inequalities A.2 Convergence of Steklov means A.3 Some technical results needed in Section 9.2

479 479 483 484

Bibliography

489

List of Symbols

493

Index

CHAPTER 1

Dissipative and Maximal Monotone Operators In this chapter we present the basic facts on dissipative resp. monotone operators in Hilbert or Banach spaces which will be used in the following chapters. Of particular interest are results on m-dissipative and maximal monotone operators (Section 1.3 resp. 1.5). We include also perturbation results on mdissipative operators (Section 1.4) and on subdifferentials of convex functionals (Section 1.6). Whenever it is possible we give self-contained proofs and refer to the literature only for results which are outside the scope of the book. All results are taken from the large number of texts available on the subjects discussed in this chapter (see for instance [Bb2], [Bb4], [Brl], [Bro], [Crl], [Cr-Pa], [Miy], [Pav2], [Pal], [Ro2]).

1.1.

Duality mapping and directional derivatives of norms

In order to introduce the concept of a dissipative operator we need the notion of the duality mapping between a Banach space and its dual. Let X be a Banach space with norm | • |, X* be the dual space and denote by (•, •) the pairing between X and X*, i.e., (x,x*)=x*(x),

x£X,x* X* defined byx-> Fx is demi-continuous, i.e., is continuous as a mapping from X with the norm topology to X* with the weak*-topology. 1

We set B*(0, |x|) = {x* G X* | |x*| < |a;|}. 1

Chapter 1. Dissipative and Maximal Monotone

2

Operators

c) / / X* is uniformly convex, then, for all x G X, Fx is single-valued and the mapping x —> Fx is uniformly continuous on bounded subsets of X. d) If X is a Hilbert space and we identify X* with X, then Fx = {x} for all x G X, i.e., F is the identity mapping on X. Proof, a) Closedness of Fx is an easy consequence of the definition. Obviously we have Fx C B*(0,\x\). Choose x*,y* G Fx and a G (0,1). For arbitrary z e X we have (using ja;*| = \y*\ = \x\) {z,ax* + (1 - a)y*) < a\z\ \x*\ + (1 - a)\z\ \y*\ = \z\ \x\, which shows \ax* + (1 - a)y*\ < \x\. Using (x, a;*) = (a;, y*) = \x\2 we get (a;, ax* + (1 - a)y*) = a(x, x*) + (1 - a)(a;, y*) = \x\2, so that \ax* + (1 - a)y*\ = \x\. This proves ax* + (1 - a)y* G Fx. b) Choose x*,y* G Fx, a G (0,1), which by a) implies \ax* + (1 — a)j/*| = \x\. Since X* is strictly convex (i.e., balls in X* are strictly convex), we get x* = y*. Let (xn)neN C X be a sequence with a;n —• x G X. From |Fa;„| = \xn\ and the fact that closed balls in X* are w*-compact (by the Banach-Alaoglu theorem; see for instance [Wi, Theorem 9-1-12 on p. 130]) we see that there exists a w*-accumulation point y* of (Fxn)ne^. The sequence (xn)ne^ and x generate a separable subspace of X. Therefore we can assume without restriction that X itself is separable. Using the fact that for separable spaces the w*-topology on w*-compact subsets of X* is metrizable (see for instance [Wi, Theorem 9-5-3 on p. 143]) we conclude that there exists a subsequence (xnk)keN such that wMinifc-yoo Fxnk = y*. This implies lira (xnk,Fxnk)

= (x,y*).

K—tOO

From limk^.oo(xnk,Fxnk) = lim^oo |a: n J 2 = \x\2 we get (x,y*) = \x\2, which implies |a;| < \y*\. In order to prove the converse inequality we use the fact that Fxnk —*• V* implies \y*\ < liminf \Fx„k\ = lim \xnk\ = |a;|. fc—>-oo

k—foo

Thus we have \y*\ = \x\ and {x,y*) = \x\2, i.e., y* = Fx. Therefore we have w*-limn_+00 Fx„ = Fx. c) Since a uniformly convex Banach space is also strictly convex, the duality mapping F is single-valued by statement b). Assume F is not uniformly continuous on bounded subsets of X. Then there exist constants M > 0, «o > 0 and sequences (un), (vn) C X satisfying \un\<M,

|t>„|<M,

\un — v„\ —¥ 0

ra=l,2,...,

as n —>• oo,

\Fun - Fvn\ > Q 0 ,

n - 1,2,,... .

1.1. Duality mapping and directional derivatives of norms

3

Without restriction of generality we can assume that, for a constant /3 > 0, we have in addition \un\ > 0,

\vn\ >(3,

jz = 1,2,... .

Indeed, if for instance (un) has a subsequence (unk) such that unk —> 0 as k —> oo, then also vnk -» 0. By definition of the duality mapping this implies Funk —> 0 and Fvnk —>• 0 as k —• oo, a contradiction. We set a;n = u „ / | u n | and j / n = w n /|v n |. Then we have I \Fxn + Fyn\ > (xn, Fxn + Fyn) which together with (xn,Fxn

+ Fyn) = | z n | 2 + \yn\2 + = 2+ (xn -yn,Fyn)

{xn-yn,Fyn) > 2 - |x„ - y „ |

implies (1.1)

lim \Fxn +

Fyn\=2.

n—>oo

Suppose there exists an eo > 0 and a subsequence such that \Fxnk ~FyHk | > eo for k — 1,2,... . Observing \Fxnk\ = \Fynic\ = 1 and using uniform convexity of X* we conclude that there exists a (5o > 0 such that \Fxnk +Fynk\
• 0 as n -¥ oo.

This contradiction proves the result. d) Let X be a Hilbert space with inner product (-,-). For x £ X we choose / s Fx. Then we have | / | 2 = \x\2 = (x,f) and consequently \x — f\2 = (x-f,x-f) = \x\2 - 2 R e ( x , / ) + | / | 2 = 0, i.e., / = x. D 1.1. Remark. By the proof for statement b) of Proposition 1.1 we have also established the following result: Let X be a Banach space and (xn)neN C X be a sequence with i „ - i i £ X. Then, for any sequence (a;* )nen C X* with x*n G Fxn, n = 1,2,..., there exists a subsequence (a^fc)fceN and an element x* £ Fx such that w*-lim x*, = x*.

4

Chapter 1. Dissipative and Maximal Monotone

Operators

We next consider directional derivatives of the norm | • | in X and define pairings between elements of X which to some degree are analogues to the inner product in a Hilbert space: 1.2. Definition. The functions (•,•)+: X x X ->• K and (-, •)_ : X x X ->• R are defined by |a; + ay| — | i | (y,x)+ = lim '" ' " y | —, x,yeX, aJ-0

(y,x)- = hm N ' afo Furthermore, we set

"' J \x + ay\! - !\x\

— —=hmJ—!—! a «4.o a

(y,^)s = \x\(y,x)+

and

-,

(y,a;) i = \x\(y,x)_,

x.y G X.

x,y e X.

The limit in the definition of (y, x)+ exists, which can be seen as follows. First, we note that a -* a~1(\x + ay\ — \x\) is increasing on a > 0. This follows from (0 < a < (3, x,y £ X) (P - a)\x\ = |(fix + afiy) - (ax + a/3y)\ > (3\x + ay\ - a\x + /3y\, which implies I a: + 0y\ - \x[

0

\x + ay\ - \x\

~

a

From a~r(\x + ay\ — |x|) > —\y\ we see that this function is bounded below. Hence, ,.„, (1.2)

, < ,. \x + ay\-\x\ (y, x)+ = hm alO

\x + ay\-\x\ !• = inf * ^—L-l

a

a>0

-|• R is lower semi-continuous resp. upper semi-continuous2. Proof, a) The result for (•, -) + follows by taking the limit h \. 0 in the following relation: - ( \ x . + h(ax + y\ - \x\\ = -(|(1 =

T

h\

+ ah)x + hy\ - \x\)

((1 + ah)\x + Ty\ - (1 + ah)\x\ + ah\x\) I 1 + ah I /

= (T^YX\x + TTc*y\-lx])

+ alxl a>0

-

The result for (•,•)- is clear from (1.4). 2 For the definition of 'lower semi-continuous' see Definition 1.41, b). A real-valued function / is upper semi-continuous if —/ is lower semi-continuous.

Chapter 1. Dissipative and Maximal Monotone Operators b) The result for (y + z, x)+ follows from \(\x

+ h{y

+

1 . I + 1 z)\-\x\)• Y is given, then its graph GA = {{x, y)eXxY\x€ dom A and y G Ax} is a subset of X x Y. Conversely, a subset G C X xY defines an operator AQ : X —> Y (which in general is multivalued) by (i) dom AG = {x G X | There exists a y G Y such that (a;, y) G G} and (ii) y G AGX if and only if (x, y) G G. Henceforth we shall always write [x, y] G A instead of (a;, y) G A in order to indicate that A C X x y is viewed as an operator. If A is an operator X —» Y, we define the domain of A by dom A = {x G X | Ax / 0} and the range of A by ranged = \JxedomAAx. The inverse operator A"1 : Y ->• X is denned by [a;, y] G A - 1 if and only if [y, x] G A. Obviously, we have ( A - 1 ) - 1 = A. For two operators A , B : X ->• F and A e R resp. G C we define XA = {[x,\y\ \ [x,y] G A} and A + B = {[a:, j/ + 2] | [x, 2/] G A and [a;, z] G 5 } . It is clear that domXA = dom A, dom(A + B) = dom A n domB and dom A - 1 = range A. 1.5. Definition. An operator A on X, i.e., A C X x X, is called dissipative if and only if, for any [XJ, j/i] G A, i = 1,2, there exists an a;* G -F(a;i — X2) such that (1.9)

Re(yi-y2,x*) 0. (iii) {y,x)_ < 0 . (iv) (y,x)i (ii). Let x ^ 0 and choose / G Fx with Re(j/, / } < 0. Then, for any A > 0, \x\2 = {x, f) = Re{x -Xy + Xy, f) = Re(a; - Ay, / } + A Re(y, f) 0 and[xi,yi] G A, i — 1,2. (ii') There exists a Ao > 0 suc/i i/iai |ari — x2 — A(yi — y 2 )| > |#i — £2! / 0 such that

\xi -x2-

\{yi - i/2)| > (1 - Xu)\xi - x2\

for all A G (0, A0] and [xi, yt] G A, i — 1,2. (iii) (2/1 -yi,x\ -x2) \xi - x2\

for all A > 0 and \xi, Zi] G A — oil, i = 1,2. Since z £ {A — ioI)x if and only if z = y — u)x with some y G Ar, the last inequality is equivalent to (1.11)

| ( 1 + A w ) ( x i - a : 2 ) - A ( j / i - 2 f c ) | > \xi - x2\

for A > 0, [xi, j/i] £ 4 , i = l,2. It is easily seen that this is equivalent to (1.12)

\xi -x2

-//(j/i - 2/2)1 > ll-jU^Hxi -x2\,

[xi,yi]£ A, i = 1,2,

where 0 < fi < 1/w in case w > 0 and fi G (0, CXD) (J (—00, l/o;) in case a; < 0. Therefore it is obvious that (ii) of Corollary 1.7 implies statement (ii). If statement (ii') is valid, we can assume that (1.12) is valid with a [i G (0, 1/|CJ|), which implies (1.11) for some Ao > 0. • If A is an w-dissipative operator on X, then in view of Proposition 1.8 we restrict ourselves to A G (0, l/|w|). A first important consequence from dissipativity is given in the following proposition: 1.9. Proposition. Let A be an UJ-dissipative operator on X. Then, for any A G (0, l/|w|), the operator (I — XA)_1 is single-valued and, for any x,y G range(7 - \A) and A G ( 0 , 1 / M ) ,

I(/ - \A)~lx - (I - \A)~ly\ < YZJ^\X ~ V\Proof. Choose x\,x2 G dom A, 0 < A < l/\u\ and Zi e (I - XA)xt, i = 1,2. Then we have zi = Xi — Xyt for some j/i G Axi, i = 1,2. By Corollary 1.7, (ii), we get \zi - z2\ = \xi - x

2

- A(j/i - ?/2)| > (1 - Xw)\xi - x2\.

For x\ ^ x2 this implies (i" - XA)x\ C\ (I — AA)a;2 = 0- Therefore, for any y G range(7 — XA), we have (I — \A)~ly = x, where x is the unique element with y G (7 - XA)x. •

12

Chapter 1. Dissipative

and Maximal Monotone

Operators

We define the resolvent J\ resp. the Yosida approximation A\ of an wdissipative operator A by J\x = (I - XA)~lx G dom A, A\x = A _1 ( J\x — x)

for all x G dom J\ — range(7 - XA).

For an operator A on X we define (1.14)

||AE|| =inf{|j/| | ye Ax},

a; G dom A

Some fundamental properties of J\ and A\ are summarized in the following theorem: 1.10. Theorem. Let A be an u>-dissipative operator on X. A,/i < l/|w|, the following is true:

Then, for 0
J ~ 0 ~ 2 5Z^2 a i O L i Re(vi>ui - ui)

»=1 j = l n n =

o 5> ^

i=l j = l QiQ: , R e

t=i j = i

n

w

•

^ J ~~ V i ' U j ~ Ui^'

Since A is dissipative and [«», t>i] G A, i = 1 , . . . , n, we have Re(uj — Uj, Mj — Uj) < 0,

i, j = 1 , . . . , n .

Thus we have shown that V>(a°,/3) 0,

i—

l,...,n.

This proves that x(a°) G X ^ . i ^ ) , i — 1 , . . . , n, i.e., (1.19) is true.

1.3.

•

Properties of m-dissipative operators

In this section we collect some of the most important properties of mdissipative operators. 1.14. Lemma. Assume that X* is strictly convex and that A C X x X is maximal dissipative. Then, for all x G dom A, Ax is a closed convex subset of X. Proof. By Proposition 1.1, a), we see that the duality mapping F is singlevalued. In order to show that Ax is convex we choose yt, j / 2 G Ax and a G [0,1]. Since A is dissipative we have, for all [x, y] G A, Re(m/! + (1 - a)j/ 2 -y,F(x-x))

= aRe(j/i - y,F(x - x)) +

{l-a){y2-y,F{x-x))• 0, w-lim^oo y„ = y and [xn, yn] G A, n = 1,2,..., imply [a:, y] G A. 1.16. Theorem. Assume that A is an m-dissipative operator on X. Then the following is true: (i) The operator A is closed. (ii) If x\ —• x and A\X\ -> y as \ \.Q for a family [x, y] G A.

(XA)O ?/ as n ->• oo. By dissipativity of A we have (compare Corollary 1.7) (yn — y,xn - x)- < 0 for all [x,y] G A. Since {•, •)_ is lower semi-continuous, we obtain

(y-V,x-x)-

< \immi(yn-y,xn

-x)_ 0 for all [it, v] G A. By maximality of A we conclude that [a;, a;*] G A. Since x G P was arbitrary we have shown that dom A — P.

1.5. Maximal monotone

operators

35

The set A\ = {[#, y + Bx] \ [x, y] £ A} is monotone and we have dom A\ — domA = P. According to Lemma 1.35, a), with M = P and x* = 0 there exists a n i E P such that (u - x, w) > 0

for all [u, w] £ Ax

or, equivalently, (1.48)

{u-x,v

+ Bu)>0

for all [u, v] £ i .

We fix [u, v] e A From u,x £ P and convexity of P we see that wt = x + £(M — x) £ P = domA for all t G [0,1]. Moreover, for any t e [0,1] there exists a i>t such that [ut,vt} £ A. Inequality (1.48) implies 0 < {ut - x,vt + But) = t(u-

x,vt + But),

t e [0,1],

and consequently (1.49)

(u-x,vt

+ But)>0,

0,

iG[0,l).

This inequality and (1.49) show that {u-x,v

+ B(x + t(u -x)))

>0,

te (0,1).

Hemi-continuity of B implies, for t J. 0, (u — x, v + Bx) > 0 for all [u, v] £ A, which proves the result for finite A. Note that this part of the proof did not use the assumption 0 £ domA. b) Let A satisfy the assumptions of the theorem and denote by A? a monotone extension of A which is maximal in K x X*, A C A-z C K x X*. We define Q = {G C AI I G is monotone and finite with 0 £ domG}. For each G £ Q we set BG = {[x,Bx] \x£

K, (u-x,v

+ Bx) > 0 for all [u,v] £ G}.

Part a) of the proof shows that BQ is not empty. For x £ dom BQ we have (x,Bx)

< {u,Bx) - (x,v),

[u,v] £ G.

We choose u = 0 and v £ GO, which implies —Ax,Bx)

< \v\

for all x £ dom Be-

36

Chapter 1. Dissipative and Maximal Monotone

Operators

From (1.46) with xo = 0 we see that domBc is bounded. This implies that BG is bounded, because B maps bounded sets onto bounded sets. Since X is reflexive, boundedness of BQ implies that the weak closure BQ of BQ is weakly compact. From BQX n BQ2 = BQ1UG2, GI,G% G Q, we see that for any finite number G\,..., Gn of sets in Q we have pl?=i BGi =£ 0- Note that, for G\,G2 G Q, the finite operator G\ U Gi is monotone, because G\ U G 0 for all [u, v] G A 2 .

We claim that [xo, — XQ] € ^ 2 - To this end we remind the reader that [x, Bx] G Bg if and only if (1.51)

(u -x,-v-

Bx) < 0 for all [u, v] G G.

For [xi, Bxi] G BG, i = l,...,n, and Xi > 0, i = 1 , . . . , n, with JZ™=1 A* = 1 we set [x,y] = Yl7=i ^i[xi,Bxi\. Then easy computations show that n

n

- ^jAjXj, - v — } i=l 1

n

XjBxj)

j=l n n

n

— ^ ^ A i A j ( w - x^ —v - BXJ) i—1 j=l

= r 5Z 5Z ^^' ((u ~ xi' ~v ~ Bx^

+

(xi ~ Xi>Bxi ~ Bx^

+ (u — Xi, —v - Bxi) J < 0. Here we have used (1.51) and monotonicity of B. Thus we have shown that (u — x, -v - y) < 0 for all [x, y] G co .BG and [u, v] G G. It is clear that this inequality is also true for all [x,y] G COBQ- Since [XO,XQ] G BQ C c o B G — coBG, we have proved that (1.52)

{u - x 0 , -v - XQ) < 0 for all [u, v] G G.

Obviously (1.52) is true for all G G Q. Therefore A2 = \JGeg G implies (u — xo, v + XQ) > 0 for all [u, v] G A%. This implies [xo, —x^] G A^, because A-z is maximal monotone in K x X* and x 0 G K. We choose w G A2O and define, for any [u, v] G A2, the set Go G Q by Go = {[0,U/],[M,U],[X 0 ,-XO"]}.

1.5. Maximal monotone

operators

37

For any x G dom BG0 we have in particular (1.53)

(u - x, v + Bx) > 0,

(x0 - x, -x*0 + Bx) > 0.

We set ut = Xo + t(u — xo) G K, t G (0,1). Observing monotonicity of B and ut - x = (1 — t)(x0 — x) + t(u — x) we get, for t 6 (0,1), 0 < {ut - x, But - Bx) = t(u - x, But - Bx) + (1 - t)(x0 - x, But -

Bx).

This together with (1.53) implies 0 < t(u-x,v

+ But) + (1 -t)(x0-x,

-xl + But),

i G (0,1), x £

domBGo.

For ar = xo this gives (w-a;o,w + B(a;o + t(u-a;o))) > 0,

i € (0,1), [u,v] 0 for all [u, v] e A^,

which is (1.50).

•

The next theorem characterizes maximal monotone operators in real Banach spaces4 by a range condition (see [Min] for the Hilbert space case which is Theorem 1.12, c), and [Bro] for the general case): 1.37. Theorem. Assume that X, X* are reflexive and strictly convex. Let F denote the duality mapping of X and assume that A c X x X* is monotone. Then A is maximal monotone if and only if (1.54)

range(AF + A) = X*

for all A > 0 (or, equivalently, for some A > 0). Proof, a) Assume that (1.54) is satisfied for some A > 0 and let [a:o,2/o] € X x X* be such that (1.55)

(x - xo, y - yo) > 0 for all [x, y] G A.

From (1.54) we see that there exists an element [xi,yi] G A with (1.56)

AFm + yi = XFx0 + y0.

From (1.55) and (1.56) we obtain, for [x, y] = [xi,yi], (Xi - X0, FX0 - Fxi) > 0. By monotonicity of F we have also the converse inequality, so that (xi - x0, Fxi - Fx0) = 0. 4

Note that throughout this section we assume that X is real.

38

Chapter 1. Dissipative and Maximal Monotone

In view of (1.47) this implies t h a t |Xo| = | x i | a n d (xi,Fxo) | ceo j 2 - The last two equations imply FXQ = Fx\ and (xuFx0}

= (x0,Fx0}

— \xi\2,

Operators

{XQ,FX\)

—

= \x0\2 = \Fx0\2.

If we denote by F* the duality mapping of X* (which is also single-valued), then the last equation implies XQ = x\ = F*(Fxo). This and (1.56) imply that [£0)2/0] = [xi)2/i] ^ A, which proves that A is maximal monotone, b) Assume that A is maximal monotone and that 0 G dom A. We fix x* G X*, A > 0 and set K — X. The operator B defined by Bx = \Fx — x*, x G X, satisfies the assumptions of Theorem 1.36. Note that demi-continuity implies hemi-continuity. Therefore this theorem implies that there exists an x G X such that (1.57)

(u - x, XFx - x* + v) > 0 for all [u, v] G A.

Since A is maximal monotone, this implies [x,x* — XFx] G A, i.e., x G domj4 and x* G Ax + XFx. Thus we have range(A + XF) — X*. If 0 ^ dom A, we choose tin G dom A and define the operators A= {[u,v] 6 l x l *

[u + Uo,v]

GA}

and Bx — XF(x + «n) — x*, x G X. It is easy to see that A and B satisfy the assumptions of Theorem 1.36 for K = X. Consequently there exists an x G X with (u - x, XF(x + UQ) - x* + v) > 0 for all [u, v] G A, which is equivalent to (1.57) with x = x + UQ.

D

A consequence of Theorems 1.36 and 1.37 is the following corollary: 1.38. Corollary. LetX be a reflexive Banach space and A c 1 x 1 * be a maximal monotone operator. Furthermore assume that the operator B : X —> X* is monotone, hemi-continuous and bounded (i.e., rangeB is bounded). Then A + B is maximal monotone. Proof. By Asplund's renorming theorem 5 we can assume that X and X* are strictly convex. Obviously monotonicity and maximality are invariant with respect to changes to equivalent norms. We can also assume that 0 G dom A. Otherwise we choose UQ G dom A and introduce the operators A = {[u, v] G X x X* | [u + u0, V] G A} and Bx = B(x + u0), x G X, which satisfy also the 5 See [Asp, Theorem 2], where it is stated that on a reflexive Banach space there exists an equivalent norm which is rotund (i.e., strictly convex) and smooth (i.e., the dual norm on X* is strictly convex). Another reference is [Die, Corollary 2 on p. 148], where it is proved that any weakly completely generated Banach space - reflexive Banach spaces are such spaces - has an equivalent norm such that X and X* are strictly convex.

1.5. Maximal monotone

operators

39

assumptions of the theorem. Moreover, A + B is maximal monotone if A + B is. We choose x* £ X* and define the operator B : X —» X* by Bx = Bx + Fx - x*,

x £ X.

It is easy to see that B satisfies the assumptions of Theorem 1.36 for B. We set K = co(dom A) and get from Theorem 1.36 that there exists an x G K such that (u - x, Fx + Bx - x* + v) > 0 for all [u, v] £ A. Since A is maximal monotone, this implies that x* £ Ax + Bx + Fx — (A + B)x + Fx, i.e., we have range(A + B + F) = X*. By Theorem 1.37 the result follows. • Since A = {[u, 0] G X x X* | u € X} is maximal monotone, we get from Corollary 1.38 that a bounded, monotone and hemi-continuous operator is maximal monotone. The next result shows that boundedness is not necessary. 1.39. Theorem. Let X be reflexive. a) / / the operator A : X -* X* is monotone and hemi-continuous, then A is maximal monotone. b) If A 0 for all x G X.

We choose x G X and set xt = txo + (1 — t)x, 0 < t < 1. Then we have 0< {xt- x0, Axt - y0) -(l-t)(xx0, Axt - y0) so that (x - x0, Axt - y0) > 0

for all x £ X, 0 < t < 1.

Hemi-continuity of A implies, for 111, (1.59)

(x - x0, Ax0 - yo) > 0 for all x G X.

This implies AXQ = yo> which contradicts the assumption that A is not maximal monotone. b) We choose XQ G X*. By Asplund's renorming theorem we can assume that X and X* are strictly convex. Prom Theorem 1.37, for any A > 0, there exist an x\ £ dom A and & y\ £ Ax\ with (1.60)

x*0 = yx + XFxx.

40

Chapter 1. Dissipative and Maximal Monotone

Operators

Let XQ £ X be an element such that (1.46) is true. Then we get from (1.60) that {x\ - X0,XQ) = {x\ - x0,yx) + X\xx\2 - X(x0,Fxx) resp. (x\-x0,y\)

, .. , (x\ -X0,XQ) + X\xx\ = ; i \xx\

\x\\

x

(x0,Fx\) h Ax

\x\\

x

^ \ o\ + I—i\ o\ \ o\ + A|z 0 |.

\xx\

Coercivity of A implies that \x\\ is bounded as A 4- 0. Therefore there exists a subsequence {x\n) and an XQ € X such that x0 = w- lim

x\n.

n—too

From (1.60) and boundedness of |:EA| as A 4- 0 we get, taking A = An and n —> oo, XQ

= lim yx„. n—too

Using monotonicity of A we have, for any [u, v] G A, (xXtl -u,yXn

-v)

> 0,

n=l,2,....

Taking n —^ oo we have (£o — u, xl — v) > 0 for all [u, v] e A, which implies [io^o]

e

A i-e-' ^o

G

range A, because A is maximal monotone.

• 1.40. Corollary. Let X be reflexive. Then the duality mapping is maximal monotone and range F = X*.

1.6.

Convex functionals and subdifferentials

Important examples of maximal monotone operators are subdifferentials of lower semi-continuous convex functionals. Therefore we present in this section the basic properties of such functionals. As in the previous section we assume also for this section that the Banach spaces considered are real. 1.41. Definition. Let

- 0 0 for all x € X. The set ~Deff(ip) = {x € X I (p(x) < 00} is called the effective domain of

.

1.6. Convex functionals

and

subdifferentials

41

b)

0 there exists a y\ such that lim^o \y\\ = 0 and x + Ay + Ay* G # .

1.6. Convex functionals

and subdifferentials

47

Then for any x* G (9ix(x) we have (v + V\,x*)0,

which for A 4.0 implies (y,x*)0 for all x G X or, equivalently, (1 66)

^

~ ^X^

" ^ ~ X °' X o) + 21 210 ' 2 ~ 2 ^ > (x — x0,XQ) — (x — xo,Fx),

x G X,

where we have also used the fact that F is the subdifferential of x —» \x\2/2 (see Example 1.51). For arbitrary u G X let xt = x0 + t(u — Xo), 0 < t < 1. Then we get from (1.66) and convexity of ip p{u)

-P(XQ)

> -(p(xt)-p(x0)) = (u -

XQ, XQ)

> -(xt-xo,x*0) - (u - X 0 ,

Fxt).

- -{xt

-x0,Fxt)

Chapter 1. Dissipative and Maximal Monotone

48

Operators

Observing that F is demi-continuous (see Proposition 1.1, b)) we get, for 11 0, the inequality (u — XQ,XQ — FXQ) for all u £ X, which proves XQ - Fx0 G dip(x0), i.e., XQ G {F + dip)x0. • 1.54. Proposition. Let

0 such that [u,v] G dtp and \v\ < a imply \u\ < /3 (i.e. dom(dip)-1 = X* and (dtp)-1 maps bounded subsets of X* onto bounded subsets of X). Proof. Assume first that ip is radially unbounded and let [x„,x*] G dip, n = 1,2,..., be a sequence with limn-^oo |x„| = oo. By definition of dip we have (y — xn, x*) for all y £ X. We choose a y0 G Deg(.oo(^n - yo,x*n)/\xn\ = oo. Assume now that dip is coercive and choose a > 0. Since dip is maximal monotone (see Theorem 1.53) we get from Theorem 1.39, b), that range dip = X*. Assume that there exist [u n ;^n] G dip such that \vn\ < a for all n and linin^oo \un\ = oo. Then the estimate (un - x0,vn) i—j \un\

\un - x 0 | < —;—:—a —> a \un\

as n -4 oo

gives a contradiction to (1.46). Therefore (iii) holds. Finally we assume that (iii) is true. Let (j/n)neN C X be a sequence with lim„_4.oo \yn\ = oo- For fixed a > 0 we define the elements vn = a|j/„| _ 1 Fyn, n = 1,2,..., where F is the duality map X -> X*. Then for any n there exists a un G X with [un,vn] G dip. Since \vn\ — a, we get |u n | < (3, n — 1,2,... . For j/o G X we have ip(y0) - ip(un) > (y0 - un, vn), which implies ip(un) < ip(y0) + a(3+ \yo\a, n = 1,2,... . This and Lemma 1.44 together with |u„| < (3, n = 1,2,..., prove that the sequence (ip(un)) N is bounded. From the definition of dip we get tp(yn) - ip(un) > {yn - un, vn) = a\yn\ - (un, vn) > <x\Vn\ ~ a0, which implies jg(jfa) ^ ip(u) - a/3 + |j/n|

n =

1 2

|j/n|

For n -> oo we obtain liminfn_+00 v?(y n )/|j/ n | > a. Since a was arbitrary, we get the result. •

1.6. Convex junctionals

and

subdifferentials

49

1.55. Theorem. Assume that X is a real Hilbert space with inner product (•, •) and that A is a maximal monotone operator on X. Let if be a proper, convex and lower semi-continuous functional on X satisfying dom ACidomdip ^ 0 and (1.67)

0, x e D eff (p),

where M is some non-negative constant. Then the operator A + dip is maximal monotone and6 \A°x\ < \(A + dip)°x\ + M1/2,

Z € dom,4 nDeff()•

Proof. According to Theorem 1.53 the operator dp is maximal monotone on X, so that —A and — dip are m-dissipative. By Lemma 1.19 the minimal section A0 of A is single-valued and dom A 0 = dom A (note that dom A0 = dom(-A) 0 and (—A)°x = — A°x, x £ dom A 0 ). According to Lemma 1.23 there exists, for any A > 0 and any y £ X, an element xx £ domdip such that y£xx-

(-A)xxx

+ dip{xx).

Moreover, \xx\ is bounded on A > 0 by Theorem 1.24, a). Observing y — xx + {-A}xXx £ dip(xx) we obtain, for any z £ X, (1.68)

ip{z) - ip(xx) >(z-xx,y-xx l

We take z — (I + \A)~ Xx, (1.68) \({-A)xxx,

y-xx

+

(~A)xxx).

so that z — xx = A(—A)xXx-

+ {-A)xxx)

< ip{{I + XA^xx)

Then we get from

- ip(xx) < AM,

A > 0,

and 1(-A)xxx\2

< \(-A)xxx\

\y - xx\ + M.

Since \xx\ is bounded on A > 0, the last estimate shows that |(—A)xXx\ is also bounded on A > 0. According to Theorem 1.24 the operator —A - dip is m-dissipative or, equivalently, A + dip is maximal monotone. For x € Deff( 0, because |^4°ar| = inf ze ^ x \z\ and Ax is 6

Recall that A0 denotes the minimal section of A (see Definition 1.18).

50

Chapter 1. Dissipative

and Maximal Monotone

convex (see Lemma 1.14). Therefore we have (A°x,yi) we get

Operators

> |yl0a;|2. Using (1.69)

(A°x, y) = (A°x, yi) + (A°x, y2) > \A°x\2 - M and |,4°:r| 2 < \A°x\\(A + d