Co-Semigroups and Applications

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ISBN: ISSN:

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To my wife Veronica

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Contents

Preface

xi

C h a p t e r 1. P r e l i m i n a r i e s 1.1. Vector-Valued Measurable Functions 1.2. The Bochner Integral 1.3. Basic Function Spaces 1.4. Functions of Bounded Variation 1.5. Sobolev Spaces 1.6. Unbounded Linear Operators 1.7. Elements of Spectral Analysis 1.8. Functional Calculus for Bounded Operators 1.9. Functional Calculus for Unbounded Operators Problems Notes

1 1 4 9 12 15 20 24 27 31 33 34

C h a p t e r 2. S e m i g r o u p s of Linear O p e r a t o r s 2.1. Uniformly Continuous Semigroups 2.2. Generators of Uniformly Continuous Semigroups 2.3. C0-Semigroups. General Properties 2.4. The Infinitesimal Generator Problems Notes

35 35 38 41 44 48 50

C h a p t e r 3. G e n e r a t i o n T h e o r e m s 3.1. The Hille-Yosida Theorem. Necessity 3.2. The Hille-Yosida Theorem. Sufficiency 3.3. The Feller-Miyadera-Phillips Theorem 3.4. The Lumer-Phillips Theorem 3.5. Some Consequences 3.6. Examples 3.7. The Dual of a C0-Semigroup 3.8. The Sun Dual of a C0-Semigroup

51 51 54 56 58 61 63 67 70

vii

viii 3.9.

Contents

Stone Theorem

Problems Notes Chapter 4. 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.

The The The The The The The The The

Differential Operators Generating C0Semigroups Laplace Operator with Dirichlet Boundary Condition Laplace Operator with Neumann Boundary Condition Maxwell Operator Directional Derivative SchrSdinger Operator Wave Operator Airy Operator Equations of Linear Thermoelasticity Equations of Linear Viscoelasticity

Problems Notes Chapter 5. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

Approximation Problems and Applications

The Continuity of A ~ e t A The Chernoff and Lie-Trotter Formulae A Perturbation Result The Central Limit Theorem Feynman Formula The Mean Ergodic Theorem

Problems Notes Chapter 6. 6.1. 6.2. 6.3. 6.4. 6.5.

Some Special Classes of C0-Semigroups

Equicontinuous Semigroups Compact Semigroups Differentiable Semigroups Semigroups with Symmetric Generators The Linear Delay Equation

Problems Notes Chapter 7. 7.1. 7.2. 7.3. 7.4.

Analytic Semigroups

Definition and Characterizations The Heat Equation The Stokes Equation A Parabolic Problem with Dynamic Boundary Conditions

72 74 75

77 77 83 84 87 90 91 95 96 98 101 103 105 105 110 113 114 117 121 126 127 129 129 133 137 144 147 149 150 151 151 156 162 166

Contents 7.5. An Elliptic Problem with Dynamic Boundary Conditions 7.6. Fractional Powers of Closed Operators 7.7. Further Investigations in the Analytic Case Problems Notes

C h a p t e r 8. T h e N o n h o m o g e n e o u s Cauchy P r o b l e m 8.1. The Cauchy Problem u' = Au + f , u(a) = 8.2. Smoothing Effect. The Hilbert Space Case 8.3. An Approximation Result 8.4. Compactness of the Solution Operator from LP(a, b ; X ) 8.5. The Case when ( ~ I - A) -1 is Compact 8.6. Compactness of the Solution Operator from L 1(a, b; X) Problems Notes Linear E v o l u t i o n P r o b l e m s w i t h M e a s u r e s as Data 9.1. The Problem du = { A u } dt + dg, u(a) = 9.2. Regularity of L~ 9.3. A Characterization of L~ 9.4. Compactness of the L~ Operator 9.5. Evolution Equations with "Spatial" Measures as Data Problems Notes

ix 168 170 177 180 181 183 183 189 192 193 197 199 202 204

C h a p t e r 9.

205 205 210 213 216 220 223 225

C h a p t e r 10. S o m e N o n l i n e a r C a u c h y P r o b l e m s 10.1. Peano's Local Existence Theorem 10.2. The Problem u~= f (t, u ) + g(t, u) 10.3. Saturated Solutions 10.4. The Klein-Gordon Equation 10.5. An Application to a Problem in Mechanics Problems Notes

227 227 231 236 242 245 247 248

Chapter 11.1. 11.2. 11.3. 11.4. 11.5. 11.6.

249 249 253 255 261 264 265

11. T h e C a u c h y P r o b l e m for S e m i l i n e a r E q u a t i o n s The Problem u' = A u + f ( t , u) with f Lipschitz The Problem u ~ - A u + f (t, u) with f Continuous Saturated Solutions Asymptotic Behavior The Klein-Gordon Equation Revisited A Parabolic Semilinear Equation

x

Contents

Problems Notes

267 268

Chapter 12. Semilinear Equations Involving Measures 12.1. The Problem du = { A n } dt + dg~ with u ~ gu Lipschitz 12.2. The Problem du - { A n } dt + dgu with u ~ gu Continuous 12.3. Saturated L~-Solutions 12.4. The Case of Spatial Measures 12.5. Two Examples 12.6. One More Example Problems Notes

269 269 273 276 282 284 286 288 290

Appendix A. Compactness Results A.1. Compact Operators A.2. Compactness in C([a, b]; X) A.3. Compactness in C([a, b]; Xw) A.4. Compactness in LP(a, b ; X ) A.5. Compactness in LP(a, b ; X ) . Continued A.6. The Superposition Operator Problems Notes

291 291 295 301 304 308 312 315 318

Solutions

319

Bibliography

361

List of Symbols

368

Subject Index

371

Preface

This book is an entirely rewritten English version of the lecture notes of an advanced course I taught during the last eleven years at the Faculty of Mathematics of "A1. I. Cuza" University of Ia~i. Lecture notes appeared in 2001 in Romanian. The idea was to give a unified and systematic presentation of a fundamental branch of operator theory: the linear semigroups. The existence of several very good books on this topic such as: Ahmed [2], Belleni-Morante [24], Butzer and Berens [32], Davies [45], Engel and Nagel et al [51], Goldstein [61], Haraux [68], Hille and Phillips [70], McBride [89], and Pazy [101] made this task very hard to accomplish. Nevertheless, I decided to accept it, simply because there are several particular topics which have not found their place into a monograph until now, mainly because they are very new. This book, although containing the main parts of the classical theory of C0-semigroups, as the Hille-Yosida theory, illustrated by a wealth of applications of both traditional and non-standard mathematical models, also includes some new, or even unpublished results. We refer here to: the characterization in terms of real regular values of both differentiable and analytic semigroups, the study of elliptic and parabolic systems with dynamic boundary conditions, the study of linear and semilinear differential equations with distributed measures, as well as a finite-dimensional like treatment of semilinear hyperbolic equations, mainly due to the author. As far as I know, some other topics appear for the first time in a book form here: the equations of linear thermoelasticity, the equations of linear viscoelasticity and the characterization of generators of equicontinuous and of compact semigroups, being the most important ones. Besides, the last part of the book contains detailed solutions to all the problems included at the end of each chapter. There are some interesting topics which, although useful, were not discussed in this book. In this respect I would like to mention the spectral mapping theorems and a thorough study of the asymptotic behavior of solutions. Moreover, in order to avoid some slight complications, most of the results in this book refer only to C0-semigroups of contractions, xi

Preface

xii

although they hold true for general C0-semigroups, i.e. of type (M,w). However, I assume that the interested readers will be able to fill in this gap, if necessary. I believe that someone who has some acquaintance with functional analysis and differential equations can read the book. Therefore, I hope that it will be found useful not only by graduate students and researchers in Mathematics to whom it is primarily addressed, but also by physicists and engineers interested in deterministic mathematical models expressed in terms of differential equations. I am greatly thankful to my former professors and students, as well as to my colleagues and friends who helped me to clarify many ideas and to organize the presentation. More specifically, I am grateful to professor Viorel Barbu for the courses he taught, which had a decisive influence on my further evolution, and for his unceasing interest in my efforts. The discussions with Professor Dorin Ie~an were of great help to me in order to clarify some aspects concerning the examples in Mechanics presented in Sections 4.8, 4.9 and 10.5. The writing of this book was facilitated by a very careful reading of the manuscript followed by many suggestions and comments by Professors Ovidiu C~rj~, Mihai Necula and Constantin Z~linescu, by Dr. Corneliu Ursescu, Senior Researcher at The "Octav Mayer" Institute of Mathematics of the Romanian Academy, as well as by Dr. Silvia-Otilia Corduneanu. Both Professor C~t~lin Lefter and my former student Eugen V~rv~ruc~ read the entire Romanian version of the manuscript and made several useful remarks I took into account in the presentation. Dr. Ioana Slrbu from SUNY at Buffalo was of great help to make the English read smoothly. It is a great pleasure to express my appreciation to all of them.

Ia~i, November

12th, 2002

Ioan I. Vrabie

CHAPTER

1

Preliminaries

The aim of this chapter is to give a brief presentation of some auxiliary notions and results which are needed for a good understanding of the whole book. In the first two sections, we define and study the class of vector-valued measurable functions as well as the integral of such functions with respect to a a-finite and complete measure. In the third section, we recall the definition of the spaces LP(~, #; X) and LP(~, tt; X), with (~, E, #) a a-finite and complete measure space, and X a Banach space, and we recall their most remarkable properties. Also here, we present some properties of wk'P(a, b;X) and Ak'p(a, b;X). The fourth section is devoted to a short presentation of the space BV([a, b]; X) of functions of bounded variation from [a, b] to X, while in the fifth section, we collect several results referring to Sobolev spaces, exactly in form they will be used later in the book. The sixth section contains some basic facts concerning unbounded linear operators in Banach spaces, with main emphasis on self-adjoint and respectively skew-adjoint operators acting in Hilbert spaces. In the seventh section, we include several spectral analysis results with regards to unbounded, closed linear operators on Banach spaces, while in the last two sections, we introduce and study the Dunford integral in order to offer an elegant way to define the value of an analytic scalar function at such an operator.

1.1. V e c t o r - V a l u e d M e a s u r a b l e F u n c t i o n s Let X be a real B a n a c h space a n d (f~, E, #) a a-finite a n d c o m p l e t e m e a s u r e space. We recall t h a t # : E -+ IR+ is a-finite if t h e r e exists a family {f~n; n C N} C E such t h a t P(f~n) < + o c for each n E N a n d ft = UncN~n. T h e m e a s u r e p is called complete if each subset of a null p - m e a s u r e set is m e a s u r a b l e (belongs to E).

D e f i n i t i o n 1 . 1 . 1 . A function x " f~ -+ X is called: (i)

countably-valued if t h e r e exist two families: {f~n; n C N} C E a n d {Xn; n C N} c X , w i t h ~k n ~p - O for each k r p, ~ - Un~o~n, a n d such t h a t x(w) = Xn for all w C ~ n ;

2

Preliminaries (ii) almost separably-valued if there exists a p-null set gt0 such t h a t x(~t \ ~0) is separable; (iii) strongly measurable if there exists a sequence of countably-valued functions convergent to x p-a.e, on ~t; (iv) weakly measurable if, for each x* C X*, the function x* ( x ) : ~ --+ IR is measurable 1.

D e f i n i t i o n 1.1.2. A subset A in X* is called determining set for X if for each x E X we have

Ilxll- sup{ Ix* (x) l; x* c n}. R e m a r k 1.1.1. If A is a determining set for X then its elements have the n o r m at most equal to 1. This is a consequence of the definition of the usual s u p - n o r m on X*. T h e o r e m 1.1.1. Each separable Banach space has at least one countable determining set. Proof. have

Let {Xn; n C N} a dense subset in X. Since for each x E X we

Ilxll

= sup{ Ix* (x) l; x* e X*,

IIx*ll

= 1},

it follows t h a t there exists a family {X~,m; n, m C H} in the unit closed ball in X*, such that, for each n C H,

lim

IX*,m(X n)] -- I]Xn II"

As Ilxn,miI - 1 for n, m e H, ]]Xni] -- s u p { I X n , m ( X n ) l ; n C H a n d {xu; n C H} is dense in X, we deduce t h a t [[xI[ - sup{[X~,m(X)[;

m

e 1~} for each

e H}

for each x C X. The proof is complete.

El

T h e o r e m 1.1.2. If X admits a countable determining set A and x : ~ --+ X is weakly measurable, then ]]xII : ~ -+ IR+ is measurable. P r o o f . Since the s u p r e m u m of a countable family of real m e a s u r a b l e functions is a measurable function and

llx( )ll =

x* e A}

for each w C ~, where each function x * ( x ) is measurable, it follows t h a t ]]xiI has the same p r o p e r t y and this achieves the proof. [:] 1Some authors prefer the term scalarly measurable instead of weakly measurable, keeping the latter term for those functions x : ~ -+ X with the property that, for each weakly open subset D in X, x -1 (D) E E.

Vector- Valued Measurable Functions

3

T h e o r e m 1.1.3. (Pettis) A function x" ft -+ X is strongly measurable if and only if it is weakly measurable and almost separably-valued. P r o o f . Necessity. As x is strongly measurable there exists a sequence (xn)ncN of countably-valued functions and a #-null set f~0, such that lim zn(w) - x(w)

n---+ (x)

(1.1.1)

for each w E f~ \ ft0. But each function in the sequence is at most countablyvalued, and thus Un>o{Xn(W); w E f~} is at most countable and dense in x(ft \ ft0). Hence x is almost separably-valued. From (1.1.1) we conclude that, for each x* E X* and w E ft \ ft0, we have lim x* (xn (w) ) - x * (x(w) ). n--).oo

Taking into account that the functions x* (Xn) a r e almost countably-valued, and thus measurable, it follows that x* (x) is measurable. Sufficiency. Since x is almost separably-valued, we may assume with no loss of generality that X is separable. Indeed, if X is not separable, let us consider the p-null set f~0 such that x(f~ \ ft0) is separable and let Y be the closed linear subspace spanned by x(ft \ ft0). Obviously this is separable and, in addition, x coincides #-a.e. with a function y defined on ft and taking values in Y. It is easy to see that y is strongly measurable if and only if x enjoys the same property. Similarly, x is weakly measurable if and only if y is weakly measurable, since, by virtue of the Hahn-Banach theorem (see Theorem 2.7.1, p. 29 in Hille and Phillips [70]), each linear bounded functional on Y coincides with the restriction of a linear bounded functional on X. So, let {Xn; n C N*} be a dense subset in X and let e > O. We define

f~+ - {w C Ft; x(w) sr O} and f t E n - { w C f t + ; Iix(w)-xnlI <_r for n C N*. From Theorems 1.1.1 and 1.1.2, it follows that both f~+ and D~ are measurable. Since {Xn; n C N*} is dense in X, we deduce that, for each e > 0, U ft~ - ft+.

(1.1.2)

n>l

Indeed, if we assume by contradiction that this is not the case, then there exist e > 0 and w C ft+ such that I I x ( w ) - Xnll > e for each n C N*. But the inequalities above show that x(w) does not belong to the closure of the set {Xn; n C N*} which coincides with X. This contradiction can be

4

Preliminaries

eliminated only if (1.1.2) holds. Let us define now n--1

E~-~

and E n - ~ t ~ \

U~

for n - 2 , 3 , . . .

k=l

and let us observe that all the sets En~ are measurable and and E ~ N E p - O

UE~-~t+

for k r

n>l

Let x~ :~t -+ X be defined by

x~(w) -

Obviously

Xn 0

is ount bly-v lued

if w E E ~ if w E ~ \ ~t+.

IIx(

)-

The proof is complete.

-< c for [3

R e m a r k 1.1.2. The definition of x~ in the proof of Theorem 1.1.3 shows that a function x : ~t --+ X is strongly measurable if and only if there exists a sequence of countably-valued functions from ~t to X which is uniformly #-a.e. convergent on ~ to x.

1.2. The Bochner Integral As in the preceding section, let X be a real Banach space, ( ~ , E , # ) a a-finite and complete measure space and let x : ~ --+ X be a countablyvalued function. Then there exist {~n; n C N} C Z and {Xn; n C N} C X , satisfying ~kN~p = ~ for each k r p, ~t - Un>0~tn, and such that x(w) = Xn for each n C N and each w C ~n- Obviously, the two families {gtn; n C N) and {Xn; n C N) which define a countably-valued function are not unique. For this reason, in all that follows, a pair of sets ({~n; n e N}, {Xn; n e N}) enjoying the above properties is called a representation of the countablyvalued function x. Inasmuch as ~t has a-finite measure, each countablyvalued function x : ~ -+ X admits at least one representation with the property that, for each n E I~, P ( ~ n ) < -~-C~. Such a representation is called a-finite representation. D e f i n i t i o n 1.2.1. Let x : ~ --+ X be a countably-valued function and let - ({~tn; n C N}, {Xn; n C N}) be one of its a-finite representations. We say that :~ is Bochner integrable (B-integrable) on ~t with respect to #, if (X)

E #(~n)llXnl I < --~-CX~. n--0

The Bochner Integral

5

R e m a r k 1.2.1. If 9~ and ~ are two a-finite representations of a countablyvalued function x 9~ ~ X, the series ~-~'-n=0C~P (~n)Xn and ~-~nCC=op(~t~n)X~ n are either both convergent, or both divergent, in the norm topology of X, and, in the former case, they have the same sum. Accordingly, 9~ is Bintegrable on gt with respect to # if and only if 9~~ enjoys the same property. This remark enables us to introduce: D e f i n i t i o n 1.2.2. The countably-valued function x : ~ --+ X is Bochner integrable on ~ with respect to p if it has a a-finite representation

which is B-integrable on ~ with respect to #. In this case, the vector (:X3

E#(~n)Xn--/ n:O

x(w) d#(o3) -- ~ x d # ,

which does not depend on the choice of 9~ (see Remark 1.2.1), is called the Bochner integral on ~t of the function x with respect to #. D e f i n i t i o n 1.2.3. A function x" ~ -+ X is Bochner integrable on ~ with respect to # if it is strongly measurable and there exists a sequence of countably-valued functions (Xk)kEN, Bochner integrable on ~t with respect to #, such that y.

lim /

k---~cxDJ~

]Ix(w) - Xk (w)II dp(w) - O.

P r o p o s i t i o n 1.2.1. If x " ~ --~ X is Bochner integrable on ~ with respect to It and (xk)kcN is a sequence with the properties in Definition 1.2.3, then there exists y 8

lim ] xk d# k--+cx~ J~ in the norm topology of X . In addition, if (Yk)kEN is another sequence of countably-valued functions with the property that lim f IIx(w) - Yk(w)ll dp(~o) - O, Ja

k--+oc

then lim L x k ( w ) d # ( w ) -

k--+cx~

lim j ~ yk(w) dp(w).

k~cx~

P r o o f . Let e > 0 and let k(e) E N be such that

L IIx-xkll

<

6

Preliminaries

for each k >_ k(c). Let us observe that

/ ; xk dp - / xp d# < ~ ,,xk - Xp,, d# < / 'lxk - x" dP + / "X - Xp" d# 9 This inequality and the preceding one show that, for each k > k(r p > k(z), we have

and

/ x k dp - ~ xp d# < c. Accordingly, (f~ xk d#)keN is a Cauchy sequence and consequently it is convergent. Let now (Ya)acN be another sequence with the specified property. We have

/ xk dp - /2 yk d# < / ,,xk -- yk,, dp < / ,,xk - x,, dp + / ,,x - yk,, dp, which completes the proof.

[:]

D e f i n i t i o n 1.2.4. Let x" ~ -~ X be a Bochner integrable function on ~. The vector lim

k-+ cx3

fa xk d # - f ~ x(w) dp(w) - f~ x d#

which, according to Proposition 1.2.1, exists and does not depend on the choice of the sequence ( X k ) k E N in Definition 1.2.3, is called the Bochner integral of the function x on ~ with respect to #. 1.2.1. (Bochner) A function x" f~ --4 X is Bochner integrable on f~ with respect to # if and only if x is strongly measurable and the real function IIx]] is integrable on f~ with respect to #. Theorem

P r o o f . Necessity. Let x be Bochner integrable on f~ with respect to # and let (Xk)k~N be a sequence as in Definition 1.2.3. Then, by Lebesgue theorem (see Proposition 14, p. 126 in Dinculeanu [47]), it follows that, at least on subsequence (denoted for simplicity again by (Xk)kcN), we have lim

k-+ cr

Xk(W) -- x(w)

a.e. for w C f~. So x is strongly measurable. The fact that ]]xll is integrable on ft with respect to # follows from the obvious inequality

The Bochner Integral

7

Sufficiency. Let x be a strongly #-measurable function on ~t such that ]]x[] is #-integrable on Ft. Since f~ has or-finite measure, there exists a family {~2n; n C N} such that #(f~n) < +oo and U~>0f~n = f~. In addition, we may assume with no loss of generality that f~k N f~p = 0 for each k ~ p. Let c > 0. From R e m a r k 1.1.2, it follows that, for each n C I~ for which #(f~n) ~ 0, there exists a countably-valued function x~ 9 f~n -+ X such that C IIx(w) -- X~n(W)ll ~ 2 n + l # ( ~ n

)

(1.2.1)

a.e. for w C ~n. Let E = Un>oEn where En is the set of all elements in ~n for which the above inequality does not hold. Obviously E is negligible. Let x~ : ~ --+ X be defined by x~ (o3)

_ f x~(~) 0

if W C ~ n \ E n if w c E .

It is easy to see that x~ is a countably-valued function. In addition, the function xE is integrable because it is measurable, bounded from above by the function IIx~-xll+llxll , Ilxll is integrable and, in view of (1.2.1), IIx~-xll is integrable too, inasmuch as CX:)

CX:)

IIx~- xll d~ ~ ~ n=O

IIx~- xll d~ ~ ~ ~(~n)2n+l#(~n ) n

--C.

n=O

Since c is arbitrary, from the inequality above~ it follows t h a t x is Bochner integrable on ~t with respect to # and this achieves the proof. [--1 The next consequence shows that each Bochner integrable function x can be approximated, as in the proof of Theorem 1.1.1, by countably-valued functions whose values belong to the range of x except 0. C o r o l l a r y 1.2.1. Let x : ~ --9 X be Bochner integrable function on ~. Then, for each c > O, there exists a partition {~n; n C N} C E of the set ~+ = {w C ~; x(w) ~ 0}, such that, for each choice of the elements Wn E ~n, n = O, 1 , . . . , the function x~ : ~ ~ X , defined by

xc(w) - ~ X(Wn) 0 L

forwC~n for w C ~ \ ~t+,

is countably-valued, Bochner integrable on ~t and IIx(~) - x~(~)ll d ~ ( ~ ) _< ~.

8

Preliminaries

In addition, the inequality above holds for each refinement of the partition considered. 2 P r o o f . Let {fin; n C N} be the class of all subsets of ft on which the

XnC's, defined as in the proof of Theorem 1.2.1, are constant. Redefining the functions x cn as suggested in the statement of Corollary 1.2.1, we observe that, by multiplying the right-hand side of (1.2.1) by 2 if necessary, the inequality thus obtained holds true. It is easy to see that the partition {fin; n C N} has all the required properties for c ~ = 2c. The proof is complete. E] We conclude this section with a result which will prove useful in the sequel. We recall that an operator acting between two Banach spaces X and Y is called closed if its graph is closed. T h e o r e m 1.2.2. (Hille) Let A: D(A) C_ X -+ Y be a linear closed operator

and let x : f~ ~ D(A). Then

/f Ax(w) dp(w) - A / x(w) d.(w) whenever both sides of the above equality are well-defined. P r o o f . Let c > 0. We apply Corollary 1.2.1 to both x and Ax in order to obtain two partitions of ft+ = {w E f~; IIx(w)ll > 0}, the first one defining an e-approximation for x and the second one for Ax. Let {ftn; n E N} be a refinement for these partitions. We define xe : ft --+ X by X(COn) for w e f~n X~ (03) 0 for w C Ft \ f~+. -

-

Then we have

IIAx(co) - Ax~(w)[[ dp(w) < e.

~ llx(w) - x~(w)l I dp(w) <_ c and But (x)

/

- Z

m

n a) -

m--+oo

x~(w~)tt(a~ N a),

n=O

n--O

and

lira ~

oo

/fl Ax~(w) dp(w)

E

Ax~(w~)p(gtn N gt)

n----O

2By a refinement of the p a r t i t i o n {f~n; n E N} C E of the set fi we m e a n a p a r t i t i o n {Ek; k E N} C E of the same set with the p r o p e r t y t h a t for each k E N t h e r e exists n E N such t h a t Ek C_ fin.

Basic FunctionSpaces

9

m

- lira ~

AxE(O.)n)#(~n A ~).

n--O Since A is closed, it follows that

/~ xE(w)d#(w) E D(A)

and A

( / x~(w)d#(w)) - / Ax~(w) dp(w).

Let (ek)kcN be a sequence convergent to 0. By Proposition 1.2.1, we have lim

k--+oc

~ xek (w) dp(w) - ~ x(w) dp(w)

and lim A / ~ x~k (co)dp(w) - l ikm~

/~Axsk(w) d#(w) - / Ax(w)dp(w).

k---~o~

We conclude the proof by using once again the closedness of A.

[-1

1.3. Basic F u n c t i o n Spaces Let X be a real Banach space, (~t, E, #) a a-finite measure space and p > 1. We denote by LP(~,# ; X ) the set of all functions f 9 ~t --+ X with the property that f is strongly measurable on ~ and IlflIp is integrable on with respect to #. Let us define II 9 II~p(~,~;x) " LP(~, #;X) --+IR+ by ;x) -

Ilfll p

f E LP(~,p;X) and let us observe that this is a seminorm on LP(~,# ;X). We define the relation "~" by f ~ g if f(w) - g(w) p-a.e. w C ~. Clearly "~" is an equivalence on LP(~, p ; X ) . Let LP(~, p ; X ) be the quotient space LP(~,p;X)/ ..~ and let us remark that if f ~ g then Iifiin,(~,~;x) -Iigii~p(~,~;x). So, ]I.IiLp(a,p;x).Lp(~, # ; x ) --+ R+, given for each

by

IifiiLp(~,#

;x)

-

IlfliPdp

for each ] e LP(~,# ; X ) , is well-defined (i.e. it does not depend on the choice of f e ]) and, in addition, is a norm on LP(~, #;X), in respect to which this is a Banach space. Next, let Lcc(gt, # ; X ) be the space of all functions f 9 gt ~ X satisfying Iifll~(~,,;x) -inf{a

C ~; I]f(w)l] <_ a,

a.e. w C ~t} < +co.

Preliminaries

10

The mapping [I " IIL~(~,p;x) " L ~ ( ~t, # ; X ) --+ I~+, defined as above, is a seminorm. Let L ~ ( ~ t , p ; X ) - L ~ ( ~ , # ; X ) / ~, where "~" is the #-a.e. equality on ~t and let I I . I I L ~ ( ~ , p ; x ) . L ~ ( ~ , p ; X ) --+ ]~+, given by

for each ] C L~ ;X). Obviously ]]" ]lL~(~,p;X)is well-defined and, in addition, is a norm on L~ p ;X), with respect to which this is a Banach space. For simplicity, in that follows, we denote by f both a fixed element in LP(~, # ; X ) , and its corresponding equivalence class in LP(~, # ; X ) . The next properties are either well-known, or follow directly from their specific counterparts corresponding to the case X - ~. Theorem

1.3.1. Let (~t, E, #) be a finite measure space. 3

(i) If f E LP(~,#; ]R) and g C L q ( ~ , # ; ]R), where p C (1, +oc) and p ~ 1, then

If gl d , <

]fIPd,

]glqd,

(Hhlder's inequality).

(ii) If f c L I ( ~ , # ; I ~ ) and g C L ~ ( ~ , p ; I R ) , then

f l/gl

<

(iii) If X is an arbitrary Banach space, and 1 < p < r < +oc, then L r (~, # ;X) C LP(~, # ; X ) , with continuous imbedding; (iv) If, in addition, ~t is a compact subset in IRn, p is the Lebesgue measure on ~ and p e [ 1, +cx~], then C(~; X) C L P ( ~ , t t ; X ) , with continuous imbedding 4. We state, again without proof, the following result which gives a simple but precise description of the topological dual of an L p space, description well-known in the classical case X - ]~. T h e o r e m 1.3.2. If either X is reflexive, or X* is separable, then, for each p e [1,+oc), (LP(f~, # ; X))* ~ Lq(f~, p ; Z*), 1 where -~ + ~1

_

1 if p > l and q - - cxD if p - - 1 .

3We recall that (~t, E, #) is a finite measure space if #(~) < +c~. 4Here C(~; X) is endowed with the uniform convergence topology, or equivalently with the sup-norm topology.

Basic Function Spaces

11

See Dinculeanu [48], Corollary 1, p. 252. Some extensions and variants of Theorem 1.3.2 can be found in Edwards [50], Theorem 8.18.2, p. 588, Remarks, p. 589 and Theorem 8.20.5, p. 607. A remarkable consequence of Theorem 1.3.2 is stated below. C o r o l l a r y 1.3.1. If X is reflexive then, for each p C (1, +co), LP(~, # ; X) is reflexive. If X is separable, then, for each p C [ 1 , +co), Lp(~,p ;X) is

separable. The next density result will prove useful in what follows. We notice that in the theorem below, p is the Lebesgue measure on R n. 1.3.3. Let X be Banach space and let 99 C CC~(]l~n; ]~) with ~(w) - O , fop each w e ]~n \ ~(0~ 1) and fR, ~(w) dw - 1. Let c > 0 and

Theorem

let ~ : R n --+ IR be given by l(w) for each w C I~n. Let p E [ 1, +oc), f C LP(IRn, # ; X ) and let re: IRn -+ X be given by f~(x) - / ~

~ ( x - w) f (w) dw

for each x E ]~n. Then fe C C~

X)V1Lc~(]Rn,#;X) and lira f~ - f

~-+0

in the norm of LP(]Rn, i t ; X ) . See Barbu [21], Lemma 1.1, p. 14. D e f i n i t i o n 1.3.1. A function ~ with the properties in Theorem 1.3.3 is called mollifier, and the function fE is called the c-mollified of f. We present next some basic results concerning vector-valued distributions of one variable. Let (a, b) be an open interval and let 9 b) the set of C ~ real functions defined on (a, b) with compact support in (a, b). We recall that the support of a function ~ : (a,b) --+ ~ is the set supp~p = {t e (a,b); ~ ( t ) ~ 0}. By definition, a sequence (~n)n~N in this space is convergent to ~ if there exists a compact set K C (a, b) such that ~n(t) = 0 for each n E N and each t C (a, b) \ K and, for each k C N, we have lim ~(k)(t) -- ~(k)(t) n---+(x)

uniformly for t C K. Let X be a Banach space. We denote by 9 b;X) the set of linear continuous operators from 9 b) to X and we call the

Preliminaries

12

elements of this space X-valued distributions on (a, b). If f E 9 b;X) and k E N, we denote by f(k) the kth-order derivative of f in the sense of X-valued distributions on (a, b), i.e. f ( k ) ( W ) _ (_l)kf(w(k)) for each ~ E 9 b), where ~(k) is the classical kth-order derivative of ~. Let k,p E N and let us denote by Wk,V(a, b; X ) the set of all X-valued distributions f on (a, b) satisfying

f(m) C L p(a, b ; X ) for each m measure. We denote derivatives and belong

- 0, 1 , . . . , k, where the set (a, b) is endowed with the Lebesgue by Ak'P(a, b ; X ) the set of all u 9 [a, b] -+ X, whose ruth-order u (m) are absolutely continuous on [ a, b] for m - 0, 1 , . . . , k - 1, to LP(a, b ; X ) , for m - 0, 1 , . . . , k. By convention, u (~ - u.

1.3.4. If X is reflexive then each function u E Al'P(a, b; X ) is a.e. differentiable on (a, b) and for each t E [ a , b]

Theorem

-

+

See Barbu and Precupanu [22], Theorem 3.4, p. a3. T h e o r e m 1.3.5. If X is a Banach space, 1 <_p <_ ec and f E LP(a, b; X ) , then f E Wk'P(a, b ; X ) if and only if there exists u E Ak'P(a, b ; X ) with f (t) - u(t) a.e. on (a, b). See Brezis [28], Proposition A.6, p. 154. 1.4. F u n c t i o n s o f B o u n d e d V a r i a t i o n

In this section we recall some basic concepts and results concerning vectorvalued functions of bounded variation. Let ~[ a, b] be the set of all partitions of the interval [a, b]. We recall that, i f g ' [ a , b ] --+ X, then, for each ~P E ~[a,b], ~P" a - to < t~ < ... < tk -- b, the number k

Vary(g, [a, b]) - ~

IIg(t )

- g(ti-1)ll

i=1

is called the variation of the function g with respect to the partition T. If sup

Vary(g, [a, b]) < +co,

Functions o/ Bounded Variation

13

then g is said to be of bounded variation 5 and the number Var(g,[a,b])=

sup

Var~(g,[a,b])

is called the variation of the function g on the interval [a, b ]. Similarly, we define Vat (g, (a, b ]), Var (g, [a, b)) and Var (g, (a, b)). Of course, in each of the above three cases, we have to consider only partitions ~P satisfying either a < to or tk < b or both, as required by the type of the interval considered. We denote by B V ( [ a , b]; X) the vector space of all functions of bounded variation from [a, b] to X. The mapping g ~ Var (g, [a, b]) is a seminorm on B V ( [ a, b ]; X), while g ~-+ IIg(a)I1 + Var (g, [a, b]) is a norm, denoted by l]'[[BV([a,b];X) and, in respect to which, B V ( [ a , b ] ; X ) is a Banach space. In the proposition below, we collect without proof some simple but useful properties of the functions of bounded variation. P r o p o s i t i o n 1.4.1. Let g C B V ( [ a, b ]; X ) . Then: (i) if [c,d] C [a,b] then g e B V ( [ c , d ] ; X ) and Var (g, [ c, d l) _< Var (g, [ a, b l); (ii) the function t ~-~ Var (g, [a, t]) is nondecreasing on [a, b]; (iii) for each c E (a, b) we have Var (g, [a, b]) = Var (g, [a, c ]) + Var (g, [c, b]). The next extension of L e m m a 16, p. 140 in Dunford and Schwartz [49] will prove useful in the sequel. Lemma have

1.4.1. For each g E B V ( [ a , b ] ; X ) , t e [a,b), and s E (a,b], we lim Var (g, (t, t + h ]) - 0, h$0

(1.4.1)

lim Var (g, [s - h, s)) = 0. h$0

(1.4.2)

and

P r o o f . Since the proofs of both (1.4.1) and (1.4.2) we confine ourselves only to the proof of (1.4.1). Let us diction that this is not the case. Then there exist c > 0 such that Var (g, (t, t + hi]) > c. So, there exists at least o n e ~ 1 " t < to1 < tl < ... < t~(1)

are quite similar, assume by contraand hi C (0, b - t)

- t + hi such that

Var[p I (g, (t, t -~- hi ]) > c. 5Some authors, as for instance Hille and Phillips [70], p. 59, refer to this kind of functions as to functions of strong bounded variation.

Preliminaries

14

Repeating the same reasoning on (t, t~ ], we find an h2 C (0, t~ - t l, and a partition T 2 " t < to2 < t~ < . . . < t k(2) 2 -- t~ of (t, t~ ], such that, for the very same e > 0, we have Var~ 2 (g, (t, t + h2 ]) > e. 2 < t~ < tl < ... < tk(1) - t + hi is a partition Clearly, [P1,2 " t~ < tl2 ... t k(2) of (t, t + hi ], and Vary1, 2 (g, (t, t + hi ]) > 2e. By an inductive argument, we get that Var (g, (t, t + hi ]) -- -~-cx~, which contradicts the hypotheses. This contradiction can be eliminated only if (1.4.1) holds, and this achieves the proof. D P r o p o s i t i o n 1.4.2. If g E B V ( [ a, b ]; X ) , then g is piecewise continuous on [a, b], i.e. there exists an at most countable subset E of[a, b ], such that g is continuous on [a,b] \ E and, at each t E E N [a,b) (s C E A (a,b]),

limit g(t + o)

0)).

P r o o f . Let t C [a, b). Then, for each t < s < ~- < b, we have IIg(~) - g(s)II -< V~r (g, [s, ~ 3) _< Var (g, (t, T ]). By virtue of Lemma 1.4.1, this inequality shows that 0 ~ g(O) satisfies the C a t c h y Criterion for the existence of the one-sided limit g(t + 0). Since the existence of g(s - 0 ) follows by a very similar argument, the proof is complete. [Z] We say that exists M > Similarly, a if the set of

a subset 9 in B V ( [ a, b ]; X) is of equibounded variation if there 0 such that, for each g E 9, we have Var (g, [a,b]) _< M. sequence (gk)k~N in B V ( [ a , b ] ; X ) is of equibounded variation its terms enjoys this property.

T h e o r e m 1.4.6. (Helly-Bray) Let (gk)k~N be a sequence o f f unctions with equibounded variation in B V ( [ a , b l), and let us assume that, for each t in [ a, b], we have lim gk(t) = g(t). k-+ c~

Then g C B V ( [ a, b ]) and, for each f C C([ a, b]), we have lim k--'t e ~

/a

f (s) dgk(s) --

/a

f (s) dg(s).

See Graves [63], Theorem 2.3, p. 283. An extension of this result which we shall need later is presented below. Let X ~ be a closed subspace in X* with the property that, for each x E X, we have

Ilxll- sup{l(x, xG)l ; x | ~ X ~ Ilx|

_< 1}.

Sobolev @aces

15

In other words, the closed unit ball in X ~ is a determining set for X. On X we consider a ( X , X ~ which, from now on, we shall refer as to the weak-| topology which, in general, is weaker t h a n the weak topology on X.

T h e o r e m 1.4.7. Let (gk)kEN be a sequence in B V ( [ a, b ]; X ) of equibounded variation, let g: [a, b] --+ X , and let us assume that lim (gk(t) x G) - ( g ( t ) k---~ c ~

x G)

~

for each x G E X ~ and t E [a,b]. f E C ( [ a , b ] ; X G ) , we have

Then g E BV([a,b]; X ) and, for each

lim

(dgk(s) f (s)) -

k----~ (x)

~

(dg(s) f (s)).

P r o o f . First, let us observe that, whenever limk__~(Xk,X ~ = ( x , x ~ for each x ~ E X ~ we have [Ixll _< liminfk__~ Ilxkll _< l i m s u P k _ ~ Ilxkll and this clearly shows that g E B V ( [ a, b ]; X). Next, let us observe that, by the classical Helly-Bray Theorem 1.4.6, for each ~i E C([a, b]) and x~ E X * with i = 1 , 2 ~ . . . , n , we have

dga(s)

lira k--+c~

~i(s)x~

-

i=1

= Z

~i(s) d(gk(s) x~)

lira i--1

)

-

i=1

So, the conclusion holds for any function f E C([a, b ] ; X ~ of the form f ( t ) - Y~i~l ~i(t)x~. Since the set of functions of the form as above is dense in C([ a, b l; X), this achieves the proof. K]

1.5. S o b o l e v Spaces Let ~t be a nonempty and open subset in R n and 99 : f~ --+ R. As in the one-dimensional case, the set supp ~ defined by s u p p ~ = {x E ~; ~(a)

0},

is called the support of the function 99. Let 9 be the set of C ~ functions from f~ to R with compact supports included in f~. Let a E N n be a multi-index, a = ( a l , a 2 , . . . , a n ) and 99 E 9 (f~). We define 0 0 ~ 1 -[-O~ 2 -[- " "-'[-OL n (~)

V a ~ - Ox~l Ox~2 . . . OXen'~ "

Preliminaries

16

One may easily see t h a t 9 is a vector space (even an algebra) over R. We endow this space with a convergence structure as follows. D e f i n i t i o n 1.5.1. We say t h a t the sequence (~n)nEN is convergent in 9 to ~ and we write ~n ~ ( ~ ~, if (i) there exists a compact subset K C ~ such that, for each n E N, s u p p ~n C K ; (ii) for each multi-index c~ we have lira D~g~n - Daqp uniformly on u+oc Q, or equivalently on K. D e f i n i t i o n 1.5.2. By a distribution on 9 continuous functional defined on 9

we mean a real-valued, linear

I f u ~ ~)'(~) and We denote by 9 the set of all distributions on 9 qo E 9 we denote (u, qo) - u(~). Similarly, we may consider the space 9 C) of all C c~ functions from ~ to C and we may define complex-valued distributions on 9 C).

D e f i n i t i o n 1.5.3. Let c~ E N n be a multi-index and u " ~ --+ IR a locally integrable function. By definition the derivative of order c~ of the function u in the sense of distributions over 9 is the distribution 9 defined by /

.

(9176 qp) - ( - 1 ) ~ / ~ uDaqodw ta,

for each ~ E 9 multi-index c~.

(1.5.1)

~&

where I c ~ ] - c~, + c~2 + . . .

+ c~n is the length of the

Let us observe that, if the function u is a.e. differentiable of order c~ on ~ in the classical sense and D~u is locally integrable, then 9 can be identified with Dan by means of the equality (9

~) - ./o D a u ~ dw

for each ~ E 9 equality obtained by integrating ]c~l-times (1.5.1) by parts. Let m E N, 1 _< p _< + c o and let us define

wm,~(~)Theorem

{~ e L~(~); 9 ~

~ L~(~) for 0 _ I~l-<

"~}.

1.5.1. The mapping I]" [[m,p " Wm'p(~) --+ I~+, defined by

112) U[[Lp(~)

m~x II~~llL~r o<_l~l<_m

I

if 1 _< p < +oc

i f p - c~

Sobolev @aces

17

for each u E wm'p(f~), is a norm with respect to which wm'p(f~) is a real Banach space.

For the proof see Adams [1], Theorem 3.2, p. 45. The space Wm,P(f~) is called the Sobolev space of integer order m and exponent p. Let H m ' p ( ~ ) the completion of the space {u C cm(~t); ]]U[Im, p < +(2~} with respect to I1" t[m,p. Theorem

1.5.2. (Meyers, Serrin) For any n o n e m p t y and open subset of IRn, any m E N and l < p < +ec, we have w m , P ( ~ ) -- Hrn,P(Q).

For the proof see Adams [1], Theorem 3.16, p. 52. Let us define now the

sp~r Wo '~(~) ~s being the dosu~ of ~ ( ~ ) in W~,~(n). We ~lso d~note this space by H o 'p(Q). It is easy to see that

Wo,~(n) c w~,~(n)c L~(n), both imbeddings being continuous. Moreover, let us observe that, for each 1 _< p _< ~ , we have W ~ - L P ( ~ ) . If 1 _< p < + ~ , we also have W ~ (~) - L p (Q), the last equality being a simple consequence of the fact that 9 (f~) is dense in L p (~) if and only if p < +oc. Let 1 _< q _< +oc and let us denote by w - m ' q ( Q ) the set of all distributions u C 9 of the form ~t

I~l<m with fa E Lq(~). T h e o r e m 1.5.3. For each 1 <_ p < cx~, the topological dual of Wo'P(f~) is w - m ' q ( f ~ ) , where 1/p + 1/q = 1 if I < p < +oc and q = oc if p = 1.

See Adams [1], Theorem 3.8, p. 48. In all that follows we shall denote the spaces w m ' 2 ( ~ ) - H m , 2 ( a ) and W o '2 (f~) - H~n'2 (f~) by H TM (f~) and by H~n(f~) respectively. Clearly these are Hilbert spaces with respect to the inner product

<~,v>~-

Z s 9~v~vd~. al<m

T h e o r e m 1.5.4. (Sobolev, Rellich, Kondrachov) Let us assume that ~ is a nonempty, open and bounded subset in I~n whose boundary is of class C 1, m E N a n d p , q C [1, +oc). ~v , then w m ' v ( ~ ) is compactly imbedded in (i) If m p < n and q < n-my

c~(~).

Preliminaries

18

(ii) If mp - n and q e [1, +co), then wm'p(Q) is compactly imbedded in Lq(Q). (iii) If mp > n, then Wm'P(Q) is compactly imbedded in C ( ~ ) . Se A d a m s [1], T h e o r e m 5.4, p. 97 and T h e o r e m 6.2, p. 144. In order to define the Sobolev space of fractional order s >_ 0 and exponent 2, we shall use the Fourier transform. More precisely, if s _> 0, we define

H s ( R n) - {u E L2(Rn); (l+II~l12)s/25(~) E L 2 ( R n ) } , where 5 is the Fourier transform of the function u, i.e.

1

jf

-i(~,~)u(w) dw.

On the space H s(]~n), let us define the n o r m [[. [[s by

]l li - I1(1 + II ll2) /2ellL (R ). From Plancherel theorem (see Stein and Weiss [116], T h e o r e m s 2.3~ 2.4, p. 17), it follows that if s - m is natural, then Hs(IR n) - Hm(~t) with - R n. See, for instance, Lions and Magenes [85]~ Th~or~me 1.2, p. 7. In order to define the space HS(~) for every s C R+, let us assume t h a t ft is a n o n e m p t y and open subset whose b o u n d a r y F is an n - l-dimensional C 1 manifold. Moreover, let us assume t h a t ~t is locally on one side of F. We define b o t h H s (~), as the set of restrictions to ~ of all elements in H s (R n), and ][. I[s " H S ( ~ ) -+ IR+ as [[u[[s

inf{[[U[[s; U C H ~(~), U n - u}.

Next, let {X~; i E J} be a family of local charts on F, and {0~; i C J} a s u b o r d i n a t e d finite partition of the unity. For each u C L2(F) we have

u - ~

Oiu.

iCJ

Let ui - Xi o Oiu. We define H s (F) as the space of all functions u C L2(F) with the property t h a t ui C H s ( R n - l ) for each i C :J. One may easily conclude t h a t Hs(F)~ endowed with the inner product

iCJ

is a real Hilbert space. More t h a t this, b o t h HS(F) and the inner p r o d u c t are independent of b o t h the choice of the family of local charts and the corresponding partition of unity. By definition, for s < 0, H s (F) - (H -~ (F))*.

Sobolev @aces

19

Taking advantage of Riesz theorem, we agree to identify H ~ - L2(F) with its own dual. If u C C I ( ~ ) , we denote by u . the outward normal derivative of u on F, i.e.

= (-, Vu), where u is exterior unit normal at the current point of F. T h e o r e m 1.5.5. Let ~ be a nonempty and open subset in ]~n whose boundary F is of class C 1 and let a > 1/2. The function u ~ (uiF) from C(~) to C(F), has a unique linear bounded extension ~o " HS(~) -+ For s - 1, the kernel of this application is H 1(~). Similarly, if (~ > 3/2, the function u ~ u , , from C1(~) to C(F), has a unique linear bounded extension ~l " HS(~) --+ Theorem 1.5.5 enables us to give a sense to the outward normal derivative of u, whenever u C HS(~), with s > 3/2. More precisely, for u E HS(~), we set u . = 9~1u. We conclude this section with some general results in the theory of elliptic equations. T h e o r e m 1.5.6. (Friedrichs) Let ~ be a nonempty and open subset in ~n whose boundary F is of class C 1. Then there exists kl > 0 such that, for each u C H I ( ~ ) , we have

See Nebas [94], Thdor~me 1.9, p. 20. T h e o r e m 1.5.7. Let ~t be a nonempty open and bounded subset in ~n whose boundary r is of class C 1. Then [[" ll: H I ( ~ ) --~ R+, defined by

[[Ull_ (][~TUl[22(Vt)+IlU[FII~2(F))1/2 for each u E H I ( ~ ) , is a norm on HI(a) equivalent with the usual one. In particular, the restriction of this norm to H I ( ~ ) , i.e. [[. [[o " H~(~) -+ ]~+, defined by

II llo -IIWlIL ( ), for each u e Hl(~t), is a norm on H~(~) (called the gradient norm) equivalent with the usual one. In respect with this norm the application D" HI(~t) ~ H - I ( ~ ) , defined by

(V, DU)H~(~),H-I(~

) --

V u V v dw,

20

Preliminaries

is the canonical isomorphism between H~(~t) and its dual H-I(f~). The restriction of this application to H2(f~) coincides with - A , where A is the Laplace operator in the sense of distributions over 9

P r o o f . The conclusion follows from Theorems 1.5.5 and 1.5.6.

[2]

R e m a r k 1.5.1. Theorem 1.5.7 justifies why, the canonical isomorphism D between H~(~t), endowed with the gradient norm ]]. II0, and H-l(f~), is by now on denoted by - A .

Corollary 1.5.1. For any )~ > 0 and f E H - l ( ~ ) the equation ) ~ u - A u - f has a unique solution u C H 1(~).

T h e o r e m 1.5.8. The application I - A "

H~(f~) --+ H-I(f~) is the canonical isomorphism between H~(~), endowed with the usual norm on Hl(f~), and its dual H -l(f~), endowed with the usual dual norm. In addition, for each u e H~(a) and each v E L2(a), we have

(U~ V)L2(t~) -- (U,

V)H~(~),H-I(~ ).

1.6. U n b o u n d e d Linear Operators Let X be a Banach space over K E {R, C} whose norm is denoted by II" II.

Definition 1.6.1. By a linear operator in X we mean an ordered pair (D, A), where D is a subspace of X and A 9D -+ X is a linear application, i.e." A ( a x + ~y) - a A x + flAy for each x, y C D and each a,/3 C K. The operator (D, A) is called bounded if sup{iiAxiI; x c D, IixiI _< 1} < +oc. Otherwise, the operator ( D , A ) i s said to be unbounded.

Definition 1.6.2. Let (D,A) be a linear operator. The sets D - D(A), A(D) - R(A), {(x,y) e X x X; x E D, y - Ax} - g r a p h ( A ) a n d {x C D; Ax - 0} - ker(A) are called" the domain, the range, the graph and respectively the kernel of the operator (D, A). In the sequel, from traditional reasons, we shall denote an operator (D, A) by A " D(A) C X --+ X and we shall say that D(A) is its domain, or that A is defined on D(A), R(A) is its image and graph (A) is its graph, instead of the domain of (D, A), the image of (D, A) and respectively the graph of (D, d). Also, we shall write A - B instead of graph (A) - graph (B). We denote by L ( X ) the set of all bounded linear operators, from X to X. On this space we define II" IlL(x)" X -~ it(+ by IITIIL(x)- sup{IITxI[; x e X, IIx]l <_ 1}

Unbounded Linear Operators

21

for each T C L ( X ) . It is easy to see that II. II <x) is a norm on L ( X ) , called the operator norm, with respect to which this is a Banach space and that, for T E L ( X ) ,

IITII

(x)- sup{llTzll; z c x, Ilxll- 1}.

The norm topology on L ( X ) is called the uniform operator topology. R e m a r k 1.6.1. Let ~ E K. One may easily verify that ( M - A ) -1 c / 5 ( X ) if and only if, for each f C X, the equation ~ u - Au - f has a unique solution u - u(~, f) and the mapping f ~ u(~, f) is continuous. Let X be a Banach space and A 9 D(A) C_ X --+ X a linear operator (bounded or not) and let X ( A ) be the closed linear subspace in X spanned by D(A). Obviously X ( A ) - X if and only if D(A) is dense in X. We recall that the adjoint of the linear operator A 9 D(A) C_ X --+ X is the operator A*- D(A*) C_ X* --+ X(A)* defined by

D(A*) - {x* e X*; 3C > 0,Vy e D(A), I(x*,Ay)l < CllYll} and, for each x* C D(A*), A'x* is the unique continuous extension to X ( A ) of the linear continuous mapping y ~ (x*, Ay) from D(A) to K. D e f i n i t i o n 1.6.3. Let H be a real Hilbert space identified with its own topological dual. The operator A ' D ( A ) C_ H ~ H is called: (i) (ii) (iii) (iv)

self-adjoint if A - A*; skew-adjoint if A - - A * ; symmetric if (Ax, y} - (x, Ay} for each x, y e D(A); skew-symmetric if {Ax, y} - - { x , Ay} for each x, y C D(A).

R e m a r k 1.6.2. If A ' D ( A ) c_ H -+ H is symmetric (skew-symmetric), then graph (A) C_ graph (A*) (graph (A) C_ graph (-A*)). R e m a r k 1.6.3. We may easily see that an operator A" D(A) C H -+ H is skew-symmetric if and only if, for each x E D(A), (Ax, x) - O. Indeed, if A is skew-symmetric we have (Ax, x} - - ( x , Ax}, which implies {Ax, x} - 0 for each x E D(A). Conversely, if {Ax, x} - 0 for each x C D(A), we have

0 - (A(x + y), x + y} - (Ax, x) + (Ax, y} + (Ay, x} + lAy, y} = {Ax, y} + (x, Ay} for each x, y C D(A) and thus A is skew-symmetric. Let A be self-adjoint. Then its graph coincides with the graph of A* and consequently A is symmetric. Similarly, if A is skew-adjoint, then it is skew-symmetric. The next lemma shows that, for a sufficiently large class of operators, the converse statements are true.

22

Preliminaries

L e m m a 1.6.1. Let H be a Hilbert space and A : D ( A ) C H --+ H a densely defined operator. We have: (i) if ( I - A) -1 C L ( H ) , then A is self-adjoint if and only if A is symmetric; (ii) if (I i A) -1 C L ( H ) , then A is skew-adjoint if and only if A is skew-symmetric. P r o o f . (i) Let us assume that A is symmetric. In view of Remark 1.6.2, the graph of A is included in the graph of A*. To complete the proof is suffices to show that D(A*) C_ D(A). Since D ( A ) is dense in H, it follows that, for each y e D(A*), there exists z e H such that (Ax, y) = (x, z) for each x E D ( A ) . Since I - A is surjective, there exists w C D ( A ) such that y - z = w - Aw. On the other hand, for each x C D(A), we have (Ax - x, y - w) = (Ax, y) - (Ax, w) - (x, y} + (x, w) = (x, z) - (Ax, w) - (x, y} + (x, w) = (x, y - w + Aw) - (Ax, w} - (x, y} + (x, w) - O.

But I - A is surjective and accordingly, the above equality proves that y - w, which implies y C D ( A ) and Ay - A*y. So (i) holds. Since (ii) follows by using similar arguments, the proof is complete. [--1 Let H be a complex Banach space for which there exists a complex inner product, i.e. a mapping [ . , . ] : H x H -+ C satisfying: (i) (ii) (iii) (iv)

[x,y] = [y,x] for each x , y C H ; [c~x+fly, z] = c ~ [ x , z ] + ~ [ y , z ] for each x , y , z C H and c~,/~ E C; Ix, x] > 0 a n d [ x , x ] = 0 i f a n d o n l y i f x = 0 ; [[xl]- V/[x, x ] for each x e g .

We may easily verify that the mapping (., .} : H • H --+ IR defined by

(x, y) =

x, y ],

for each x, y C H, is a real inner product on H. In addition, endowed with this real inner product, H is a real Hilbert space. From now on, a Banach space H endowed with a complex inner product will be called a complex Hilbert space. L e m m a 1.6.2. Let H be a complex Hilbert space, let A : D ( A ) C H -+ H be a densely defined, C-linear operator and let iA be defined by D(iA)- D(A), (iA)x- iAx

/or each x e D(iA).

Unbounded Linear Operators

23

Then A* is C-linear and (iA)* = - i A * . In particular, A is self-adjoint if and only if iA is skew-adjoint. Analogously, A is skew-adjoint if and only if iA is self-adjoint.

R e m a r k 1.6.4. Essentially, Lemma 1.6.2 asserts that the adjoint of A, on the complex Hilbert space H identified with its own topological dual Hc, coincides with the adjoint of A, on the real Hilbert space H identified with its own topological dual H~. P r o o f . Let x C D(A*). For each y C D ( A ) and ~ C C we have (s

x, y) - (A* x, s

- (x, A()~y))

= (x,)~Ay) - ()~x, Ay) - (A* ()~x), y).

Since D ( A ) is dense in H and y E D ( A ) is arbitrary, from the above equality it follows that A*()~x) = )~A*x, which proves that A* is C-linear. Next, let x E D(A*) and y C D ( A ) . We have ( - i A * x , y) = ( A ' x , iy) = (x, A(iy)) = (x, NAy) = ((iA)*x, y).

So x c n ( ( i A ) * ) , and graph ( - i A * ) C graph ((iA)*). Substituting A by iA, we deduce that graph ( - i ( i A ) * ) C graph ( - A * ) . From the C-linearity, we conclude that graph (iA)* C_ graph ( - i A * ) and thus it follows that graph (iA)* = graph ( - i A * ) . Finally, if A is self-adjoint, from the previous considerations~ we conclude that (iA)* = - i A * = - i A , which shows that iA is skew-adjoint. Conversely, if iA is skew-adjoint, we have A* = ( - i ( i A ) ) * = i(iA)* = - i ( i A ) = A and hence A is self-adjoint. Similarly, if A is skew-adjoint then (iA)* = - i A * = iA

and so iA is self-adjoint. Conversely, if iA is self-adjoint then A* = ( - i ( i A ) ) * = i((iA)*) = i(iA) = - A .

The proof is complete.

D

L e m m a 1.6.3. Let A : D ( A ) C_ H --+ H be a linear operator. If A is densely defined, there exists its adjoint A* : D(A*) C_ H --+ H, and: (i) graph (A*) = {(x, f) C H xH; (f, y) = (x, g), V(y, g) C graph (A)} ; (iN) (x, f) e graph (A*) if and only if ( - f , x) E (graph (A))-L ; (iii) graph (A*) is closed in U x U.

Preliminaries

24

P r o o f . Let G* - {(x, f) e H x H; (f, y} - (x, g), V(y, g) e graph (A)}. If (x, f) C G*, we have (f, y} - (x, Ay} for any y E D(A) and therefore I(x, Ay}[ < I]f[[llyl]. It follows that x e D(A*) and f - A ' x , and accordingly G* C_ graph (A*). Next, let x E D(A*) and let f - A*x. Clearly, we have (f,y} - (x, Ay} for each x e D(A). But, this last relation shows that (f, y) - (x, g} for each (y, g) C graph (A) and hence graph (A*) C_ G*, which proves (i). Obviously (ii) follows from (i), while (iii) from (ii) simply because for each subset B in H its orthogonal complement B • is a linear closed subspace. We conclude this section with a useful necessary and sufficient condition in order that a densely defined, symmetric operator be self-adjoint.

Theorem 1.6.1. Let A : D(A) C_ H -~ H be a linear, densely defined, symmetric operator. Then A is self-adjoint if and only if (I+iA) -1 C L ( H ) . See Taylor [119], Proposition 8.5, p. 513.

1.7. Elements of Spectral Analysis Let X be a complex Banach space and A" D(A) C_ X ~ X a linear closed operator. We recall that the resolvent set of A is the set of all A C C for which the range of A I - A is dense in X and ( A I - A) -1 9 R ( ) J - A) ~ X is continuous. We denote this set by p(A) and we call its elements regular values of the operator A. For ~ E p(A), denote by R(~; A) - ( ~ I - A) -1. We also recall that the spectrum of the operator A, denoted by a(A), is defined by C \ p(A). In the next theorems we assume that X is a complex Banach space.

Theorem 1.7.1. Let A " D(A) C X --> X be a linear closed operator. Then, for each ~ C p(A), R(A; A) C L ( X ) . P r o o f . If ~ C p(A) then R ( / ~ I - A ) there exists a constant c > 0 such that

D ( ( ~ I - A ) -1) is dense in X and

ii()~i- A)xll > cllxll

(1.7.1)

for each x C D(A). In order to show that R ( A I - A ) - X, let y C X and let (Xn)nEN such that lim ( A I - A)xn - y . From (1.7.1) it follows that there n---~ o o

exists lim Xn - x. Since A is closed, we conclude that A I - A is closed too n----~ o o

and consequently x C D ( A I - A) and ( A I - A)x - y. But R ( A I and accordingly R ( A I - A) - X. The proof is complete.

A) - X D

Let A " D(A) C_ X ~ X be a linear and closed operator. The mapping R(. ; A ) ' p ( A ) ~ L ( X ) , defined by R(A; A ) - ( A I - A ) -1 for each ~ C p(A), is called the resolvent function of A.

25

Elements o~ Spectral Analysis

T h e o r e m 1.7.2. Let A 9 D(A) C_ X --+ X be a linear closed operator. Then p(A) is an open subset in C and, on each connected component of p(A), the resolvent function )~ ~-+ R()~; A) is analytic. P r o o f . Let # C p(A) and let us consider the series S()~) - R(#; A)

I + ~(#

- s

A) n

(1.7.2)

.

n--1

Clearly, for each ~ e C for which ] ~ - #[]IR(#; A)ii~(x) < 1, the series in (1.7.2) is convergent in the operator norm topology to an analytic function. Since { S()~)()~I- A) - S(A)[(A - # ) I + ( # I - A)] - I

()~I- A)S()~) -

[(A -

#)I + ( # I -

A)]S(A)

-

I,

it follows that S()~) - R()~; A). In conclusion, for each # e p(A), there exists a disk centered in # included in p(A), disk on which R(. ;A) is analytic and this achieves the proof. E] T h e o r e m 1.7.3. Let A" D(A) C_ X --+ X be a linear operator. If )~, # are regular values, i.e. ~, # E p(A), and R(~; A ) , R ( # ; A) C L ( X ) , then R(A; A) - R(#; A) - (# - A)R(A; A)R(#; A).

(1.7.3)

The relation (1.7.3) is known as the Resolvent equation. P r o o f . Let us observe that R(~; A) - R(A; A ) ( # I - A ) R ( , ; A) - R(A; A ) { ( , - A)I +(s

- A)}R(#;

A) -

(# - s163

A)R(#; A) + R(#; A).

The proof is complete.

O

T h e o r e m 1.7.4. Let X be a complex Banach space and A C L ( X ) . Then there exists 1/n lim IIAnIIL(x) - r~(A) (1.7.4) n - - + (x)

and r~(A) <_ IIAllL(x). (1.7.5) In addition, we have {A E C; ]A] > rz(A)} C_ p(A) and, for each )~ E C with ]A] > rz(A), we have O0

R(A; A) - E

i~-nAn-l"

(1.7.6)

n:l

D e f i n i t i o n 1.7.1. The number ra(A) given by (1.7.4) is called the spectral radius of the operator A.

Preliminaries

26 P r o o f . Let r -

1/n inf [[Anllc(x). In order to prove (1.7.4) it suffices to n>l

show that

1/n lim sup IIA n II~(x)

- ~.

n---~oo

1/m Let e > 0 and m C N such that IlAmll~(x) <_ r + e. Obviously, each n C N can be uniquely written as n - mp + q with 0 _< q _< m - 1. Since, for each A, B C L ( X ) , we have IIABII~(x) <_ IIAII~(x)IIBII~(x), we deduce

q/n 1/n < [[A TM p/n [[An[[L(X) ~(x)[[AII c ( x ) As lim m p / n -

1 and lim q / n -

n--+ oo

< (r + -

g)mp/n [[A [ q/n

~(x)"

O, it follows that

n--+ oo

1/n limsup IlAnlljz(x) < r + e. n--+oo

Since e > 0 is arbitrary, the preceding inequality shows that 1/n limsup IIAnll~(x) _< ~, n---+ cx)

which proves (1.7.4).

In order to prove (1.7.5) it suffices to observe that IlAnll~(x) <_ llAIl~(x) and then to make use of (1.7.4). Next, if I)~l > r~(A), there exists e > 0 such that IA[ _ r , ( A ) + e . But (1.7.4) implies that, for large n, II)~-nAnllc(x) _< (r~(A) + e)-n(r~(A) + e/2) n. Hence the series in (1.7.6) is convergent. Multiplying this series, either on the left, or on the right, by ( M - A ) we obtain the identity in X. So (1.7.6) holds and the proof is complete. D C o r o l l a r y 1.7.1. Let X be a complex Banach space and A C L ( X ) . p(A) is nonempty.

Then

1.7.5. Let X be a complex Banach space and A E L ( X ) .

Then

Theorem

r~(A)-

sup I~1. t,c~(A)

(1.7.7)

P r o o f . From (1.7.6) it follows that

r~(A) >

sup

#ca(A)

I # 1 - IAI.

To prove the converse inequality we recall that, by the inequality above and Theorem 1.7.2, it follows that R(A; A) is analytic for IAI > IAI. So, it can be expanded as a Laurent series which converges in the operator norm topology for IAI > IAI. Since this expansion is unique, from Theorem 1.7.4, it must coincide with the series on the right-hand side of (1.7.6). Consequently,

Functional Calculus for Bounded Operators

27

for each IAI > IAI, lim IIA-nAn[ls(x) - O. Then, for each c > 0, we have n---~ (x)

IIAnll (x) <_ (c + I11) n

if n is sufficiently large. By consequence

1/n rz(A) -- limcr IIAnllc(x) <_ IA[ -

sup I 1, E]

which proves (1.7.7).

C o r o l l a r y 1.7.2. For each ~ C C with I)~l < r~(A), the series in (1.7.6) diverges. P r o o f . Let r be the smallest positive number with the property that (X)

the series y ~ A-nAn-1 converges for each A E C with I)~1 > r. Then, for n=0

>

we

have lim n---~ cx)

II -nAnll (x)

-

the previous proof, we deduce that

and, by repeating the reasoning in 1/n lim IIAnll (xl <_ ~. Hence r~(A) <_ r 0

n---~ o o

and the proof is complete.

D

1.8. F u n c t i o n a l C a l c u l u s for B o u n d e d O p e r a t o r s Let X a complex Banach space, D a domain in C whose boundary F is a finite union of rectifiable Jordan arcs oriented in the positive sense, let f : D -+ C be an analytic function and A E L ( X ) . The problem which we may raise in this context is to give a sense to the expression f ( A ) . A first and very natural idea which comes to mind is to define f ( A ) by f (A) = f (~)I + n--1

f(n)()~) ( A - s n!

~

where ~ C I[} is fixed. One can prove that, for each A E L ( X ) satisfying a(A) C D, the series on the right-hand side of the above relation is welldefined and normal convergent, i.e. all the terms are properly defined and the series of the corresponding norms is convergent. Unfortunately, this definition cannot be extended easily to the case in which A is unbounded (when D(A) r X , which clearly implies that the domain of the operator ( A - ~I) n depends on n C N). So, in this case, the idea to define the operator f (A) by means of the above equality is no longer applicable in its simplest form as suggested by its "A-bounded" counterpart. One way to overcome this difficulty is to find a possibility to do this without making use of the natural powers of A - M. This way has been suggested by the celebrated Cauchy integral formula which asserts that, for each z C I[3, we

Preliminaries

28 have

I(z)

1 fr f(A)

dA.

By observing that, at least formally, A ~ R(A; A) - ( A I - A ) , ~ corresponds to the function A ~-+ ~l-~_a,it is quite natural the define f ( A ) by

f (A) - 27d

f(A)R(A; A)dA,

(1.8.1)

whenever, of course, a(A) C D. We recall that a(A) is the spectrum of the operator A, which is defined by C \ p(A), where p(A) is the resolvent set of A, i.e. the set those A C C for which A I - A is invertible with bounded inverse. See Theorem 1.7.1. R e m a r k 1.8.1. In the case when A is bounded, by Theorem 1.7.4 we know that a(A) is bounded and, so, the set D can be chosen bounded. Moreover, if f is an analytic function on C, then the domain lI} can be chosen simple connected and bounded (for instance, the disk centered at 0 and of radius r~(A) + 1). R e m a r k 1.8.2. One can easily verify that the integral on the right-hand side of (1.8.1) depends only on f and A, but not on the domain D, with the condition that the spectrum of the operator A is contained in D. This follows by observing that, by virtue of the classical Cauchy theorem, for each x e X and each x* e X*, the integral fr f(A)(R(A;A)x,x*)d)~ does not depend on D. D e f i n i t i o n 1.8.1. The integral on the right-hand side of (1.8.1) is called the Dunford integral associated to f and A. If A E L ( X ) , we denote by if(A) the set of all complex functions, which are analytic in a certain neighborhood of a(A). Theorem Then :

1.8.1. (Dunford) Let A C L ( X ) , f , g E if(A) and ~,~ C C.

(i) c~f +/~g C if(A) and ~ I (A) § ~g(A) = (~ f +/~g)(A) ; (ii) f g e if(A) and f (A)g(A) = (f g)(A) ; oo

(iii) if f(A) (x)

f (A) - E

E

Cn/~n On an open neighborhood ]D of (7(A),

then

n--O

cnAn in the norm of f~(X);

n--O

(iv) if (fn)ncN are analytic on an open neighborhood D of a(A) and lim fn = f uniformly on D, then lim fn(A) = f (A) in the norm n-+~c

oy

n--+oc

(x) ;

Functional Calculus for Bounded Operators

29

(v) f e 2=(A*) and f (A*) = (f (A))* (vi) f (a(A)) = a ( f (A)). Proof. The fact that 9"(A) is a vector space is obvious and so (i) holds. In order to prove (ii), let ]I}1 and D2 two open neighborhoods of or(A) whose boundaries F1 and F2 are finite unions of Jordan rectifiable arcs with the property that D 1 (..JF1 C ID)2 and D2 U F2 is included in the common domain of analyticity of both functions f and g. Then, by the resolvent equation (1.7.3) and the Cauchy integral formula, it follows that

f (A)g(A) 4~-2 -

47r2

1

~r f(s 1

f ()~)R()~; A) d)~ fr 2 g(p)R(p;A)d# # - s163

A) - R(p; A)) d)~d#

2

1 f r f(s163 27ri 1

A) { ~ 1 f r (, _ )~)-l g(,) d , } d)~ 2

1 jfr g(p)R(p; A) { ~ 1 f r (, _ )~)-l f ()~) ds } d , 2~ri 2 1

= 27ri

f()~)g()~)R(;~; A) d)~ - (fg)(A), 1

which completes the proof of (ii). To prove (iii), let us observe that, by virtue of Theorem 1.7.4, there exists e > 0 such that B(0, r~(A) + e) c II3, where r~(A) is the spectral radius of the operator A. See Definition 1.7.1. Since f is analytic on II3, the series f (A) - ~n~=OCn)~n is uniformly convergent on B(0, rz(A) + e). From the Cauchy integral formula and the Laurent expansion of the function ~ R(A; A) (see (1.7.6)), we deduce

1 ~~ ck f r A k R ( A ; A ) d A 1 ;(k~__~~ ck Ak) R(/~;A) d A - 2~i f (A) - 2~i k=O

= 27ri l E~ ck ~~ fr ,~k-nA n- 1 d)~ k=0

n=l

ck A k, k=0

which proves (iii). Obviously, (iv) follows from (1.8.1), while (v) is a consequence of (1.8.1) and Phillips theorem which says that, for each densely defined linear and closed operator A : D(A) C X -+ X, we have

p(A) = p(A*) and R(~; A*)= R()~; A)*

30

Preliminaries

for each A C p(A). See Yosida [136], Theorem 2, p. 225. It remains to show (vi). To this aim, fix A E a(A) and let us observe that A C D. Let us define the function g :D --+ C by

for f'(A)

#

for p -

)~.

Obviously g C 9"(A) and so, from Theorem 1.8.1 (i), it follows that

f ()~)I - f (A) = ()~I - A)g(A). So, if f ( ) ~ ) I - f ( A ) has a bounded inverse B, then g(A)B, which in accordance with the relation above is the inverse of ( A I - A), is bounded too. Hence, if A E a(A), then f(A) C a ( f ( A ) ) , implication which is equivalent with f ( a ( A ) ) C a ( f ( A ) ) . Now, let us assume by contradiction that there exists A e a ( f ( A ) ) such that A ~ f ( a ( A ) ) . In view of Remark 1.8.1 and Theorem 1.7.2, f ( a ( A ) ) is a compact set and therefore, in these circumstances, the function h(#) - ( f ( # ) - A) -1 is in 5(A), because f ( # ) - A 5r 0 for each # C a(A) and f is analytic on an open neighborhood of a(A). By virtue of (ii) in Theorem 1.8.1, we have h ( A ) ( f ( A ) - )~I) = I, relation which contradicts the supposition )~ E a ( f ( A ) ) . This contradiction can be V1 eliminated only if a ( f ( A ) ) C f ( a ( A ) ) . The proof is complete. 1.8.2. Let A C L ( X ) , f : D ~ C in 9:(A) and g E 9=(f(A)). Then the function h: D --+ C, h()~) = g ( f ()~)) for each )~ C ID, is in ~(A) and h(A) = g ( f (A)).

Theorem

P r o o f . From (vi) in Theorem 1.8.1 it follows that h C 9:(A). Let D1 be an open neighborhood of cr(f(A)) whose boundary F1 consists from a finite number of Jordan rectifiable arcs and such that D1 U F1 is included in the domain of analyticity of the function g. Let D2 be another open neighborhood of a(A) whose boundary F2 consists from a finite number of Jordan rectifiable arcs and such that D2 t2 F2 is included in the domain of analyticity of the function f with property that f(D2 U F2) C_ D1. Then, for each A C D1, we have 1 f r (A - f ( ~ ) ) - l R ( # ; R(~; f (A)) - 27ri 2

A) d#

this because, by virtue of (i) in Theorem 1.8.2, the operator S defined by the integral on the right-hand side satisfies ()~I- f (A))S = S ( A I - f (A)) = I. Then, from the Cauchy integral formula, we have

g(f (A)) - 27ri

g(A)R(A; f (A)) d)~ 1

Functional Calculus for Unbounded Operators =

47~2

~

31

2g(A)(A- f(#))-lR(tt;A)d#dA

= 4~ 2

g(f(p))R(#; A) d# - h(A). 2

[:]

The proof is complete. 1 . 9 . F u n c t i o n a l C a l c u l u s for U n b o u n d e d

Operators

Let X be a complex Banach space and A" D(A) C X ~ X a closed linear operator. As we already have seen in the preceding section, for A C L ( X ) and f analytic in an open neighborhood D of or(A) whose b o u n d a r y F consists of a finite union of Jordan rectifiable arcs, oriented in the positive sense, we can define f ( A ) by

f (A) - 2~i

/(A)R(A; A)dA,

(1.9.1)

where F is the b o u n d a r y of D. In the case of an unbounded operator A, one may happen t h a t or(A) = C, and therefore, the integral on the righthand side of (1.9.1) is meaningless, because F = 0. More t h a n this, even in the case in which or(A) # C, or(A) might be unbounded. Consequently, in (1.9.1), one should add the contribution of the point at oc. This explains why, throughout this section, we shall assume that a(A) r C and we shall show what are the modifications we have to do to (1.9.1) in order to give it a sense in the case in which both A and or(A) are unbounded. Let A : D(A) C_ X ~ X be a closed linear operator and let 9"(n) the set of complex functions which are analytic in a neighborhood of the set or(A) and at oc. Let c~ E p(A) be fixed and let us define the mapping P~ : X --+ D(A) by ga = ( A - aI) -1 = - R ( a ; A). Obviously P~, which is a bijection from X to D(A), belongs to L ( X ) and thus, by using the arguments in the preceding section, we can define f(Pa). We shall show in the sequel that, by means of Pa, we can define an operator which does not depend on a and which is a quite appropriate candidate for f ( A ) . More precisely, let ~ = C U {o c} be the complex unit ball and let Ca : S -+ S be the homeomorphism defined by (), -

0 (3O

for A r c~ and X r oc for A - ec for A - a.

(1.9.2)

1.9.1. If ~ e p(A), then C a ( a ( A ) U {co}) - a ( P a ) and f ~ h~, defined by ha(p) - f(r is a bijection from ~=(A) to 5:(Pa).

Lemma

Preliminaries

32

P r o o f . Let A c p ( A ) , A : o~. We prove that I t - ( A - a ) -1 C p(Pa). To this aim, let us observe that # - ( A - c~)-1 ~ 0 and

(A-oJ)(A-

AI)-I - [ ( A - AI) + I I ] ( A - A / ) - 1 -

On the other hand, we have

(A

-

-

aI) ( A

-

AI) -

(A - a I ) -

1 _

11

p~

Pc~

#

1

[

(A -

otI)

-

- #(#I-

I ( A - AI)-I +

! I

#

]-

1

Pc~) -1

which shows that (A -

I)-1 -

.2

- P

)-I -

(1.9.3)

So # c p(P~). Conversely, if # E p(P~), # 7/=O, then

(pI- g~)-lg~

[pffl (#i

pa)]-I

2)-1

I(A_

AI)-I

P p~--1 _ A- aI and therefore A C p(A). Obviously 0 C a(P~) b e c a u s e is unbounded. Since the last assertion follows from the definition of the D function r this achieves the proof.

T h e o r e m 1.9.1. Let A 9 D(A) C X -+ X be a closed linear operator with p(A) ~ 0 and let f C ~(A). Let a C p(A), Pa - ( A - a I ) -1 and ~a - f ( r where Ca is defined by means of (1.9.2). Let D be an open neighborhood (possibly unbounded) of a(A) whose boundary F is the union of a finite number of Jordan arcs oriented in the positive sense. Then

~ ( P ~ ) - f (oc)I + ~

f(A)R(A; A) dA.

(1.9.4)

P r o o f . We may assume without loss of generality that a ~ D U F. Indeed, if this is not the case, thanks to Cauchy theorem, we can modify the domain D without altering the value of the integral in (1.9.4) in order to have a ~ DU F. Then Da - q~-I (]]~)) is an open set which includes a(Pa) and whose boundary Fa - r is a union of a finite number of Jordan rectifiable arcs positively oriented. In addition, ~a - f ( r is analytic on D. Since ~a(0) - f(oo) and 0 C a(Pa), from (1.9.3), observing that d A - - d p / p 2, we get 1 f r f(A)R(A; A) dA 27ri =

1 fr 2~i

(fla ( # ) [ - # - 1 1 4- R(#; Pa)] d#

Problems

33

which completes the proof. D e f i n i t i o n 1.9.1. Let with p(A) # ~, let 113 be the union of a finite set be an analytic function

VI

A 9 D(A) C_ X ---+ X be a closed linear operator an open neighborhood of a(A) whose b o u n d a r y is of Jordan rectifiable arcs positively oriented, let f on D and at oc and let c~ E p(A). By definition

f(A) -~(P~). In view of Theorem 1.9.1, the operator f ( A ) is well-defined in the sense that it does not depend on a E p(A) and, in addition f (A) - f (oc)I + ~

1;

f (s

A) ds

From Theorem 1.8.1 we deduce" 1.9.2. (Dunford) Let A " D(A) C X ~ X be a closed linear operator with p(A) 7/=O, let f, g E 5(A) and let a, ~ E C. Then" (i) a f + ~g E ~(A) and a f (A) + ~g(A) - ( a f +/3g)(A) ; (ii) f 9 E Y(A) and f (A)g(A ) - (f g)(A) ; (iii) if (fn)nEN are analytic on an open neighborhood D of a(A) and lim fn - f uniformly on ID, then lim fn(A) - f ( A ) in the norm

Theorem

n---+cx~

oS

n---->~

(x) ;

(iv) f E 9=(A*) and f (a*) - (f (A))* ; (v) f ( a ( A ) U {cxz}) - a(f(A)). Problems P r o b l e m 1.1. Let X be a Banach space and which is weakly continuous from the right on [ a, measurable. Does this conclusion hold true if continuous function, f~ being a locally compact with a positive Radon measure?

x : [a, b] --+ X a function b ]. Prove that x is strongly x : f~ -+ X is any weakly topological space endowed

P r o b l e m 1.2. Prove that, for each p E (1, +oc ], W I'p (0, 7r) is compactly imbedded in C([ 0, 7r ]). Give an example of bounded sequence in W 1'1(0~ 71-) having no convergent subsequences in C([0, 7r]). Show that W1'1(0, 7r) is continuously imbedded in C([ 0, 7r]), but not compactly imbedded. P r o b l e m 1.3. Let X - L2(0, 7r) and A ' D ( A )

C_ X ~ X defined by

D(A) - {u E H2(0, 7T); u(0) - u'(0) - u(Tr) - u'(Tr) - 0} A u - u" for u E D(A). Show that A is symmetric. Find A* and D(A*). Show that A is not selfadjoint. This is Exercise 1.8.5, p. 99 in Brezis and Cazenave [31].

34

Preliminaries

P r o b l e m 1.4. Let X - L2(0, 7r) and A ' D ( A )

C_ X -+ X defined by

D(A) - (u C H2(0, Tr); u(0) - u'(Tr) - 0} A u - u" for u C D(A). Show that A is symmetric. Find A* and D(A*). Show that A is self-adjoint. P r o b l e m 1.5. Let X - L2(0, 77) and A ' D ( A )

C_ X -+ X defined by

D(A) - {u C Wl'2(0,~r); u(0) - 0} A u - - u ' for u C D(A). Find A*. This is Exercise 1.8.6, p. 99 in Brezis and Cazenave [31]. N o t e s . The results in this chapter are by now classical, and can be found in several well-known monographs and treatises. The first four sections are adaptations upon Adams [1], Edwards [50], Hille and Phillips [70] and Lions and Magenes [85], while the other ones, are gathered from Brezis [29], Brezis and Cazenave [31], Dunford and Schwartz [49], and also from Yosida [136]. The compactness assertions in Theorem 1.5.4 are due to Rellich and Kondrachov, while the continuous imbeddings where proved by Sobolev. The systematic study of spectral analysis problems for unbounded operators in Hilbert spaces was initiated by Von Neumann in the third decade of the twentieth century. The fundamentals of the functional calculus for unbounded operators in Banach spaces was initiated in the fourth decade of the twentieth century by Dunford~ starting from Peano's idea in 1887 (see [102]) to define the exponential of a matrix, and using the theory of analytic functions of one complex variable with values in Banach algebras. The problems in this chapter are standard and, in one form or another, can be found in the usual textbooks and monographs.

CHAPTER 2

Semigroups

of Linear

Operators

The concept of semigroup of linear bounded operators has its roots in the simple remark that the Cauchy functional equation f ( t + s) = f ( t ) f ( s ) has as continuous nontrivial solutions only functions of the form e ta, with a C It{, combined with the fundamental idea of Peano [102], [103] to define the exponential function of 1 A k , in order to solve explicitly the first-order linear a matrix A by e A - Y~k=0 V., vector differential equation u' = Au + f by means of the variation of constants formula

~(t) - ~A~(0)+

f0 t ~(~-~)A f (~) d~

Roughly speaking, the notion of semigroup of linear operators is a quite natural extension of the exponential of a matrix to the exponential of a possible unbounded operator. Taking advantage of the powerful Functional Analysis' machinery, the Theory of Linear Semigroups tremendously emerged between 1930- 1960 through the major contributions of Stone, Hille, Yosida, Feller, Lumer, Miyadera, Phillips. The aim of this chapter is to introduce the concept of semigroup of linear bounded operators on a Banach space and to present some of its remarkable properties.

2.1. Uniformly Continuous Semigroups Let X be a Banach space and let L ( X ) be the set of all linear bounded operators from X to X. Endowed with the operator norm II " IlL(x), defined by

If/ll~(x)- s~p ffUxll xll_
for each U E L ( X ) , L ( X ) is a Banach space.

Definition 2.1.1. A family {S(t); t _> O} in L ( X ) is a semigroup of linear operators on X , or simply semigroup if" (i) S(O) - I (ii) S(t + s) - S ( t ) S ( s ) for each t, s _> O. 35

36

Semigroups of Linear Operators

If, in addition, it satisfies the continuity condition at t - 0 lim S ( t ) - I, t40

in the norm topology of L(X), the semigroup is called uniformly continuous. E x a m p l e 2.1.1. A first significant example of uniformly continuous semigroup is given by t ~ etA, where e tA is the exponential of the matrix tA. Namely, let A E 2M2~• and let S(t) - e tA for each t > 0, where o~

etA _ ~-~ tnAn n=O

n!

We can easily see that {S(t); t _> O} is a uniformly continuous semigroup of linear operators. More that this, straightforward computations show that that t ~ S(t) is of class C ~ from [0, +oc) to X, and satisfies the first-order differential equation d dt (S(t)) - AS(t) - S(t)A,

(2.1.1)

for each t _> O. Actually, {S(t) ; t _> O} is nothing else than the fundamental matrix of the first-order vector differential equation 1

ttm

Au

which satisfies S(0) - I. We shall see later that all uniformly continuous semigroups are of the form etA, with A C L ( X ) and satisfy (2.1.1) The next example shows that there exist semigroups which are not uniformly continuous. E x a m p l e 2.1.2. Let X = Cub(R+) be the space of all bounded and uniformly continuous functions from R+ to R, endowed with the sup-norm I1" I1~, and let {S(t); t _> 0} c_ L ( X ) be defined by [S(t)f](s) = f ( t + s) for each f C X and each t, s C R+. One may easily verify that {S(t); t _> 0} satisfies (i) and (ii) in Definition 2.1.1, and therefore it is a semigroup of linear operators. As in this specific case, the uniform continuity of the semigroup is equivalent to the equicontinuity of the unit ball in X, property which obviously is not satisfied, the semigroup is not uniformly continuous. D e f i n i t i o n 2.1.2. The infinitesimal generator, or generator of the semigroup of linear operators {S(t) ; t > 0} is the operator A" D(A) C_ X --+ X , defined by D(A) - { x C X ; 3 lim l ( s ( t ) x t~o t

x)}

37

UniSormly Continuous Semigroups

and A x - lim 1

x)

Equivalently, we say that A generates {S(t) ; t _> 0}. R e m a r k 2 . 1 . 1 . If A : D ( A ) C_ X ~ X is the infinitesimal generator of a semigroup of linear operators then D ( A ) is a vector subspace of X and A is a possibly unbounded linear operator.

R e m a r k 2.1.2. It is easy to see that the infinitesimal generator of the semigroup in Example 2.1.1 is A E L(]~n), defined by A x = Ax. This remark clarifies the relationship between semigroups of linear operators and first-order linear differential equations. E x a m p l e 2.1.3. The generator of the semigroup in Example 2.1.2 is given by D(A)-{fcX;

3 lim lt~0~v ( f (t + .) - f ) - f ' strongly in X } ,

and A f - f'. Let us remark that, if f E D ( A ) , then u ( t , s ) satisfies the first-order partial differential equation %t t

~

[S(t)f](s) -

f ( t + s)

%t s -

Accordingly, in this case, we have the following pointwise variant of (2.1.1) d (S(t)x) - A S ( t ) x - S ( t ) A x dt for each x C D ( A ) and each t > 0.

P r o p o s i t i o n 2.1.1. If {S(t) ; t _> 0} is a uniformly continuous semigroup of linear operators then, for each t >_ O, S(t) is invertible. P r o o f . Inasmuch as lim S(t) - I - 0, t40 in the norm topology of L(X), there exists (f > 0 such that IIS(t) - IIlr

< 1

for each t C (0, 17]. Thus, for each t E (0, 5 ], S(t) is invertible. Let t > 5. Then there exist n E N* and 77 E [0, 5) such that t - n5 + r/. Therefore S(t) - S(5)ns(~7), and so S(t) is invertible. The proof is complete, rq

Semigroups of Linear Operators

38

D e f i n i t i o n 2.1.3. A family of operators {G(t); t E I~} in L ( X ) is called a group of linear operators on X if(i) C(O) - I (ii) G(t § s) - G(t)G(s) for each t, s C ~.

If, in addition, lira G(t) - I, t-~0 in the norm topology of L ( X ) , the group is called uniformly continuous. R e m a r k 2.1.3. If {S(t) ; t _> 0} is a uniformly continuous semigroup of linear operators then it can be extended to a uniformly continuous group of linear operators. More precisely there exists a group of linear operators {G(t) ; t c I~} such that G(t) - S(t) for each t _> 0. Indeed, by virtue of Proposition 2.1.1, we can define G(t)" X -+ X by

G(t)

_ f

[ S ( - t ) ] -1

S(t)

ift O.

One may easily see that {G(t) ; t c R} is a uniformly continuous group of linear operators which extends {S(t) ; t _> 0}. The proof of the uniform continuity of this group is left to the reader. C o r o l l a r y 2.1.1. If {S(t) ; t > 0} is a uniformly continuous semigroup of linear operators then the mapping t ~ S(t) is continuous from [0, +oc) to L ( X ) endowed with the operator norm. P r o o f . Let {G(t) ; t C IR} be the uniformly continuous group of linear operators which extends {S(t) ; t > 0} and let t > 0. See Remark 2.1.3. Then lim IIS(t + h) - S(t)ll

h-+0

lim

(x)

h-+0

IIS(t)ll

(x)llG(h)

-

IIl

(x) - o.

As at t - 0 the continuity follows from Definition 2.1.3, this achieves the proof. [:] 2.2. G e n e r a t o r s

of U n i f o r m l y C o n t i n u o u s S e m i g r o u p s

2.2.1. A linear operator A" D(A) C X --+ X is the generator of a uniformly continuous semigroup if and only if D(A) - X and A C L ( X ) .

Theorem

Proof. Since

The "only if" part. Let {S(t) ; t > 0} be uniformly continuous. lim S(t) - I t$0

Generators of Uniformly Continuous Semigroups L(X),

in the norm topology of

39

there exists p > 0 such that

P1 L P S(t) dt - I I

~(x)

< 1.

We notice that the integral here is the Riemann integral of the continuous function S " [O,p] --+ L(X), which is defined by a simple analogy with its scalar counterpart. Consequently, the operator -~fd S(t)dt is invertible and accordingly fP S(t)dt has the same property. Let h > O. Let us remark that

1

-il

(s(h) - I ) L p s ( t )

I ~oP

dt -

-i

1LP

S ( t + h ) d t - -~

S ( t ) dr.

The change of variable t + h - s in the first integral on the right-hand side yields

1

-h (S(h) -

Lp

I)

S(t) dt

i f p+h -h Jh s ( ~ ) &

-

i f '+~S(~) d~ -

= -h~,

-

1LP -h

S(~)d~

~l~0h S(~) d~.

Then

l(s(h) -h

i)

-

(h fP+h ~p

l~oh S ( s ) d s -~

S(s) ds -

) (~0 p S ( t )

dt

)-1

.

But, the right-hand side of the equality above converges for h tending to 0 by positive values, and thus, the left-hand side enjoys the same property. As the convergence in the uniform operator topology of L(X) implies the pointwise convergence, letting h to tend to 0 by positive values, we deduce

A - (S(p) - I)

(jfOp S(t) dt )-1

.

A C L(X), which proves the necessity. The "if" part. Let A C L(X), t >_0 and let

Hence

s(t)

tnA~ ' ~!

- ~ nzO

where A n - A. A... A n times and A ~ - I. We can easily see that {S(t) ; t _> O} is a semigroup of linear operators. In order to prove that this semigroup is uniformly continuous let us remark that Ils(t)

-

zll~(x)

-

~ t ~ n!A n _

n:O

I ~(x)

40

Semigroups of Linear Operators --

tnAn ~.

< t L(X)

n=l

n!

IIAII n ~(x)"

n=l

Since cc t n _ 1

IIAIl~(x) ~

IIAIle t IAll

n=l

we conclude that lim S(t) = I, t$0

in the norm topology of L ( X ) , and thus {S(t); t _> 0} is a uniformly continuous semigroup. To achieve the proof we have to show that A is the infinitesimal generator of this semigroup. To this aim it suffices to verify that lim T(S(t) 1 - I) - A t$O

b

- O. ~(x)

But this follows from the obvious inequality 1 7 ( S ( t ) - I) - A

- t ~(x)

~

t n-2

~=2

A~

< tllAii2et A I ~(x) D

thereby completing the proof. Let us consider now the Cauchy problem u ' - Au + f u(a) - ~,

(e~)

where A C L (X) and f E C([ 0, T ]; X). T h e o r e m 2.2.2. For any (a, ~, f ) C [0, T) x X x C([ a, T]; X), (CiP) has a unique solution u E C l([a,T]; X ) given by the so-called variation of constants, or Duhamel, formula u(t, a, ~, f ) - S(t - a)~ +

fo~ ~ach t c [a, T], ~ h ~ generated by A.

S(t - s) f (s) ds

{S(t); t >__0}, i~ th~ ~nifo~,~ly ~o~tin~o~ g~o~p

P r o o f . A simple computational argument shows that the function u, defined as above, is of class C 1 and verifies u'(t, a, ~, f ) = Au(t, a, ~, f ) + f ( t ) for all t C [a, T ], and u(a, a, ~, f) = ~. The uniqueness follows as in the case X = R n, by observing that the mapping x ~ A x is Lipschitz continuous. See for instance Corduneanu [39], (4.36), p. 65. The proof is complete. [3

41

Co-semigroups. General Properties

2.3. C0-Semigroups. General Properties In this section we introduce a class of semigroups of linear operators, strictly larger than that of uniformly continuous semigroups, class which proves very useful in the study of many partial differential equations of parabolic or hyperbolic type. D e f i n i t i o n 2.3.1. A semigroup of linear operators (S(t) ; t > 0} is called a semigroup of class Co, or Co-semigroup if for each x E X we have lim S ( t ) x - x. t$0

Remark 2.3.1. Each uniformly continuous semigroup is of class Co but not conversely as we can state from the example below.

Example 2.3.1. Let X - Cub(R+) be the space of all functions which are uniformly continuous and bounded from R+ to I~, endowed with the sup-norm II" II~, and let {S(t) ; t _> 0} be defined by [S(t)f](s) - f ( t + s)

for each f C X and each t, s E I~+. We know from Example 2.1.2 that {S(t); t >_ 0} is a semigroup. In addition, this is of class Co. On the other hand, as we mentioned in Example 2.1.2, it is not uniformly continuous because the unit ball in X is not equicontinuous. See Definition A.2.1.

Theorem 2.3.1. If {S(t); t > 0} is a Co-semigroup, M > 1, and w C ~ such that IIS(t)li~(x ) <_ M e t~

then there exist

(2.3.1)

for each t >_ O.

P r o o f . First, we will show that there exist r / > 0 and M _> 1 such that

IlS(t)ll~(x) _< M

(2.3.2)

for each t C [0, r/]. To this aim, let us assume by contradiction that this is not the case. Then there exists at least one C0-semigroup {S(t); t _> 0} with the property that, for each r/ > 0 and each M _> 1, there exists t~,M C [0, ~/], such that

IIS(t~,M)ll~(x) >

M.

Taking r / - 1/n, M - n and denoting t~,M -- tn for n C H*, we deduce

IlS(tn)ll~(x) >

n,

(2.3.3)

where tn E [0, 1/n] for each n C H*. Recalling that, for each x C X, limn-+~ S(tn)X - x, it follows that the family {S(tn) ; n C H*} of linear

Semigroups of Linear Operators

42

bounded operators is pointwise bounded, i.e., for each x C X, the set {S(tn)x ; n c N*} is bounded. By the uniform boundedness principle (see Dunford and Schwartz [49], Corollary 21, p. 66), it follows that this family is bounded in the uniform operator norm II. II (x) which contradicts (2.3.3). This contradiction can be eliminated only if (2.3.2) holds. Next, let t > 0. Then there exist n E N* and ~ C [0, 7/), such that t - nr/+(~. We have

ils(t)llc(x ) -Ilsn(rl)S(5)llc(x) <_ IlS(rl)ll~(x)lls(5)llc(x) <_ M M n. t

But n - t-~ < t_ and thus IIs(t)llc(x) < M M ~ - Me t~ where w - 1 ln M. The proof is complete. D R e m a r k 2.3.2. If {S(t); t >_ 0} is a uniformly continuous semigroup whose generator is A, then (2.3.1) holds with M - 1 and w - I I A I I c ( x ) . D e f i n i t i o n 2.3.2. A C0-semigroup, {S(t) ; t >_ 0} is called of type (M,w) with M >_ 1 and w C R, if for each t > 0, we have

IlS(t)llc(x) <_ Me t~. A C0-semigroup {S(t) ; t _> 0} is called a Co-semigroup of contractions, or of nonezpansive operators, if it is of type (1, 0), i.e., if for each t >_ 0, we have

IlS(t)ll~(x) <_ 1. We shall use also the term of contraction semigroup. C o r o l l a r y 2.3.1. If {S(t) ; t _> 0} is a Co-semigroup, then the mapping (t,x) ~ S(t)x is jointly continuous from [0, +oc) x X to X. P r o o f . Let x, y E X, t _> 0 and h C R* with t + h _> 0. We distinguish between two cases" h > 0, or h < 0. If h > 0, we have IIS(t + h ) y - S(t)xll

<_ IIS(t + h ) y - S(t + h)xll + IIS(t + h ) x - s(t)xll <__[[S(t + h)l[s ) [lY- xll + IIS(t + h)x - S(t)xll <_ Me(t+h)~l]x -- yll + IIS(t)ll~(x)llS(h)x - ~11, which shows that lim S(~)y(~,y)--~(t+0,~) If h < 0, by Theorem 2.3.1, we deduce

S(t)x.

IIS(t + h ) y - S ( t ) x l l - IlS(t + h ) y - S(t + h)S(-h)x[[ _< IIS(t + h)ll~(x)lly- S(-h)xll

43

Co-semigroups. General Properties <- Me(t+h)~ (IIY -- xll + ] l S ( - h ) x - xi]),

which implies that lim S ( 7 ) y - S(t)x. (~,y)-~(t-0,~) The proof is complete.

D

Some basic properties of C0-semigroups are listed below. T h e o r e m 2.3.2. Let A " D ( A ) C_ X --+ X be the infinitesimal generator of a Co-semigroup {S(t) ; t _> 0}. Then (i) for each x E X and each t >_ 0, we have lim 1

ft+h

S ( T ) x dT -- S ( t ) x ;

h~o -h j t

(ii) for each x E X and each t > O, we have

/o

S(7)x d7 e D ( d ) and d

(/o

S(7)x d7

)

- S(t)x-x

;

(iii) for each x E D ( A ) and each t >_ O, we have S ( t ) x E D ( A ) . In addition, the mapping t ~ S ( t ) x is of class C 1 on [0, +co), and satisfies d dt ( S ( t ) x ) - g s ( t ) x - S ( t ) d x ;

(iv) for each x E D ( A ) and each 0 ~ s ~ t < +oc, we have A S ( T ) x dT --

S ( T ) A x dT -- S ( t ) x - S ( s ) x .

Proof. In order to prove (i), let us observe that S(T)X d T - S ( t ) x

< ~

IIS(T)X -- S(t)xll tiT.

h Jr

The conclusion follows from Corollary 2.3.1. Let x C X, t > 0 and h > 0. We remark that 1 / i t S(T)X dT. 1 (S(h) - I) f0 t S(T)X dT -- -fl f 0 t S(T + h)x dT -~ h The change of variable T + h -- s in the first integral on the right-hand side yields 1 fot lft+h -h (S(h) - I) S(~')x dT -- -h Jh =

1/o

h

lfot S(s)xds S ( s ) x d s - -h

S ( s ) x ds + -~ Jt

S(s)xds.

44

Semigroups oS Linear Operators

From this equality and from (i), we deduce that there exists lim h+o 1

(S(h)

-

,)

f0 t

-

which proves (ii). Next, let x E D ( A ) , t > 0 and h > 0. We have 1 -~ ( S ( t + h)x - S ( t ) x ) - S ( t ) A x

1 <_ IIS(t)ll~(x ) -~ ( S ( h ) x - x) - A x

,

inequality which proves that S ( t ) x C D(A), t ~ S ( t ) x is differentiable from the right, and that d+ dt ( S ( t ) x ) - A S ( t ) x - S ( t ) A x .

(2.3.4)

On the other hand, for each t > 0 and h < 0 with t + h > 0, we have 1 -~ ( S ( t + h)x - S ( t ) x ) - S ( t ) A x

1 _< IIS(t + h)ll~(x ) -~ (x - S ( - h ) x ) <_ IIS(t + h)ll~(x)

( 1

--~ (S(-h)x

- x) - A x

- S(-h)Ax

+ I ] S ( - h ) A x - Axll

)

.

This inequality shows that t ~ S ( t ) x is differentiable from the left as well. From (2.3.4) and the continuity of the function t ~-~ S ( t ) A x on [0, +c~), we deduce that t ~ S ( t ) x is of class C 1 on [0, + ~ ) , which completes the proof of (iii). Since (iv) follows from (iii) by integrating from s to t both sides in (2.3.4), the proof is complete. [3 2.4. The Infinitesimal Generator

In this section we shall prove two basic properties of the generator of a C0-semigroup: the density of the domain and the closedness of the graph. First, let us recall the following: D e f i n i t i o n 2.4.1. An operator A " D ( A ) C_ X --+ X is called closed, if its graph is closed in X • X. T h e o r e m 2.4.1. Let A " D ( A ) C X -+ X be the infinitesimal generator of a Co-semigroup {S(t) ; t _> 0}. Then D ( A ) is dense in X , and A is a closed operator.

45

The Infinitesimal Generator

P r o o f . Let x E X and e > 0. Then, by virtue of (i) in Theorem 2.3.2, we have

lf0~

-

S ( T ) x dT e D ( A )

C

and lim -1 f0 c S ( ~ - ) x d T - - z . e$O c Consequently D ( A ) is dense in X. Next, let (Xn)ncN* be a sequence in D ( A ) such that lim X n - X

and

lim A X n - y .

n----~ o o

n---~ r

From (iv) in Theorem 2.3.2, it follows that S(h)~.

- x.

-

S(~)A~.

e~-

for each n C N* and each h > 0. Passing to the limit in this equality, we obtain S(h)x

- 9 -

/0

S(~)y

d~.

By virtue of (i) in Theorem 2.3.2, it follows that there exists lim h;0 h1 f0 h S(~-)y dT -- y. From this relation, and the preceding one, we deduce that x E D ( A ) and A x - y. The proof is complete. D We continue with a uniqueness result. T h e o r e m 2.4.2. If A " D ( A ) C X --+ X is the infinitesimal generator of two Co-semigroups {S(t) ; t > 0}, and {T(t) ; t > 0}, then S(t) - T(t) for each t > O. P r o o f . Let x E D ( A ) , t > 0 and let f ' [ 0 , t] --+ X be given by f (~) - s ( t - ~ ) T ( ~ ) ~

for each s C [0, t ]. By (ii) in Theorem 2.4.2, it follows f is differentiable on [0, t l, and that f'(s) - -AS(t= -AS(t-

s)T(s)x + S(t-

s)T(s)x + AS(t-

s)AT(s)x

s)T(s)x - 0

for each s C [0, t]. Thus f is constant. Hence we have f(0) - f ( t ) , or equivalently S ( t ) x - T ( t ) x for each x e D ( A ) . Since D ( A ) is dense in

46

Semigroups of Linear Operators

X, and S(t), T(t) are linear bounded operators, we easily conclude that S ( t ) x - T ( t ) x for each x C X, which completes the proof. D R e m a r k 2.4.1. Theorem 2.3.2 (iii), and Theorem 2.4.2, assert that, for each ~ E D ( A ) , the function u "[ 0, +oc) --+ X, defined by u(t) - S(t)~ for each t _> 0, is the unique classical solution of the Cauchy problem u'-

Au

(2.4.1)

-

The example below shows that, if ~ E X, but ~ ~t D ( A ) , the function u, defined as above, is not necessarily differentiable on [0, +co). E x a m p l e 2.4.1. Let X - Cub(R+) be the space of all functions which are uniformly continuous and bounded from R+ to R, endowed with the sup-norm I1" I1~, and let us define the linear operator A ' D ( A ) C_ X -+ X by D ( A ) - {u E X ; u' E X } , and A u - u' for each u C D ( A ) . Taking any nondifferentiable function ~ C Cub(R+), we easily observe that (2.4.1) has no classical solution. Indeed, in this case (e2) may be rewritten as Ut~U

s -

whose only classical solution (if there exists any) is given by u(t, s) - ~(t+s) for each t > 0 and s C R. Let A " D ( A ) C_ X --+ X be a linear operator and n C N. We define the nth-order power of A by: A~

'

A 1 -A

and D ( A n) -- {x e D ( A n - 1 ) ; A n - i x E D(A)} A n _ AAn-1

for n > 2. T h e o r e m 2.4.3. Let A " D ( A ) C_ X -~ X be the infinitesimal generator of a Co-semigroup. Then Nn>oD(A n) is dense in X . P r o o f . Let us remark that, for each n C N, D ( A n) is a vector subspace in X. Accordingly, Nn>0D(A n) is also a vector subspace in X. Let x E X, and let ~ : R --+ R+ be a C ~ function for which there exists an interval [a, b] C (0, +co) such that ~ ( t ) = 0 for each t ~ [ a , b]. We define x(~) -

~(t)S(t)xdt,

47

The Infinitesimal Generator

and we remark that lim ~(S(h) 1 - I)x(~) h$O

= l i m e (f0 +~ ~ ( t ) S ( t + h)x d t - f0 +~ ~ ( t ) S ( t ) x dt ) = l h$O i m l-h (Jh +~ ~(t - h ) S ( t ) x at - fO+~ ~ ( t ) S ( t ) x dt ) . For s e (-cxz, a), we have ~(s)= 0, and consequently lim -~(S(h) 1 - I)x(~) h$O

= l ih40 m e-h (f0 +~ ~(t - h ) S ( t ) x dt -

~ ( t ) S ( t ) x dt )

- fo +~ lim ~(t - h) - ~(t) S ( t ) x d t - - x ( ~ ' ) . h.[.0 h Accordingly, x(9) e D ( A ) and A x ( ~ ) = -x(~'). Repeating the above arguments, one may prove by mathematical induction that, for each n C N, x(~) e D ( A n) and, in addition, A n x ( ~ ) - - ( - 1 ) n x ( ~ ( n ) ) . Consequently, x(~p) e Nn>oD(An).

Next, let ~ be a function as above, such that 11+~ ~ ( t ) dt 1, let a > O, and let us define ~ :R -~ (0, +co) by ~(t)-

~

.

Obviously, ~e is of class C ~, qDe(t) = 0 for t ~ [ca, cb], and o+~ ~ ( t ) d t -

1.

Let us remark that IIx(~) - xll-

1 ~ +~

~

~

(~)

1 ~ ~b ( t ) _< C

~ a

-

IIS(t)x- xlldt <

C

(S(t)x - x ) d t

sup

tC[ea,eb]

This inequality shows that limx(~) = x e$o

IIS(t)x-xll.

Semigroups o] Linear Operators

48

and consequently we have x E Nn>oD(An). The proof is complete.

V1

We conclude this section with both a consequence and a completion of (iii) in Theorem 2.3.2. C o r o l l a r y 2.4.1. Let n C H*. Then, for each ~ E D ( A n) and each t > O, we have S(t)~ C D(An), the function u "[0, +~c) -+ X , u(t) - S(t)~ is of class C n and is a solution of the Cauchy problem

u(n)(t)-Anu(t), u(k)(0) - Ak~,

t>__O k - 0, 1 , . . . ~ n -

1.

P r o o f . We proceed by m a t h e m a t i c a l induction. Let us remark that for n - 1 the conclusion follows from (iii) in Theorem 2.3.2. Let us assume that the property in question holds for n E N* and let ( C D(An+I). Since D ( A n+l) C_ D(An), the inductive hypothesis yields u(n)(t) - Anu(t) for eacht>0. Lett>0andhEI~witht+h>0. We have

1 (AnS(t -~- h)~ _ An S(t)~) lh ( ~t(n)(t -~- h) - u (n) (t)) - -~ _ 1 (S(t + h)An~ - S(t)An~) h because ~ C D(An), while, for each ~- > 0, A n and S(T) commutes on D ( A n ) . But An~ C D ( A ) , and consequently there exists lim 1 (S(t -L- h)An~ - S(t)An~) - A S ( t ) A n ~ . h$O -h Passing to the limit for h --~ 0 in the preceding equality, and taking into account that An~ C D(A), we deduce u (n+l) (t) - A S ( t ) A n ~ - A (n+l) S(t)~. Clearly uk(O) -- Ak~ for k - 0, 1, 2 , . . . n and this completes the proof.

[]

Problems Let p C [1~ +co) and let X - lp be the space of real sequences (Xn)nEN. satisfying En%lIXnLp < -~-CX:). This space, endowed with the norm I1" lip, defined by II(Xn)ncNllp Banach space.

--

(En%l

IXnIP) lip

for each

(Xn)nCN e lp~ is a real

P r o b l e m 2.1. Let p e [1, +co), X - lp, (an)riCH* a sequence of positive real numbers, and t E R+. We define S ( t ) ' D ( A ) C_ X -~ Z by

(S(t)(Xn)nEN* )kCN* for each (Xn)neN* G lp.

--

(e-aktXk)kcN,

Problems

49

(i) Prove that {S(t) ; t _> O} is a C0-semigroup of contractions on lp. (ii) Find its infinitesimal generator. (iii) Prove that this semigroup is uniformly continuous if and only if (an)noN* is bounded. Let co be the space of real sequences vanishing at ~ . Endowed with the norm ]]. II~, defined by ]I(Xn)nCN* ]]c~ --- S U P n c N * IXn] for each (Xn)nCN. E cO, this is real Banach space. P r o b l e m 2.2. Let X = co, (an)nEN* a sequence of positive real numbers and t C R+. Let us define S(t) : D (A) C X --+ X by

for each

(S(t)(Xn)nEH* )kcN* - - (e-aktXk)kEN, (Xn)neN* C CO.

(i) Prove that {S(t) ; t > 0} is a C0-semigroup of contractions. (ii) Find its infinitesimal generator. (iii) Prove that this semigroup is uniformly continuous if and only if (an)hEN* is bounded. P r o b l e m 2.3. Let X = Cb(R) (the space of all continuous and bounded functions from R to R, which is a real Banach space with respect to the supnorm), let t e R+, and let S(t) : X --+ X defined by [S(t)f](s) = f ( t + s) for each f C X, and each s C R. Show that {S(t) ; t > 0} is a semigroup of linear operators, which is not of class Co. Find its infinitesimal generator and show that D ( A ) is not dense in X. P r o b l e m 2.4. Let X = Cub(R) endowed with the norm supremum, let t C R, ~ > O, ~ > 0 and let us define G(t) : X -+ X by (N:)

[a(t)f](x)

-

y2.

k---V-f (x -

k=0

for each f C Z and each x C N. Prove that {G(t) ; t E N} is a uniformly continuous group of isometries, whose infinitesimal generator, A : X ~ X, is defined by [Af](x) = s - 5) - f(x)] for each f E X, and each x C R. This is Exercise 9, p. 23 in Goldstein [61]. P r o b l e m 2.5. Let X = LP(]R n), let A be an n • n matrix with real entries and let us define G ( t ) " X --+ X by [G(t)f](x) - f ( e - t A x ) for each t e R, f E X and a.e. for x E R n. Prove that {G(t); t c R} is a C0-group and find its infinitesimal generator. Show that, if ~-2~in__1aii - O, then the group is of isometries. P r o b l e m 2.6. Show that, with X replaced by Cub(Rn), i.e. the space of uniformly continuous and bounded functions from ]~n t o ]~, endowed with

50

Semigroups of Linear Operators

the sup-norm, the family {G(t) ; t C R} defined as in Problem 2.5, although a group, is not a C0-group. P r o b l e m 2.7. Let {S(t) ; t >_ 0} be a semigroup of linear operators with the property that, for each x C X, we have limt;0 S ( t ) x - x in the weak topology of X. Prove that there exists M >_ 1, and w E R, such that IlS(t)Ii~(x) <_ M e tw for each t _> 0. P r o b l e m 2.8. Let {S(t) ; t >_ 0} be a semigroup of linear operators with the property that, for each x C X, we have limt40 S ( t ) x - x in the weak topology of X. Prove that {S(t) ; t _> 0} is a C0-semigroup. This is Dunford theorem. See Pazy [101], Theorem 1.4, p. 44, or Engel and Nagel et al, Theorem 5.8, p. 40. N o t e s . The main results in Sections 2.1 and 2.2, referring to uniformly continuous groups, were obtained independently by Nathan [93], Nagumo [92] and by Yosida [134], but they have their roots in the pioneering works of Peano [102], [103] concerning the exponential function of a matrix. Sections 2.3 and 2.4 contain several classical notions and results, which may be found, in one form or another in the monographs and treatises on semigroup theory mentioned in Preface. With some exceptions, the problems included are adapted from Brezis and Cazenave [31], Engel and Nagel et al [51], Goldstein [61] and Pazy [101].

CHAPTER 3

Generation Theorems

This chapter begins with the presentation of the most fundamental result within the theory of C0-semigoups as: the Hille-Yosida generation theorem. This gives a very precise delimitation of the class of linear operators A, acting in a Banach space X, that generate C0-semigroups containing only operators whose norms do not exceed 1. Next, we prove both the Lumer-Phillips generation theorem, which is a very useful reformulation of the latter, and the Feller-Miyadera-Phillips generation theorem which extends the Hille-Yosida theorem to the general case of arbitrary C0-semigroups. After presenting some useful consequences and some simple examples, we introduce and study the dual and respectively the sun dual of a C0-semigroup, and we conclude with the celebrated Stone Generation Theorem.

3 . 1 . T h e Hille-Yosida T h e o r e m . N e c e s s i t y The goal of the next two sections is to prove the most i m p o r t a n t result in the theory of C0-semigroups: the Hille-Yosida theorem. More precisely, we shall present a necessary and sufficient condition in order that a linear operator A generate a C0-semigroup of contractions. See Definition 2.3.2. We recall that, if A : D(A) C_ X --+ X is a linear operator, the resolvent set p(A) is the set of all those complex numbers A, called regular values, for which R ( A I - A) is dense in X and R(A; A) = ( A I - A ) -1 is continuous from R ( ) ~ I - A) to X.

T h e o r e m 3.1.1. (Hille-Yosida) A linear operator A : D(A) C_ X -~ X is

the infinitesimal generator of a Co-semigroup of contractions if and only

i/: (i) A is densely defined and closed and (ii) (0, +oc) C_ p(A) and for each A > 0

1

IIR(A; A)ll~(x) <_ -~. 51

52

Generation Theorems

R e m a r k 3.1.1. Since ( A I - A) -z - ) ~ - 1 ( i _ )~-IA)-I whenever at least one of the two sides of the equality is well-defined, it follows that (ii) is equivalent with: AA) -1 C L(X) and

(ii') for each A > 0 we have ( I -

II(I - AA) -111f~(x) -~ 1. Moreover, let us observe that if A is densely defined and satisfies (jj) for each A > 0, A I - A is invertible with continuous inverse and 1 then it satisfies (i) and (ii). Indeed, by (jj), we easily deduce that ( I - A ) -1 is closed. So I - A is closed and accordingly A enjoys the same property which proves that (i) holds. As (jj) clearly implies (ii) the proof is complete. We may now proceed to the proof of Theorem 3.1.1. P r o o f . We begin with the necessity. Let A : D ( A ) C_ X ~ X be the infinitesimal generator of a C0-semigroup of contractions {S(t) ; t >_ 0}. In view of Theorem 2.4.1, A is densely defined and closed. Thus (i) holds. In order to prove (ii), let A > 0, x E X, and let us define

R(~)~- fo+~

e-~ts(t)xdt.

We notice that the integral on the right-hand side of the equality above is convergent. Indeed, for each a, b _> 0, a _< b, we have

fa

be-zts(t)xdt

<_

~a b

_
~-~11~11dt

e-~a _ e-~b -

~

IIxll.

Accordingly, we are in the hypotheses of the Catchy test, and thus the integral is convergent. Clearly R(~) C L(X) and

IIR(~,)xll <_

e-:~ts(t)xdt

<_

e-~tllS(t)ll~(x)]lxLLdt

Hence 1

<_

~llxll-

The Hille-Yosida Theorem. Necessity

53

We prove next that R(A) coincides with R(A; A). To this aim we show that R(A) is both the right and the left inverse of the operator A I - A. Let xCX, A>0andh>0. We have ] :

h

1/o+

= -~

(S(h)

-

1/o+

e-~ts(t + h)x dt - -f

e ~h - 1 f + ~ h Jo

e_Ats(t)x dt

e - ~ t s ( t ) x dt

eAh ~0h e-Ats(t)x

dr.

As the right-hand side of the above equal ity converges to A R ( A ) x - x, it follows that R(A)x C D(A), and

AR(A) = AR(A) - I, which proves that ( A I - A)R()~) = I.

So, R(A) is the right inverse of ) , I - A. Next, let x C D(A). Let us remark that

R(A)Ax-

~o co e-~t-~d (S(t)x) dt fo +~ e - ~ t s ( t ) A x dt -

= t-~cclime - ~ t s ( t ) x -

x + ~ ~oo+~ e - ~ t s ( t ) x dt - AR(A)x - x.

This equality may be equivalently rewritten as

R(~)(M-A)

=I,

which shows that R(A) is the left inverse of A I - A, and this completes the proof of the necessity. E] R e m a r k 3.1.2. Using similar arguments, one may prove that, whenever A generates a C0-semigroup of contractions, then {/~ C C; Re ~ > 0} C_ p(A) and for each ~ E C with Re ~ > 0, we have

IIR( ;

1

54

Generation Theorems

3.2. T h e H i l l e - Y o s i d a T h e o r e m . Sufficiency

In order to prove the sufficiency, some preliminary lemmas are needed. First, let us observe that, by (i) and Theorem 1.7.1, it follows that, for each ~ > 0, R(~; A) e L ( X ) . D e f i n i t i o n 3.2.1. Let A ' D ( A ) C_ X --+ X be a linear operator satisfying (i) and (ii) in Theorem 3.1.1, and let ~ > 0. The operator A~ 9X ~ X, defined by A~ - )~AR()~; A), is called the Yosida approximation of A. L e m m a 3.2.1. Let A " D ( A ) C_ X --+ X be a linear operator which satisfies (i) and (ii) in Theorem 3.1.1. Then"

lim )~R($; A ) x - x

(3.2.1)

)~--+or

for each x C X ,

(3.2.2)

A ~ x - A2R(A; A ) x - Ax for each x C X , and

lira A ~ x -

(3.2.3)

Ax

/k-+ oc

for each x E D ( A ) .

P r o o f . Let x C D ( A ) and )~ > 0. We have IIAR(A;A)x - xll - IIAR(A; A)xll - IIR(s

1 <_ -~llAxll,

and consequently lim )~R(~; A ) x - x

)~-+c<)

for each x C D ( A ) . Since D ( A ) is dense in X and II)~R()~;A)II~(x) < 1, from the last relation, we deduce (3.2.1). To check (3.2.2), let us remark that we have successively )~2R()~; A) - s

- s163

A) - s 1 6 3 - A)R(s A) - )~AR(s A) - A~.

Finally, if x e D ( A ) , by (3.2.1), we have lim A ~ x )~--+ o c

lim AAR(A; A ) x )~--+ o c

lim AR(A; A ) A x - A x , ~--+ o c

which concludes the proof of Lemma 3.2.1.

Q

L e m m a 3.2.2. Let A " D ( A ) C X ~ X be a linear operator which satisfies (i) and (ii) in Theorem 3.1.1. Then, for each )~ > O, Ax is the infinitesimal generator of a uniformly continuous semigroup {etA~ ; t > 0} satisfying IletA~IIjz(x) < 1

(3.2.4)

for each t > O. In addition, for each x E X and each )~, # > O, we have -

<

tllA x - A,xLI

(3.2.5)

The Hille-Yosida Theorem. Sufficiency

55

P r o o f . As A~ E L ( X ) , by Theorem 2.2.1, it follows that it generates a uniformly continuous semigroup {etAx; t >_ 0}. In order to check (3.2.4), let us remark that, by virtue of (3.2.2) and (ii), we have

< et)~211R(~;A)llL(x)e-t)~ < et)~e-t~ -- 1. Since A~, Ap, e tax and e tA" commute each to another, we have

II A x- A xll<

~01 d (estAx (1-s)tA~

x)

/o 1

A,xll,

which completes the proof.

D

We proceed to the proof of the sufficiency of Theorem 3.1.1. Proof of Theorem 3.1.1 (continued). From (3.2.3) and (3.2.5), it follows that, for each t >_ 0, there exists a linear operator S(t) 9 D(A) C_ X --+ X such that, for each x C D(A), lim e t A x x - S(t)x

A-+(x)

uniformly on compact subsets in ]R+. By (3.2.4) we deduce that IIS(t)xll _< Ilxll for each t > 0 and x C D(A). Since D(A) is dense in X, it follows that S(t) can be extended by continuity to the whole space X. It is easy to see that the family of linear bounded operators thus obtained is a semigroup, denoted for simplicity again by {S(t) ; t >_ 0}. Clearly it satisfies

IIs(t)ll~(x)_

1.

In addition, for each t > 0 and x, y C X, we have

IIs(t)x - xll <_ IIS(t)x - s ( t ) y l l + IIS(t)y - etA~yll + IIdA~y - Yll + IlY - xll <_ IIS(t)Y

-

etA~y][ + IletA~y -- Yll + 211Y -- xll.

Let T > 0 and e > 0. Fix y - x~ E D(A), with I 1 ~ - x~ll _< c, , n d sufficiently large ~, such that

][S(t)x~ - etA~x~ll <_ e for each t C [0, T ]. By this inequality, we deduce

IIS(t)x - xll _< 3c +

IIdA~x~

-

x~ll.

(3.2.6)

56

Generation Theorems

Inasmuch a s {etAx; t >_ 0} is a uniformly continuous semigroup, for the very same s > 0, there exists g(s) > 0, such that lietAx - / l l ~ ( x ) _ c for each t E (0, 5(s)). Consequently

IIdA~x~- x~ll ~ lid A~- Ill~(x)llx~ll ~ cllx~ll for each t C (0,~(s)). Since {xE ; s > 0} is bounded, this inequality, along with (3.2.6), shows that {S(t) ; t >_ 0} is a semigroup of class Co. To conclude the proof, we have merely to show that the infinitesimal generator, B " D ( B ) C_ X --+ X , of this semigroup coincides with A" D ( A ) C_ X ~ X . To this aim, let x C D ( A ) and h > 0. We have lim

e tA~

A:xx

-

S(t)Ax

uniformly on compact subsets in R+. Indeed,

lid A~A),x - S(t)Axl[ <_ lietax A),x - e tax Ax[[ + lid A~ A x - S(t)Ax]]

IIdA~' II~(x)llA~x- Axll + lid A~'Ax - S(t)Ax[[. But this relation, along with (3.2.3) and with the partial conclusions above, proves that S(h)x

x

-

lim (ehAxx -- x'~ -- lim A~oo

\

/

A~oo

/0

e t A x A ~ at -

/0

S ( t ) A ~ dt.

Dividing both sides this equality by h and letting h tend to 0 by positive values, we deduce that x C D ( B ) and B x = Ax. Finally, we show that D ( A ) = D ( B ) . Since B is the infinitesimal generator of a C0-semigroup of contractions, from the necessity it follows that 1 E p(B). Accordingly I - B = D ( B ) . As ( I - B ) D ( A ) = ( I - A ) D ( A ) and, is invertible and ( I - B ) - I X by (ii), ( I - A ) D ( A ) = X, it follows that ( I - B ) D ( A ) = X, or equivalently ( I - B ) - I X = D(A). Hence D ( A ) = D ( B ) , which completes the proof of Theorem 3.1.1. 77 3.3.

The Feller-Miyadera-Phillips

Theorem

As concerns the case of general C0-semigroups of type (M, w), we have the following generation theorem due to Feller-Miyadera-Phillips. 3.3.1. The operator A " D ( A ) C_ X --+ X is the infinitesimal generator of a Co-semigroup of type (M, w) if and only if"

Theorem

(i) A is densely defined and closed and (ii) (w, +c~) C_ p(A) and, for each ~ > w and each n E N*, we have M IIR(A; A)nlIL(x) ~- (A -- w) n"

The Feller-Miyadera-Phillips Theorem

57

In order to prove Theorem 3.3.1 we need the following renorming lemma. L e m m a 3.3.1. Let A : D(A) C X --~ X be a linear operator satisfying both (0, +cx~) C a(A) and ]])~nR(A; A)nilL(x) <_ M

for each n E N and A > O. Then there exists a norm I'1 on X such that iixll _< Ixi <_ Milxlt

(3.3.1)

Is163 A)x i ~ Ix]

(3.3.2)

and for each x C X and )~ > O. P r o o f . For # > 0 let us define 1" I," X --+ R+ by Ixl, - sup IlpnR(#; A)nxlI. nCN

It is easy to see that

IIxlJ <

<_ Mllxii

(3.3.3)

and

A)xl

ixi .

We shall prove next that I)~R(A; A)xlt ~ <_ Ixit~

(3.3.4)

for each A e (0, # ]. Indeed, by the resolvent equation (1.7.3), we have R(A; A)x - R(#; A)(x + ( # - A)R(A; A)x) and therefore

Consequently )~iR()~; A ) x l , ~_ ]xl, which proves (3.3.4). From (3.3.3) and (3.3.4) we deduce that, for each n e 1~ and A e (0, # ], we have ]]s163

A)~xll <_ Is

A)~x], < Ixl,.

(3.3.5)

Passing to the sup for n C N on the left-hand side of the inequality above, we get Ixl~ < ]xl, for each ~ e (0, #]. Now, we can define Ix]-

lira Ix],.

pt----~ cx~

Since (3.3.1) readily follows from (3.3.3), and (3.3.2) from (3.3.5) by taking n = 1, the proof is complete D We may now pass to the proof of Theorem 3.3.1.

Generation Theorems

58

P r o o f . Necessity. We consider first the case w = 0. Let us observe that for any norm on X which is equivalent with the initial one, all the topological properties of both the generator and the semigroup preserve. So, we begin by choosing an equivalent norm with respect to which the C0-semigroup {S(t) ; t > 0} of type (M, 0) becomes of type (1,0), i.e. a semigroup of contractions, and then we apply Hille-Yosida Theorem 3.1.1. We define II1" II1: X -+ 1~+ by IIIxlI[

sup IlS(t)xll, t>0

and we observe that

Ilxll _< Illxlll < MIIxll,

(3.3.6)

for each x C X. Thus II1"III and I1" II ~re equivalent norms on X. In addition

IIIS(t)xlll - sup IIS(s)s(t)xll _< sup IIS(t)xll - IIIxii[ s>0

t>0

and thus {S(t) ; t > 0} is a C0-semigroup of contractions on (X, I1[" Ill)From Hille-Yosida Theorem 3.1.1 it follows that A is closed, densely defined,

(0, +co)C_ a(A) and 1

IIIR(A; A)III~(x)_< X

(3.3.7)

for e~ch A > 0. By the definition of II1" III, (3.3.6) and (3.3.7) we have II~nR(~; A)nxll <_ IIl~nR(~; A)nxlll <-Hlxlll ~< MIIx[I, which show that the conditions (i) and (ii) are necessary. Sufficiency Let us assume that (i) and (ii) hold. By Lemma 3.3.1 we know that there exists an equivalent norm 1" ] on X satisfying IIxLD ___ Ixl ___MIIxll and [)~R($; A)xl < Ix[ for each x c X and )~ > 0. So, on (X, l" ]), A satisfies the hypotheses of Hille-Yosida Theorem 3.1.1 and therefore it generates a C0-semigroup of contractions {S(t) ; t > 0}. Finally, since

IIS(t)xll <__Is(t)xl <~ [xl ~ MIIxll for each t >_ 0 and x E X, this achieves the proof in the case w - 0. For the general case, we have only to observe that, if {S(t) ; t >_ 0} is a C0semigroup of type (M,w) and A is its generator, then {e-WtS(t); t >_0} is a C0-semigroup of type (M, 0) and A - wI is its generator. [:]

3.4. T h e L u m e r - P h i l l i p s T h e o r e m Let X be a real, or complex Banach space with norm ]1" II, and let us recall that the duality mapping F" X -+ 2 X* is defined by

F ( x ) - {x* E X * ; ( x , x * ) - [[x]l2 -[[x*[[ 2}

The Lumer-Phillips Theorem

59

for each x E X. In view of the Hahn-Banach theorem, it follows that, for each x E X , F(x) is nonempty. D e f i n i t i o n 3.4.1. A linear operator A " D(A) C_ X ~ X is dissipative if for each x C X there exists x* E F(x) such that Re(Ax, x*) < 0. T h e o r e m 3.4.1. A linear operator A " D(A) C X --+ X is dissipative if and only if, for each x C D(A) and ik > O, we have

P r o o f . If A is dissipative, then, for each x E D(A) and )~ > 0 there exists x* C F(x) such that R e ( A x - Ax, x*) <_ O. Therefore

llxll 2 _<

-

Ax, x*) <_ I( x - A x , x*)l <_

- Axllllxll

and this completes the proof of the "only if' part. Next, let x C D(A) and A > 0. Let y~ C F ( ) ~ x - Ax) and, let us observe that, by virtue of ( 3 . 4 . 1 ) , ) ~ x - A x - 0 implies x - 0. So, in this case, we clearly have Re(x*, A x - Ax) - O. Therefore, let us assume that ) ~ x - A x ~ O. As a consequence, y~ J: 0, and thus z~ - Y~/IlY~II lies on the unit ball, i.e. 1. So, we have llxll _< II x - A

II-

-

= ARe(x, z~) - Re(Ax, z~') _~ AIIxll - Re(Ax, z~). Hence Re(Ax, z~,) _< 0 and 1

Re(z~,x) > Ilxll--~llAxll. Now, let us recall that the closed unit ball in X* is weakly-star compact. Thus, the net (z~):~>0 has at least one weak-star cluster point z* C X* with IIz*ll < 1. From the inequalities above, it follows that Re(Ax, z*) ~ 0 and Re(x,z*) > Ilxll. Since Re(x,z*) < I(x,z*)l ~ Ilxll, it follows that (x,z*) - I]xll. Hence x* - I l x l ] z * e F(x) and Re(Ax, x*) ~ O, and this completes the proof. [::] R e m a r k 3.4.1. In the case in which X - H is a real Hilbert space with inner product (-,-), a linear operator A ' D ( A ) C_ H -+ H is dissipative if, for each x E D(A), we have (x, Ax) <_ O. This is an easy consequence of Definition 3.4.1 combined with the observation that, if we identify H with its own dual, the corresponding duality mapping is given by F(x) - {x} for each x C H.

Generation Theorems

60

T h e o r e m 3.4.2. (Lumer-Phillips) Let A " D(A) c X --+ X be a densely defined operator. Then A generates a Co-semigroup of contractions on X if and only if (i) A is dissipative; (ii) there exists )~ > 0 such that ) ~ I - A is surjective.

Moreover, if A generates a Co-semigroup of contractions, then ) ~ I - A is surjective for any )~ > O, and we have Re(Ax, x*) < 0 for each x C D(A) and each x* C F(x). P r o o f . Necessity If A is the infinitesimal generator of a C0-semigroup of contractions {S(t); t > 0}, by Hille-Yosida Theorem 3.1.1, we have (0, +c~) C_ p(A) and therefore ) ~ I - A is surjective for each ~ > 0. Next, if x e D(A) and x* e F(x), we have

I(S(t)x,x*)[ <_ IIx*llllS(t)

ll _< Ilxll 2.

$o R e ( S ( t ) x - x,x*) <__Re(x, x * ) - Ilxll~ _< 0. Dividing by t > 0 and letting t $ 0, we deduce Re(Ax, x*) <_ 0 and this completes the proof of the necessity. Sufficiency Since A is dissipative~ for each A > 0 and x C D(A), we have

llxll _< I1( 1- A)xll.

(3.4.2)

Inasmuch as ~ I - A is surjective for some ~, say ~ - #, from (3.4.2), it follows that # I - A has a bounded inverse, and thus it is closed. Hence A is closed. We will prove next that ~ I - A is surjective for each ~ > 0. To this aim, let us consider the set A - {~; 0 < )~ < +co, ) ~ I - A is surjective}. So, it suffices to check that

A - (0,

(3.4.3)

In view of (3.4.2), it follows that A C p(A). Recalling that, by virtue of Theorem 1.7.2, p(A) is open, it follows that, for each ~ C A there exists an open neighborhood of A, V C C, which is contained in p(A). The intersection of V with the real line is clearly included in A, and accordingly A is open. To complete the proof of (3.4.3), let (An)oN be a sequence in A with l i m n ~ / ~ n - - /~ > 0 . Then, for each n E N and y C X, there exists Xn E D(A) so that / ~ n X n - - Axn - y. (3.4.4)

Some Consequences

61

By virtue of (3.4.2), it follows that there exists C > 0 such that

llxnll <_

1

I111 ___ C.

Also from (3.4.2), we get

)Xm]]Xn -- Xm[I ~ II~m(Xn -- Xm) -- A(x~ - xm)ll --I/~n -- Amlllxnll ~ c l a n - Aml. Consequently (x~)ner~ is a Cauchy sequence. Let x - limn-~oo Xn and let us observe that, in view of (3.4.4), we have limn-~oo Axn - ,Xx - y. Since A is closed, if follows that x E D(A) and )~x - Ax - y. Thus A E A and so A is both open and closed. Since it is nonempty (we recall that p E A), we necessarily have A - (0, + ~ ) and this completes the proof. D

3.5. S o m e C o n s e q u e n c e s We begin with the following remarkable consequence. C o r o l l a r y 3.5.1. Let A " D(A) C X --+ X be the infinitesimal generator of a Co-semigroup and let ]].IID(A)'D(A) -+ N+ and]. ID(A)'D(A) -+ R+ be defined by llxllD(A) --llxl] + IIAxll, and respectively by IXID(A) - - I I x - Axl], for each x E D(A). Then: (i) ]] " lID(A) is a norm on D(A), called the graph norm, with respect to which D(A) is a Banach space; (ii) D(A) endowed with the norm II'[ID(A) is continuously imbedded in X; (iii) A E L ( D ( A ) , X ) (where D(A) is endowed with I[ " lID(A)) ; (iv) I" ]D(A) is a norm on D(A) equivalent with [[ 9 lID(A) ; (v) I - A is an isometry from (D(A), I" ID(A)) to (X, l[" ll); (vi) for each x E D(A), S(. )x E C([0, +c~); D(A))NCI([o, +c~); X) 1. P r o o f . (i) Inasmuch as d is linear, we may easily verify that ][" liD(A) is a norm on D ( d ) . Let (Xn)ncN be a Cauchy sequence in the norm II " liD(A)" Then both (Xn)nEN and (Axu)neN are Cauchy sequences in the norm of X. As X is complete and A is closed, it follows that (Xn)nEN is convergent to an element x E D ( d ) . Thus D(A), endowed with the norm ]]'[[D(A), is complete and thus a Banach space. We remark that (ii) and (iii) follow from the inequalities: ]]x[] _< IIxiID(A) and [[dxil _< Ilxl[D(A) for x E D(A). By Theorem 3.1.1, Ilxl] <_ IxlD(A). 1Here C([0, +c~); D(A)) is the space of all continuous functions from [0, +cx~) in D(A), the latter being endowed with the graph norm I]" lID(A)-

Generation Theorems

62

So, JXID(A) ~ IIxIID(A) and Ilxllo(A) ~ 211xl[ + IXID(A) ~ 3IXID(A) for each x 6 D(A), from where it follows (iv). -1 - I, we have I(I--A)-lXlD(A) --Ilxll, for each Since ( I - A ) - I - A ( I - A ) x C X, which implies that I - A is an isometry from D(A) to X. Thus (v) holds. Finally, (vi) is a consequence of (iii) in Theorem 2.3.2 which shows that, for each x C D(A), A S ( . )x C C([ 0, +co); X). From the definition of the graph norm, we finally deduce S ( - ) x c C([0, +co); D(A)) and so (vi) holds. The proof is complete. [] The following theorem shows that the restriction of a generator of a C0semigroup of contractions to its domain, endowed with the graph-norm, is the generator of a C0-semigroup of contractions too. Theorem

3.5.1. Let A : D(A) C_ X --~ X be the infinitesimal generator

of a Co-semigroup of contractions and let X~ = D(A) endowed with the graph-norm 1. ID(A) " X1 ~ 1t~§ defined by lUlO(A) - - I l u - Aull /or u e X l . Then the operator A(1) " D (A(1)) c_ X1 --+ X1 defined by D(A(1)) - {x E X 1 ; A x C X1} A ( 1 ) x - Ax, for x E D(A(1)) ,

is the infinitesimal generator of a Co-semigroup of contractions on X1. P r o o f . Let A > 0 and f C X1 and let us consider the equation

~ u - Au = f. Inasmuch as A generates a C0-semigroup of contractions, by Theorem 3.1.1, it follows that this equation has a unique solution u E D(A). As f C X1, we conclude that Au C D(A), and thus u C D(A(1)). Thus A n - A ( 1 ) u - f. On the other hand we have I ( A I - A(1))-l flD(A) - - I 1 ( I - A ) ( A I - A ) - l f l l 1

1

which shows that A(1 ) satisfies (ii) in Theorem 3.1.1. Moreover, it follows that A(1) is closed in X1. Indeed, as ( k I - A(1)) -1 C L(X1), it is closed, and consequently k I - A(1) enjoys the same property which proves that A(.1) is closed too. Next, let x C X l , k > 0 and let x~ - A x - A(1)x. Clearly x~ E D(A(1)), and, in addition, lim [x:x - X[D(A) -- O.

A--+oc

63

Examples

Indeed, this simply follows from (3.2.3) in Lemma 3.2.1. Thus D(A(1)) is dense in X1 and, by virtue of Theorem 3.1.1, A(1) generates a C0-semigroup [~ of contractions o n X 1. The proof is complete. R e m a r k 3.5.1. An inductive argument shows that, for each infinitesimal generator of a C0-semigroup of contractions A 9 D ( A ) C X --+ X , there exists a sequence (Xn)ncN of Banach spaces, such that "'" ~ Xn+l ~ Xn C " " C X o -

X,

all the inclusions being dense and continuous, and a family of generators of C0-semigroups of contractions (A(n))ncN , with D(A(n)) - Xn+l and Anx - A x for each x C Xn+l. Let us remark that, if A is linear continuous, then Xn - X for each n C H, while whenever A is discontinuous, for each n C H, Xn+l is strictly included in Xn. In addition, for each n C H, Xn - D ( A n) and the norm ]l. ][x~ is equivalent with the norm ]] 9 ]]D(A~). See (iv) and (v) in Corollary 3.5.1. T h e o r e m 3.5.2. The linear operator A" D ( A ) C_ X --+ X is the generator of a Co-group of isometries if and only iS" (i) A is densely defined and closed and (iii) IR* C_ p(A) and for each A E IR* 1

IIR( ; A)llz(x) _< I 1 P r o o f . Necessity. Let us assume that A is the infinitesimal generator of a C0-group of isometries {G(t) ; t C JR}. Then both operators A and - A generate a C0-semigroup of contractions {G(t) ; t >__ 0}, and respectively { G ( - t ) ; t _> 0}. Thus, by Theorem 3.1.1, it follows that A satisfies (i) and (iii). Sufficiency. If A satisfies (i) and (iii), by Theorem 3.1.1, it follows that both A and - A generate a C0-semigroup of contractions {S(t) ; t _> 0}, and respectively {T(t) ; t >__ 0}. Then the family of linear bounded operators {G(t) ; t C IR}, defined by

G(t) -

I

S(t)

if t _> 0

T(-t)

if t_<0,

is a C0-group of isometries, and this achieves the proof.

D

3.6. E x a m p l e s This section contains two simple examples which illustrate the effectiveness of both Theorems 3.1.1 and 3.5.1. For the sake of simplicity, here we confine

Generation Theorems

64

ourselves to the 1-dimensional case. We notice that, in the next chapter, we shall reconsider both these examples in the general n-dimensional setting exactly as they are encountered in the study of parabolic, or hyperbolic partial differential equations. E x a m p l e 3.6.1. In this Example we shall show that, for f~ = (0, 7r), the Laplace operator with the Dirichlet boundary condition is the infinitesimal generator of a C0-semigroup of contractions on L 2 (0, 7r). More precisely we have: P r o p o s i t i o n 3.6.1. The operator

A " D(A) C_ L2(O, ~r) -+ L2(O, 7r), defined by D(A) -//01(0, 7r) N H2(0, 7r) Au - u" for u E D (A),

is the infinitesimal generator of a Co-semigroup of contractions. P r o o f . We apply Theorem 3.1.1 in the space X - L2(0, Tr). By the definition of H2(0, 7r) and by Theorem 1.3.3, it follows that D(A) is dense in L2(O, ~r), and thus the first part in the condition (i) in Theorem 3.1.1 is satisfied. We shall prove the closedness of A by using (jj) in Remark 3.1.1. In order to check (jj), we begin by showing that, for each A > 0, the operator A I - A is bijective. To this aim, let f C L 2 (0, 7r) and let us remark that the equation ()~I- A)u = f may be equivalently rewritten as:

{ Au-u"-f

(361) - 0 .

"

9

At this point, let us recall that the general solution of the nonhomogeneous equation A u - u " = f is given by

U(X) -- C1 (X)e -x/~x -~- C2(X)e Vf-~x, where cl and c2 satisfy

{

c~ ( x ) e - ~ x + c~ (x)e,~x _ 0

_4-~d~(~)~-~x + 4 - ~ d ~ ( x ) ~ x - f(x).

See (4.50), p. 68 in Corduneanu [39]. It then follows

- kl - x + k2 '/-Xx + 2,/-2

y)f (y) dy,

where { efX(y-x) k(x, y) ev~(x_y )

ifO<_x
Examples

65

Imposing the boundary conditions u(0) = u(Tr) = 0, we obtain a system of linear equations, with the unknowns kl and k2, which has a unique solution. Thus, (3.6.1) has a unique solution u - ( A I - A ) - l f . Finally, multiplying both sides in (3.6.1) by u(x) and integrating from 0 to 7r, we obtain which implies 1

As u = ( A I - A ) - l f , the last inequality shows that 1

II(),/- A)-~fIIL~(0,~)_< ~[[fllL~(0,~), for each f C L2(0, It). Taking the supremum both sides in this inequality for ]lf[tL2(0,~r) _< 1, we deduce (jj) and implicitly (ii). See Remark 3.1.1. The proof is complete. [:] C o r o l l a r y 3.6.1. For each ~ e H~(O, rr) A H2(0, 7r), the problem

I

t,-

(t, x ) e s:+ • (0, t e ~+ 9 c (0, ~)

~xx

~(t, 0) - ~(t, ~) - 0 ~(0, x) - ~(x)

~) (3.6.2)

has a unique solution u C C I(IR+; L 2(0, 7r)). P r o o f . The conclusion follows form (iii) in Theorem 2.3.2, by observing that the problem { u' - Au ~,(o)

-

with A defined as in Proposition 3.6.1, may be rewritten as (3.6.2).

D

E x a m p l e 3.6.2. In order to understand the importance of the differential operator we will analyze in this example, let us consider the equation of an isotropic, inextensible and homogeneous string of length 7r, and having fixed endpoints

~ - ~x - 0 ~(t, 0) - ~ ( t , ~ ) ~(0, x) - ~ (x)

- 0

(t, x) e R • (0, t e R x e (0, ~)

~) (3.~.3)

Generation Theorems

66

H~ (0, ~r) x L 2 (0, it), i.e. ~ - v - 0 ~ - ~xx - o ~(t, o) - ~(t, ~) - o

(t, x) e R • (0, ~ ) (t, x) e R x (o, ~) t e R

u(O, x) -- ~I (X)

X e (O, "K)

~(o, x) - ~ ( x )

x c (o,

~).

Identifying the real functions of two real variables u, v : IR x (0, 7r) --+ IR with two functions of only one real variable with values in H~(0, 7r), and respectively in L 2 (0, 7r), i.e. u(t, x) - u(t)(x) and v(t, x) - v(t)(x) for a.e. x C (0, 7r), we observe that, in its turn, this system, may be rewritten as a first-order ordinary differential equation of the form

zt=A z z(0) - r

(3.6.4)

where z(t) = (u(t), v(t)) for each t e IR, ~ = (~1, ~2), and A is defined as in Proposition 3.6.2 below. We endow the product space Hi(0, 7r) x L2(0, 7r) with the inner product (-,.) defined by {(Ul, Vl), ('/~2, v2)) -

/0

~1! (:;C)~ (X) d z nt-

/0

v I (x)v 2 (z) dx

for each (ui, vi) C H I (0, ~-) x L 2 (0, 7r), i - 1, 2. We emphasize that, by virtue of Theorem 1.5.7, H01 (0, ~r) x L2(0, ~), equipped with this inner product, is a real Hilbert space. Proposition

3.6.2. The operator

A " D(A) C_ H~ (O, 7r) x L2 (O, 7c) --+ H~ (O, 7r) x L2 (O, 7r), defined by D(A) - (H~(0, 7r) n H2(0, 7r)) x H~(0, 7r) A(u, v) - (v, u") for (u, v) E D(A),

is the infinitesimal generator of a Co-group of isometries. P r o o f . We show that A satisfies the hypotheses of Theorem 3.5.2. Clearly D(A) is dense in H~(O, ~) x L2(0, 7~). In order to prove that A is closed, we shall use again Remark 3.1.1. Let A C IR*. To check that A I - A is a bijection, let us remark that, for each (f, g) E H~(0, ~) x L2(0, ~), the equation ( A I - A)(u, v) = (f, g) may be equivalently rewritten as

I ),u-vf s - u" - g ~(0) -~(~) - 0 .

(3.6.5)

The Dual of a Co-Semigroup

67

Multiplying the first equation by A, and adding side by side both equations, we obtain A2u-u"-Af+g -

-

0.

Repeating the arguments in the proof of Proposition 3.6.1, we conclude that, for each (f, g) E H 1 (0, 7r)• L 2 (0, 7r), this problem has a unique solution u E H01(0, 7r)A H2(0, 7r). Coming back to the first equation in (3.6.5), we easily see that this has a unique solution v E H 1(0, 7r). So, for each A E R*, ( A I - A) -1 is well-defined. By (3.6.5), we have ( A n - v, A v - u") - (f, g). Taking the inner product in H01(0, ~r) • L2(0, 7r) both sides of the equality above by (u, v), integrating by parts, and taking into account the boundary conditions v(0) = v(Tr) = 0, we obtain

A)-I(f, g)ll 2 - [ ( ( f , g), ( M - A ) - l ( f , g ) } l . By virtue of the Cauchy-Schwarz inequality, this relation proves that, for each A E R*, we have 1 ]](AI - A) -lll~(x) _< Hence A satisfies (jj) in Remark 3.1.1, and thus it satisfies (i) and (ii) in Theorem 3.5.2, from where it follows that it generates a C0-group of isometries. The proof is complete. V] C o r o l l a r y 3.6.2. For any (~1, ~2) E H2(0, 7r) N Hi(0, 7c) x H~(O, 7c), the problem (3.6.3) has a unique solution u satisfying

(U, Ut) E C 1 (1~+; H 1(0, 7r) x L 2 (0, 7r)). P r o o f . The conclusion follows from (iii) in Theorem 2.3.2, recalling that the problem (3.6.3) may be equivalently rewritten as (3.6.4). D 3.7. T h e D u a l of a C 0 - S e m i g r o u p Let X be a Banach space and {S(t) ; t > 0} a semigroup of linear operators on X. D e f i n i t i o n 3.7.1. The family {S(t)* ; t >_ 0} C L(X*), where, for each t > 0, S(t)* is the adjoint of the operator S(t), is called the dual of the

{s(t) ; t _> 0}. Our goal within this section is to present some basic properties of the dual of a semigroup of linear operators. We begin with the following Functional Analysis result. L e m m a 3.7.1. If T e L ( X ) then T* e L(X*) and IITIIz~(x)- ]]T*II~(x. )

Generation Theorems

68

P r o o f . First let us observe that, for each x* E X*, x H (Tx, x*) is a linear continuous functional on X, denoted by y*. Since (x, y*) = (Tx, x*), we have D(T*) = X*. In addition IIT*IIc(x.) -

sup IIT*z*ll - sup sup I(z, T*x*)I iix*ll<_l iix*ll
=

IIT

Ilxll_
II-Ilrllc(x/,

Ilxll
L e m m a 3.7.2. Let A : D(A) c_ X ~ X be a densely defined linear closed operator. If A E p(A), then A E p(A*), and R(A; A*) = R(A; A)*. P r o o f . We have ( A I - A)* = A I * - A * , where I* is the identity on X*. As A is closed, by Theorem 1.7.1, it follows that R(A; A) E L ( X ) . From Lemma 3.7.1, we have that R(A; A)* E L(X*). We shall show next that there exists R(A; A*) and it coincides with R(A; A)*. We begin by proving that AI* - A* is injective. Indeed, if (AI* - A*)x* - O, then

0 = (x, (M* - A*)x*) = ( ( M -

A)x, x*)

for each x E D(A). Since A E p(A), we have R ( A I - A) = X, and thus x* - 0. Hence AI* - A* is injective. If x E X and x* E D(A*), then (x, x*) = ((AI - A)R(A; A)x, x*) = (R(A; A)x, (M* - A*)x*), and consequently

R(A;A)*(AI*-A*)x*=x* for each x* E D(A*). Finally, if x* E X* and x E D(A), we have (x, x*) = (R(A; A ) ( M - A)x, x*) = ((AI - A)x, R(A; A)*x*) which shows that

(AI*-A*)R(A;A)*x* =x* for each x* E X*. So, AE p(A*), and R(A; A * ) = R(A; d)*. Let us remark that, by virtue of Lemma 3.7.1, the dual of a C0-semigroup of contractions is a semigroup of contractions, but, in general, it is not a C0semigroup because it may lack the continuity property limtl0 S(t)*x* = x* for each x* E X*, as we can easily see from the example below. Example i.e.

3.7.1. Let S(t); t E R} C

L(LI(R)) be the

[S(t)f](s) = f ( s - t)

translation group,

The Dual of a Co-Semigroup

69

for each t _> 0 each f E LI(I~), and a.e. for s E R. Recalling that the dual of L~(I~) is Lee(R), we easily deduce that, for each t E I~, the dual group S(t)* " L~(I~) ~ L~(I~) is defined by -

+ t)

for each ~ C L ~ ( R ) , and a.e. for s C I~. Indeed, this follows from

for each f C L 1(I~), and each ~ E L ~ (R). On the other hand, we can easily see that {S(t)*; t _> 0} does not satisfy the condition lim S(t)* ~ t$0

except if p is <'uniformly continuous" on R with respect to ii" iIL~(R)" R e m a r k 3.7.1. We notice that the dual semigroup {S(t)*; t >__0}, even if not a C0-semigroup in general, is weakly-, continuous at 0, i.e. lim(x, S(t)*x* - x*) - 0 t$0 for each x E X. Accordingly, if X is reflexive, the dual semigroup is weakly continuous at t - 0, and by virtue of Problem 2.8 it is a C0-semigroup. We shall give a direct proof of this remarkable result later on. L e m m a 3.7.3. Let X be reflexive, and let A 9 D ( A ) C_ X ---+X be a densely defined linear closed operator. Then A* is densely defined and closed. P r o o f . We begin by showing that A* is densely defined. We proceed by contradiction. To this aim, let us assume that there exist at least one reflexive Banach space, and at least one densely defined linear operator A " D ( A ) C X ~ X , for which D(A*) is not dense in X*. This means that there exists at least one element y** E X**, such that y** =/= 0, and (x*, y**) - 0 for each x* e D(A*). As X is reflexive, the statement above is equivalent with: there exists y E X such that y # 0 and (y,x*) - 0 for each x* E D(A*). Since the graph of A is closed in X • X, it follows that (0, y) 6 graph (A). By a consequence of the Hahn-Banach separation theorem (see Dunford and Schwartz [49], Corollary 13, p. 64), applied to graph (A) and (0, y) in X x X, it follows that there exist x~, x~ e X* such that ( x , x ~ ) - (Ax, x~) - 0 for each x e D(A), and (0, x ~ ) - (y, x~) # 0. From the second relation, it follows both x~ # 0 and (y,x~) ~= 0. On the other hand, from the first relation we deduce that x~ E D(A*) and A*x~ - x~, which implies (y, x~) - 0. This contradiction can be eliminated only if D(A*) is dense in X*. In order to prove that A* is closed, let

Generation Theorems

70

(X~,)nEN be a sequence in D(A*) with the property that limn--+oo x n - x*, and limn--+c~ A*x n- * y*. By the definition of the operator A*, we have

(Ax, x*) -

lim (Ax, Xn) n--+oo

lim (x, A*xn) - (x, y*) n---~oo

for each x E D(A). As D(A) is dense, it follows that x* E D(A*) and A'x* - y * . So, A* is closed, and the proof is complete. [3 3.7.1. Let X be reflexive, and let A " D(A) C_ X ~ X be the infinitesimal generator of a Co-semigroup of contractions {S(t) ; t _> 0}. Then {S(t)* ; t >__0} is a Co-semigroup of contractions whose infinitesimal generator is A* " D(A*) C_ X* ~ X*.

Theorem

P r o o f . We show that A* satisfies the hypotheses of Theorem 3.1.1. By Lemma 3.7.3, A* is densely defined and closed. So, A* satisfies (i) in Theorem 3.1.1. We prove now that (0, +c~) C_ p(A*) and, for each A > 0, we have IIR(A; A*)llc(x. ) _< 1/A. As A is the infinitesimal generator of a C0semigroup of contractions, we have (0, +c~) C_ p(A). But, by Lemma 3.7.2, we have p(A) c_ p(A*) and, thus, (0, +c~) C_ p(A*). Again by Lemma 3.7.2, we know that R(A;A*) - R(A;A)*, and by Lemma 3.7.1 it follows that IIR(A; A)*l]c(x. ) - IIR(A;A)IIc(x ). Recalling that IIR(A;A)II~(x ) _< 1/A for each A > 0, we have that IJR()~;A*)IJc(x.) _< 1/A for each A > 0. Hence A* is the infinitesimal generator of a C0-semigroup of contractions {T*(t); t _> 0}. To conclude the proof, we have merely to check that T*(t) - S(t)* for each t >__ 0. To this aim, let us we recall that, for each x* E D(A*), T * ( t ) x * - lira et(d*)xx *, ,k--+ + o o

where (A*)), is the Yosida approximation of the operator A*. By virtue of Lemma 3.7.1, we have (A*)~ - (A),)*. In addition, e t(Ax)* - (erAs) *, and thus, lim et(Ax)*x*--S(t)*x * lim et(d*)~x * )~--++cxa

~--++oo

for each x* E D(A*). As D(A*) is dense in X*, the proof is complete.

[:]

3 . 8 . T h e S u n D u a l of a C 0 - S e m i g r o u p D e f i n i t i o n 3.8.1. Let T " D(T) C_ X -+ X a linear operator and let X G a subspace of X. The operator T ~ 9 D ( T ~ C X G -+ X ~ defined by

D(T )-{xED(T)NX~ T Gx - T x

TxEX

is called the part of T in X e. As concerns the nonreflexive case, we have"

~

and for each x E D ( T ~

71

The Sun Dual of a Co-Semigroup

T h e o r e m 3.8.1. Let {S(t); t _> 0} a Co-semigroup of contractions on X with the infinitesimal generator A and let {S(t)*; t >_ 0} be the dual semigroup. If A* is the adjoint of A and X ~ the closure of D(A*) in X*, then the restriction S(t) ~ of S(t)* to X e is a Co-semigroup of contractions whose infinitesimal generator A ~ is the part of A* in X ~ P r o o f . Let X ~ - {x ~ E X*; limtt0 IIS(t)*x ~ x ~ l l - 0}. Clearly X | is a subspace in X* and D(A*) is dense in X G. Indeed, if x ~ E D(A*) and x E X, we have I(x, =

Accordingly

S(t)*

x ~ -

(/0' A

x~

-

S(s)xds, x ~

I(S(t)x

)

-

x,

x~

< tll~llllA*x|

lim I ( z , S ( t ) * z | - x |

.

0

t+o

uniformly for [[x[I _< 1 and therefore lira IIS(t)*x ~ - x~ tio

- O.

Thus, D(A*) C_ X ~ On the other hand, let us observe that

IIS(t)| | - x| for each x G, yO E X G. Since, by Lemma 3.7.1,

IIS(t)|174

_< IIS(t)*ll~(x*)- IIS(t)ll~(x)-

1,

the inequality above shows that X | is closed and {S(t)| ; t >__0} is a Cosemigroup of contractions on X G. Let us denote by A G its infinitesimal generator, and let us remark that D ( A ~ C_ D(A*). Thus D(A*) is dense in X G, A ~ is the part of A* in X G and S(t) G is the part of S(t)* in X G, and this achieves the proof. K] D e f i n i t i o n 3.8.2. The C0-semigroup of contractions { S ( t ) ~ Theorem 3.8.1, is called the sun dual of {S(t) ; t >_ 0}.

t >_ 0} in

R e m a r k 3.8.1. If X is reflexive, by Lemma 3.7.3, we have X ~ - X*, and the sun dual of {S(t) ; t >_ 0} coincides with its dual {S(t)* ; t >_ 0}. L e m m a 3.8.1. Let A " D ( A ) c_ X ~ X be the infinitesimal generator of a Co-semigroup of contractions. Then, for each x E X , we have

sup x* E D ( A * )

Ia*ll=l

I(x,x*)l-

sup x(D E D ( A G )

I(x,x|

Ilxll.

Generation Theorems

72

IIx~ 1} are So, both {x* E D(A*); I I x * l l - 111} and {x ~ E D ( A ~ determining sets for X , and X ~ is weakly-star dense in X*. Moreover,

IIS(t)|174

IIS(t)ll

(x).

P r o o f . In view of L e m m a 3.2.1 we have (x, x e) -

lim (nn(n; A)x, x e) n--+oc

lim (x, nR(n; A e ) x ~ n--+oc

As D ( A e) C_ D(A*), n R ( n ; d ~ ~ E D ( A e) and I I n R ( n ; d ~ 1 7 6 <_ Ilxell, this proves the first assertion. The second one is an easy consequence of the first one, while the third one follows from the second one. [3 Example i.e.

3.8.1. Let {S(t) ; t E I~} C L(LI(I~)) be the translation group,

[S(t)f](s) - f ( s - t) for each t _> 0 each f E L 1 (I~) and a.e. for s E IR. As we have already seen in Example 3.7.1, the dual group {S(t)* ; t E ~} C L ) L ~ 1 7 6 is defined by

-

+

t)

for each ~ E L~176 and a.e. for s E IR. One may easily see that, in this case, X ~ - C~b(IR). Indeed, this simply follows from the fact that ~ E X ~ if and only if ~ E L ~ (~) and limll~(. + h) - ~(')IIL~(R) -- 0. h-l.0

3.9.

Stone Theorem

We conclude this chapter with a result concerning the generation of a C0group on a complex Hilbert space H. The next theorem, due to Stone [117], concerns the case of C0-group of unitary operators on Hilbert spaces. We recall that an operator U E L ( H ) is called unitary if

UU* - U * U -

I.

3.9.1. (Stone) The necessary and sufficient condition in order that A " D ( A ) C H --+ H be the infinitesimal generator of a Co-group of unitary operators on H is that iA be self-adjoint.

Theorem

P r o o f . Necessity. If A is the infinitesimal generator of a C0-group of unitary operators {G(t) ; t > 0}, then it is densely defined and for each x E D ( A ) we have

- A x - lim t40 1t ( G ( - t ) x - x) - limt40it (G* (t)x - x) - A*x which implies A - - A * . In view of L e m m a 1.6.2 it follows t h a t iA is self-adjoint and this proves the necessity.

Stone Theorem

73

Sufficiency. If iA is self-adjoint then, by L e m m a 1.6.2, we have A = - A * . By (iii) in L e m m a 1.6.3, it follows that both A and A*, which obviously are densely defined, are closed operators. From (ii) in L e m m a 1.6.3, we deduce that, for each A E I~, A ~ 0, we have (R(s

A)) j- - {x e D(A*); s

- A*x - 0}.

Inasmuch as A is skew-adjoint, we have (Ax, x) = 0 for each x e D ( A ) . Since s A*x = 0 if and only if s + A x = 0, taking the inner product both sides by x, we deduce )~llxlI2 = 0, or x ---- 0. It then follows that (R(s • - {0}, and accordingly, R ( s is dense in H. Let )~ ~ 0, f e H, let (fn)neN and let (Xn)nCN in D ( A ) such that

{ AXn - AXn - fn lim fn - f

for each n C N in H.

u-+c<)

Then we have s - Xp) - A(Xn - Xp) = fn - fp, from where, taking into account that (Ax, x) = 0 for each x e D(A), we deduce ),liXn

-

xpll 2

-

(fn

--

fp, Xn

--

Xp) ~_

]]fn

--

fpllllx

- xpll

for each n , p E N. As A ~ 0 and (fn)~eN is convergent, it follows that (Xn)nC N is a C a t c h y sequence, and thus convergent to an element x E H. Since A is a closed operator, A I - A enjoys the same property, and therefore x C D ( A ) , and ) ~ x - A x = f . Hence A I - A is surjective. In addition, if A x - A x = 0, it follows that Aiixl]2 = 0, which proves that A I - A is injective. Thus, there exists ( A I - A) -1. By the equality I)~[[Ixi]2 - ( f , x ) for x C D ( A ) with ) ~ x - A x = f , we deduce that 1 rIR(A;A)II~(x) <_ IAI for each )~ E I~*. So, A verifies the hypotheses Theorem 3.5.2, and thus it generates a C0-group of operators {G(t) ; t C IR}. Since {G(t) ; t _> 0} is the semigroup generated by A and { G ( - t ) ; t _> 0} is the semigroup generated b y - A = A*, we have G - l ( t ) = G ( - t ) = G*(t) for each t E I~. So, the group contains only unitary operators, and this achieves the proof. [3 R e m a r k 3.9.1. As we shall see in the next chapter, Stone Theorem 3.9.1 is very useful in establishing existence and uniqueness results for homogeneous first-order and second-order hyperbolic problems, as well as for the linear SchrSdinger equation.

74

Generation Theorems Problems

P r o b l e m 3.1. Let X - 12, (an)ncN* a sequence of positive real numbers and let A " D ( A ) C_ X -+ X be defined by

{(Xn)ncN. e 12 ; (akxk)kcN* e 12} (A(Xn)nCN.)keN. -----(akXk)kCN. for (Xn)nCN* e D(A).

D(A)-

Show that A generates a C0-semigroup of contractions on X. P r o b l e m 3.2. Let X - co, (an)nCN* a sequence of positive real numbers and let A " D ( A ) C X --+ X be defined by

{(Xn)nEN* e c0; (akXk)kCN. e CO} (A(Xn)nCN*)keN. ------(akXk)kEN* for (Xn)nEN* e D(A).

D(A)-

Show that A generates a C0-semigroup of contractions on X. P r o b l e m 3.3. Let X - L2(0, ~) and A ' D ( A )

C_ X -+ X defined by

D(A) - {u e H2(0, Tr); u(0) - u'(Tr) - 0} A u - u" for u C D ( A ) . Show that A generates a C0-semigroup of contractions on X. P r o b l e m 3.4. Let X - C~r([ 0, 7r ]) the space of all continuous real functions defined on [0, 7r ] and satisfying u(0) - u(:r), endowed with the usual sup norm. Show that A" D ( A ) C_ X ---+X , defined by D(A)-{ueX; u'eX A u - - u ' for u e D ( A ) , generates a C0-semigroup on X. P r o b l e m 3.5. Prove that if A 9 D ( A ) C_ X ~ X is a densely defined linear operator and both A and A* are dissipative, then A generates a C0-semigroup of contractions. This is Corollary 4.4, p. 15 in Pazy [101]. We recall that a linear operator A is closable if the closure of its graph is the graph of a linear operator A called the closure of A. P r o b l e m 3.6. Let A " D ( A ) C_ X --+ X be a linear, dissipative operator. Then:

(i) If is (ii) If (iii) If

for some )~ > 0, ) ~ I - A is surjective, then for each A > 0, A I - A surjective. A is closable then A, the closure of A, is also dissipative. D ( A ) is dense in X, then A is closable.

This is Theorem 4.5, p. 15 in Pazy [101]

Notes

75

P r o b l e m 3.7. Let A : D ( A ) C_ X --+ X be a linear dissipative operator with I - A surjective. Show that, whenever X is reflexive, D ( A ) is dense in X. This is Theorem 4.6, p. 16 in Pazy [101]. See also Brezis and Cazenave [31], Exercise 1.8.2, p. 99. P r o b l e m 3.8. Let X = [0, 1] endowed with the by D(A) Au-

C([ 0, 1 ]) the space of real continuous functions on sup norm and let A : D ( A ) C_ X --+ X be defined - {u; u E C 1([0, 1]), u(0) - 0} - u ' for each u C D(A).

Show that A is dissipative, I - A is surjective, but D ( A ) is not dense in X. This is Example 4.7, p. 16 in Pazy [101]. N o t e s . The main result in Sections 3.1 and 3.2 Theorem 3.1.1, was proved independently by Hille [69] and Yosida [135] in 1948. The sufficiency part, as presented in Section 3.2, is due to Yosida. The Generation Theorem 3.5.1 in Section 3.3, which extends the necessary and sumcient condition in order for an operator to generate a C0-semigroup of contractions to a necessary and sufficient condition in order for an operator to generate a C0-semigroup of type (M, w), has been obtained, also independently, Feller [55] in 1953, Miyadera [91] and Phillips [104] in 1952. Theorem 3.4.2 in Section 3.4 was proved by Phillips [106]~ in the case in which X = H is a Hilbert space, and by Lumer and Phillips [87] in the general case. At this point, we notice that the dissipativity condition is very suitable to build up a powerful semigroup theory for nonlinear operators. See for instance Barbu [17]~ Showalter [112] and Vrabie [127]. The dual of a semigroup was defined and studied by Feller [53] in 1952 in the case of some parabolic problems and by Phillips [105] in 1955 in the general case. A different point of view concerning this problem, leading to analogous concepts and results, can be found in Butzer and Berens [32]. The main result in Section 3.9~ Theorem 3.9.1, was obtained by Stone [117] in 1932, and this is the first generation result of a C0-group of linear continuous operators by an unbounded operator. Concerning the general case of C0-groups of possible non unitary operators, we mention the following result due to L i t [86]: T h e o r e m (Lit) Let A " D(A) c_ H -+ H be the infinitesimal generator of a Co-semigroup. Then A generates a Co-group on H if and only if there exists ~/o > 0 such that" {A C C; ReA _> 70} c_ p ( - A ) ,

sup{[[()~I -Jr-A)-I[[A3(X)~ Re/~ ~ ")/o} < -~--oo

76

Generation Theorems

and

lim

Re A-++c~

I[(AI + A ) - l x [ [ - 0 for each x e H.

There are however situations in which an operator A does not satisfy the hypotheses of the Hille-Yosida Theorem 3.1.1 but we still can introduce a rather powerful notion of generalized solution for the differential equation u' - A u . This happens for instance if A satisfies all the hypotheses of Theorem 3.1.1 except the density of D ( A ) . In these cases, the simple observation that whenever {S(t); t _> 0} is a C0-semigroup on X, the family of operators {T(t) ; t _> 0} with T ( t ) - fo S ( T ) d T for t _> 0 satisfies" T(r)T(t) -

/o r(T(~- + t) -

T(7-)) d~- and T(0) - 0

suggests the definition of a more general concept, i.e. that of a Co-integrated semigroup as being a family {T(t); t _> 0} C_ L ( X ) satisfying the two conditions above and the continuity condition l i m s s o T ( s ) x - T ( t ) x for each x C X and t _> 0. The generator A 9 D ( A ) C_ X ~ X o f a C0integrated semigroup {T(t) ; t _> 0} is defined by x C D ( A ) and A x - y if and only if t ~ T ( t ) x is C 1 and T ' ( t ) x - x - T ( t ) y for each t _> 0. From now on, the whole classical machinery can be adapted to handle this case which has interesting applications in the study of the wave equation in L 2 (R n), as well as in that of the equations of population dynamics. See a h m e d [2], a r e n d t [8], Kellermann and Hieber [73], Thieme [120], [121], and the references therein. As already mentioned, with few exceptions, the problems here included are in fact known results or examples, and are gathered from Pazy [101].

CHAPTER 4

Differential Operators Generating C0-Semigroups

This chapter is devoted to the presentation of some partial differential operators which, defined on suitably chosen function spaces, generate either C0-semigroups of contractions, or groups of isometries. First, we refer to the Laplace operator on a domain ~t in ~n subjected to the Dirichlet boundary condition which generates Co-semigroups of contractions on each of the spaces H-1 (~), L p(~), with p in [ 1, +co), and Co(~). We also included some remarkable examples of infinitesimal generators of C0-groups of isometries as for instance: the Maxwell, the directional derivative, the SchrSdinger, the wave, as well as the Airy operators. In the last sections, we present two infinitesimal generators of C0-semigroups of contractions in the linear thermoelasticity and the linear viscoelasticity respectively.

4.1. The Laplace Operator with Dirichlet Boundary Condition Let us consider the heat equation in a domain gt in IR8

l u t - a~ u- 0 ~(O,x) -~o(x)

(t,x) ~ Q~ (t, x) ~ ~ x e ~,

where A is the Laplace operator, Q ~ - IR+ x ~ and E ~ - JR+ x F. This partial differential equation can be rewritten as an ordinary differential equation of the form u ' - Au

u(o) - uo

(0 9

in a suitably chosen infinite-dimensional Banach space X, in order that the unbounded linear operator A" D(A) C_ X ~ X generate a C0-semigroup of contractions. We present next several such classical possible choices of the space X. 77

78

Differential Operators Generating Co-Semigroups

E x a m p l e 4.1.1. The H - I ( ~ ) Setting. Let gt be a nonempty and open subset in IRa, let X - H -1(~), and let us define A ' D ( A ) C_ X -+ X by" D(A) - H ~ ( f ~ ) A u - An, for each u e D(A). In that follows, Hl(f~) is endowed with the usual norm on HI(Ft), defined by

Theorem

4.1.1. The operator A, defined as above, is the generator of a

Co-semigroup of contractions. In addition, A is self-adjoint and II'IID(A) is equivalent with the norm of the space Hl(~t). P r o o f . By virtue of Theorem 1.5.8, we know that I - A is the canonic isomorphism between H~(f~), endowed with the usual norm of HI(f~), and its dual H-l(f~). Let us denote by F - (I - A ) - ~ which is an isometry between H - I ( F t ) and H~ (f~). Consequently

(u, V)s-~(a) -- (Fu, Fv)s~(~)

(4.1.1)

for each u, v C H - l ( g t ) . Let u, v E H 1 (gt). We have

-/

+/

= I,~ u(I - A ) F ( v ) d w - (u, V)L2(~). From (4.1.1), taking into account that F ( I -

(4.1.2)

A) = I, we deduce

(--A~. V)H-I(~-~) -- (U -- A n . v) . - l (~'-~) - (u. v).-l(~'-~) From (4.1.2), we have

(/~U. V)H-I(~) -- (U, V)H-I(~ ) -- (U. U}L2(~ ). Therefore A is symmetric. But ( I - A) -1 C L(H-I(f~)), and therefore, from Lemma 1.6.1, it follows that A is self-adjoint. Taking v = u in the above equality, we obtain

(An, U)u-~(a ) -

Ilull ( ) _< 0.

(4.1.3)

Corollary 1.5.1 shows that, for A > 0, we have ( A I - A) -1 E L ( H - I ( f t ) ) , for A > 0, ( A n - Au, u)u-~(a) > ll [I 2U - l ( a ) " , --

while (4.1 9 3) implies that

The Laplace Operator with Dirichlet Boundary Condition

79

Hence [[R()~; A)[[L(H-I(~)) ~ ~. Since H~(~t) is dense in H - I ( ~ ) , we are in the hypotheses of Theorem 3.1.1, from where it follows that A generates a C0-semigroup of contractions on H -1 (~). Finally, by (iv) in Corollary 3.5.1 and (4.1.3), it follows that ]]']]D(A) is equivalent with the norm of the space H - ~(gt). The proof is complete. E x a m p l e 4.1.2. The L2(~) Setting. Let ~t be a nonempty and open subset in R n, let X - L2(f~), and let us consider the operator A on X, defined by"

D(A) - {u e H~(gt); Au e L2(f~)} A u - An, for each u E D(A). T h e o r e m 4.1.2. The linear operator A, defined above, is the infinitesimal

generator of a Co-semigroup of contractions. Moreover, A is self-adjoint, and (D(A), ]I']]D(A)) is continuously included in H](~t). If f~ is bounded with C 1 boundary, then (D(A), ]]'lID(A)) is compactly imbedded in L2(f~). P r o o f . Since C~(gt) is dense in L2(~), and C ~ ( ~ ) _C D(A), it follows that A is densely defined. Let A > 0 and f C L2(~). Since g2(~) is continuously imbedded in H - I ( ~ ) , and - A " H01(~) -+ H - I ( ~ ) is the duality mapping with respect to the gradient norm on//~(f~), we have:

(An, V)L2(a) -- - ( V u , VV)L2(~) -- (v, AU)H~(~),H_I(Vt).

(4.1.4)

By Theorem 4.1.1, we know that, for any A > 0 and any f E L2(~) (notice that L 2 ( ~ ) C H - I ( ~ ) ) , the equation ),u- Au = f

has a unique solution u~ E H~(n) C L2(Q). So, An), -- Au~--f is in L2(~), which shows that u~ C D(A), and An),- An), = f. Taking the L2(~) inner product on both sides of the equality above by u~, and taking into account that, by (4.1.4), we have (Au, u)L2(n ) <_ 0 for each u E D(A), we deduce that A[[u~[[~2(gt) _< (f,u~)L2(~) <_ IIflIL2(a)[]u~,]]L~(a), inequality which shows that [IR(A; A)]l~(x) _< ~. Finally, from (4.1.4) ~nd Lemma 1.6.1, it follows that A is self-adjoint. Inasmuch as both inclusions D(A) c H (n) c i2( t) are continuous, and the latter is compact whenever is bounded (see Theorem 1.5.4), this achieves the proof. [-7 R e m a r k 4.1.1. Obviously H ~ ( ~ ) N H2(~) C_ D(A). One can prove that, if 9t is bounded and has a C 2 boundary, or ~t - It~, then the reversed inclusion holds too, and therefore D(A) - H]

n H2

Differential Operators Generating Co-Semigroups

80

See Brezis [29], Th~or~me IX.25, p. 181. Moreover, the above equality still holds in the case in which ~t is convex and bounded. See Grisvard [64], Theorem 3.1.2.1, p. 139. E x a m p l e 4.1.3. The LP(~) Setting. Let ~ be a nonempty and open subset in ]~n and let 1 _< p < +co. L e m m a 4.1.1. Let A be the Laplace operator With the Dirichlet boundary condition defined on H - l ( a ) , let ~ > 0 and R(~;A) " H - I ( a ) --+ D(A). Then" 1 (i) R ( A ; A ) E L ( H - I ( a ) ) and IIR(A;A)II~(H-~(~)) <_ -~; (ii) R(A;A) E L ( H - ~ ( n ) , H ~ ( D ) ) ; 1 (iii) R(A; A)IH~(~ ) C L(H~(D)) and IIR(A; A)IHI(~)II~(D(A);X) <_ -~ ; (iv) lim IIR(~; A)u - ~llH-~(~) - 0 for each u e H -l(ft) ; )~--++oc

(v)

lira IIR(A;A ) u - UllH~(~) -- 0 for each u e Hl(f~). )~--+ + oc

P r o o f . The item (i) follows from Theorem 4.1.1, while (ii) from (v) in Corollary 3.5.1. To check (iii), let us observe that

IIR(~; A)XIID(A) -- IIR(~; A)xllx + IIAR(~; A)xllx 1

1

<- -(llxllx + IIAxllx) - -~IIxlIDcA). )~ The item (iv) follows from Lemma 3.2.1, and Theorem 4.1.1. Finally, (v) also follows from Lemma 3.2.1 and Theorem 4.1.1, by observing that the restriction of A to D(A), endowed with the graph-norm, i.e. the operator AID(A ) 9 D(A 2) C_ D(A) ~ D(A) satisfies the hypotheses of Theorem 3.1.1. D L e m m a 4.1.2. Let A be the Laplace operator with the Dirichlet boundary condition in H-l(f~) let )~ > O, and let R()~;A)" H - l ( f t ) --+ D(A). Let 1 < p < +ec. Then, for each f C LP(f~)A H - l ( f t ) , we have (i) R(s A) f e LP(f~) (ii) IIR(A; A)IIILp(~ ) <_ ~[I/IILp(~)" P r o o f . Let f C LP(Vt) N H - l ( f t ) . Then R()~; A ) f - u, where

~ u - Au - f. Multiplying both sides of this equality by I[ull2L-~)lulP-2u, integrating over ft, using Green's formula and the fact that u C Hl(f~), we get

~,11~112 LP(~) < - - IlflLp ( ~ ) II~IILp<~)' which completes the proof.

K]

The Laplace Operator with Dirichlet Boundary Condition

81

L e m m a 4.1.3. Let A be the Laplace operator with the Dirichlet boundary condition in H - l ( ~ t ) , let )~ > O, and 1 < p < +oc. Then there exists a unique ~ e L(LP(~)) so that ~ u - R()~;A)u for all u ~ H - I ( ~ ) N L P ( ~ t ) . In addition, ~ satisfies" (i) II~xuiiL~(a)_< ~llu]]L~(~); (ii) for each f ~ LP(~), A ~ f E LP(~) and A g ~ f - A ~ A f -- f ; (iii) for each A > O, and # > O, ~)~(LP(~)) - ~tt(LP(~)). P r o o f . Since H - I ( ~ ) n LP(~) is dense in LP(~), from Lemma 4.1.2, it follows that R(A; A) has a unique extension ~x ~ L(LP(~)) satisfying (i). Next, let ( f k ) k ~ be a sequence in 9 convergent to f in LP(~). As ~)~fk -- AA~xf~ = fk in H - I ( ~ ) , we have ~ f - )~A~xf = f in 9 from where we get (ii). Finally, let f e g - l ( f t ) a L P ( f t ) , and u - : ~ f e g - l ( ~ ) nLP(ft). For each # > 0, we have Au = f + ( # Let us denote by g the right-hand side of the equality above, i.e. g - S + (.-

and let us observe that ~ f

- u - ~ , g E H-l(~t) n LP(Q) and therefore

~ x ( H - I ( ~ ) N LP(~)) C :~tt(H-l(~t)n LP(~)). Analogously, ~ t t ( H - l ( ~ ) n LP(~)) C ~ ( H - I ( ~ )

N LP(~)),

and so ~(H-I(~)

N LP(~)) - ~ , ( H - I ( ~ ) n

LP(ft)).

Since H - I ( ~ ) n LP(~t) is dense in LP(~), and ~ , ~ are linear continuous operators in LP(~), w e deduce (iii) and this achieves the proof. D T h e o r e m 4.1.3. Let 1 < p < +oc, and ~1 C L(LB(~)) as in Lemma 4.1.3. Then, for each u C ~I(LP(~)), we have A u C LP(~), and the operator A" D(A) c_ LP(~) --+ LP(~), defined by D(A) - :~1 (LB(~)) An-An for u c D(A), is the generator of a Co-semigroup of contractions. P r o o f . The fact that, for each u E ~I(LB(~)), we have Au E LP(~), follows from (ii) in Lemma 4.1.3. So, let u C D(A), let A > 0, and let us denote f = A u - An. From (iii) in Lemma 4.1.3, we know that there exists g C LP(~) such that u = ~),g. We then conclude that g = )~u- An, and so f - g. Then A e p(A) and R(A;A) - ( A I - A ) -1 - ~x. This

82

Differential Operators Generating Co-Semigroups

relation and (i) in Lemma 4.1.3 show that [IR(,~; A)ul[Lp(a ) <_ --}llUllLp(a). Thus A satisfies (ii) in Theorem 3.1.1. To complete the proof, we have and merely to show D(A) is dense in LP(f't). To this aim, let u C 9 f = u - Au C 9 Obviously u = J ~ f , and therefore 9 C_ D(A). Hence D(A) is dense in LP(f~), which achieves the proof. [5] R e m a r k 4.1.2. Clearly we have

Wo 'p (ft) N W 2,p(ft) C_ D(A). If 1 < p < + e c and ft is bounded and has a C 2 boundary, then the reversed inclusion holds too, and we have

D(A) - W~ 'p (f~) N W 2'p (~). For the proof see Pazy [101], Theorem 3.1, p. 212. R e m a r k 4.1.3. If p -

1 and f~ is bounded and has a C 2 boundary, then

D(A) - {u e W 1'1(~); A~t e L l(f~)}. In this case D(A) is not included in w2'l(f~). See for instance Brezis and Cazenave [31], Remark 1.2.30, p. 28. However, we have D(A) C_ w~'q(f~) for each 1 <_ q < qn, where ql - c~ and qn - n / ( n - 1), for n _> 2. See [88]. E x a m p l e 4.1.4. The Co(f~) Setting. Let f~ be a nonempty and open subset in R n, let X - C0(f~) be the space of all continuous functions from f~ to R vanishing on the boundary, and let us define A" D(A) C_ X --+ X by: D(A) - {u C C 0 ( ~ ) ~ H I ( a ) , Au E C0(~)}

A u - Au, for each u C D(A). T h e o r e m 4.1.4. If ft has a Cl-boundary, then the operator A, defined as

above, is the generator of a Co-semigroup of contractions on Co(ft). P r o o f . Since 9 is dense in X and 9 C D(A), it follows that A has dense domain. On the other hand, for each f C C0(f~) the problem u-

AAu- f

has a unique solution u C H01(f~). Moreover, from the classical theory of elliptic problems, this solution u E C0(f~). See for instance Gilbarg and Trudinger [50], Theorem 8.30, p. 206. Finally, since this solution u satisfies the above equation in any Lp(f~), from Theorem 4.1.3 combined with HilleYosida Theorem 3.1.1, we deduce that 1

II ttLp< >

llfllLp< )

The Laplace Operator with Neumann Boundary Condition for each p >_ 1. So, the above inequality holds also for p that 1

83

cc which shows

IIR(;~; A)llco(~) _< ~.

The conclusion follows from Theorem 3.1.1, and this completes the proof. D 4.2. The Laplace Operator w i t h N e u m a n n B o u n d a r y C o n d i t i o n Here we consider the heat equation in a domain gt in ]~3

(t,~) c Q~

l ut--Au ?.t v

--

(t,x) e r~

0

~(o, x) - ~o(x)

xE~,

where u. is the outward normal derivative of u at F, Qr162- R+ x ~ and E ~ - R+ x F. As in the preceding case, this partial differential equation can be rewritten as an ordinary differential equation of the form

u ' - Au

(0 9

u(O) - ~o

in an infinite-dimensional Banach space X, suitably chosen in order that the unbounded linear operator A" D(A) C_ X --+ X generates a C0-semigroup of contractions. E x a m p l e 4.2.1. The [Hl(~t)] * Setting. Let ~t be a nonempty and open subset in I~n with C 1 boundary F, let X - [HI(~)] *, and let us consider the operator A" D(A) C_X --+ X, defined by" D(A)= HI(~)

(An, V) Hl(a),[Hl(a)].

--

- ( V u , VV}L2(a) for each u, v C H I ( ~ ) .

Within this section, {., .}[HI(~)],,HI(~) is the duality between [H~(~)] * and HI(a).

Theorem 4.2.1. The operator A" D(A) C_ X --+ X, defined as above, is the generator of a Co-semigroup of contractions on X. P r o o f . Since HI(~) is densely imbedded in [H I (~)]*, in view of HilleYosida Theorem 3.1.1, we have merely to show that for each )~ > 0, the operator ) ~ I - A " D(A) C_X ~ X, where A is defined as above, is bijective and 1 I I ( A I - A ) - I [[~(x) But this simply follows from the obvious identity

( k u - Au,

~)[gl(~,-~)], ,gl(~,~)--- ~11~11~=(~)+ IlWll 2L2(a)

84

Differential Operators Generating Co-Semigroups D

and this achieves the proof.

E x a m p l e 4.2.2. The L2(~) Setting. Let ~ be a nonempty and open subset in I~n with C 1 boundary F, let X - L2(~t), and let us consider the operator B " D ( B ) C X --+ X , defined by: D(B) - {u e H 2 ( ~ ) ; uv - 0 B u - A u for u E D ( B ) .

on F}

T h e o r e m 4.2.2. The operator B " D ( B ) C_ X --+ X , defined as above, is the generator of a Co-semigroup of contractions on X . P r o o f . Let u C D(B). Then, for each v C Hl(~t), we have

(Au, V}HI(~),[HI(~)]. and thus, A u -

-- --(VU, VV)L2(~ ) -- (AU,

V)L2(~t)

B u for each u E D(B). In addition, (Bu, v)L~(~) - - ( V u , Vv}L~(a)

for each u, v C D(B). Thus B is symmetric and, for each A > 0, A I - B is bijective from D ( B ) to L2(~) and

1

II(AI - B) -lllc(x) <- ~Since D ( B ) is dense in X - L2(~), we are in the hypotheses of Hille-Yosida Theorem 3.1.1, and this completes the proof, ff] R e m a r k 4.2.1. For the Laplace operator A subjected to the Neumann boundary condition, either in [Hl(~t)] *, or in L2(~t), 0 is an eigenvalue, and all constant functions are the corresponding eigenfunctions. This is the first major distinction between this operator and the homonymous one with Dirichlet boundary condition.

4.3. T h e M a x w e l l Operator The aim of this section is to present a partial differential operator which generates a C0-group of unitary operators and which is, without any doubt, one of the most important operators in the Field Theory. Some other examples of infinitesimal generators of C0-groups of unitary operators, or isometries, will be discussed in the next sections. For the beginning we

The Maxwell Operator

85

recall that, if ~ e L2(]~ 3) and F e (L2(]~3)) 3, then

3 0~--+ V cp - g r a d ~ - E

~

el,

30Fi

V-F - divF -

i=1

and

i=l

V • F - curlF =

-- (

0F3

0F2

) ~1 ( +

C~F1 ( ~ F 3 ) (~2 -+- C~F2 c~F1) e3. -+

Ox3

Ox3

Oxl

Oxl

Ox2

The evolution of the intensity of both the electric field E and the magnetic field H in the space I~3 containing no matter, is described by the Maxwell system Et - -cX7 x H H t - cX7 x E V.E-0 and V - H - 0 E(O, x) - Eo(x) and H(O, x) - Ho(x)

(t, x) c I~ • I~3

(t, X) C ]~ X ]~3 (t, X) C ]~ X ]~3 XC]~ 3,

where c > 0 is constant. This system can be rewritten, in a suitably chosen Hilbert space, as u ' - An u(0)

-

u0,

where A is the generator of a C0-semigroup of contractions. Example space H -

4.3.1. (The Maxwell Operator). Let us consider the Hilbert (L2(~3)) 3 x (L2(I~3)) 3, and let us denote by

u - (E, H)

-

(El,

E2, E3, Hi:

H2,

H3)

a g e n e r i c element i n H . Let H0 - {u C H; V . E V.H-0}, where the differential operator V is considered in the sense of distributions, i.e. V . F - 0 if and only if /R V g ' F d w - 0

3

for each g C C ~ (R). This means t h a t u is in H0 if and only if it is orthogonal on each element v in H of the form v - (Vcp, V r with cp, r E H ~(R3). Let us define the Maxwell operator, A ' D ( A ) C_II ~ H, by D(A) - { ( E , H ) e H; ( - c V x H, cV x E) e H} A(E, H) - ( - c V x H, cV x E), for (E, H) e D(A). Let us observe that A maps D(A) in H0, and therefore H0 is invariant under A, because the divergence of a curl is always 0. This explains why, in all that follows, we shall consider the restriction of A to

86

Differential Operators Generating Co-Semigroups

H0, restriction which, for the sake of simplicity, we denote again by A. We emphasize that the operator A is not densely defined in H, but its restriction to H0 does, as we shall see from the proof of Theorem 4.3.1. T h e o r e m 4.3.1. The operator A, defined as above, is the generator of a

Co-group of unitary operators. P r o o f . We show that A satisfies the hypotheses of Stone Theorem 3.9.1. To this aim, let C ~ ( R n) = {F e C~(Rn); V" F = 0}. Inasmuch as C~(]R n) x C ~ ( R n) is included in D(A), and dense in H0, it follows that A is densely defined. We prove next that A is skew-adjoint. First, let us observe that, for each E, H E C~(Rn), we have (A(E, H), (E, H)) = 0.

(4.3.1)

Indeed, since the integral on R 3 of the divergence of a C 1 function with compact support is 0, we have (A(E, n ) , (E, H)) - [ d i v ( n x E ) d w JR 3

0

for each (E, H) e C ~ (Rn). Inasmuch as C ~ (Rn) x C ~ ( R n) is dense in H0, it follows that (4.3.1) holds for each ( E , H ) C D(A) and, by virtue of Remark 1.6.3, A is skew-symmetric, or equivalently, iA is symmetric. To check that A is skew-adjoint we prove that iA is self-adjoint and, to this aim, we shall use Lemma 1.6.1. More precisely, we shall prove that 1 c p(iA). Let us denote by I2I and E the Fourier transform of H and respectively of E, i.e.

1

/)(~) - (27r)3/2

1

/2/(~) _ (27r)3/2

s

_i(~,X)E(x)dx ' 3 C

/~ e- i 3

(~'X>H(x)dx.

Then the mapping (E, H) ~ (]~, I:I) is an isomorphism from H to a Hilbert space/2/analogously defined. More that this, this isomorphism maps Ho into a subspace/2/0 in/2/, subspace defined by

/:/0 - { (0,, I:I) c/:/;

E -

I:I - 0}

and it maps the operator A to the operator A" D(A) C_/2/0 --+/2/0, defined by D(A) - {(l~, I:I) e/:/0; (-c{ x I:i, c{ x E) e/2/o} x I:I, x E).

87

The Directional Derivative

Let ~r C/:/0, v - (Vl, v2), and let us consider the equation ( i I - A)6 - ~,

(4.3.2)

where/* denotes the identity operator on/:/. Obviously, 1 C p(iA) if and only if i C p(-~). But this last condition holds if and only if, for each C /:/0, the equation (4.3.2) has a unique solution R19 and there exists K > 0 such that

IIRI?II < KII, II for each 9 C/2/0. Clearly, in this case, the with R(1; A). Let us observe that, written form c~ • fi2 + ifil --C~ X s -Jr-ifi2

solution operator R1 coincides on components, (4.3.2) has the

-- ~rl

--v2,

system whose unique solution is given by ill-

1712--

--ivl+c~

• 92 all ll 2 + 1

c2[[~[[2 + 1

From the equalities above one may observe that 6 is a linear continuous function of 9. Therefore i C p(fi.), or equivalently 1 C p(iA). Analogously we deduce that - 1 C p(iA), which, according to Lemma 1.6.1, shows that A is skew-adjoint. The conclusion of Theorem 4.3.1 follows from Stone Theorem 3.9.1. The proof is complete. D 4.4.

The Directional

Derivative

The aim of this section is to prove that the well-known directional derivative operator is the infinitesimal generator of a C0-group of isometries in a suitably chosen function space. More precisely, let us consider the transport equation along the direction a C ]~3 (t, X) C ]~ X ]R3 X e 1~3.

ut + a . V u - 0

~t(0, X ) - ~t0(X )

(~'(~)

This first-order linear partial differential equation can be rewritten as an ordinary differential equation of the form u'-

Au

u(O)

-

uo

(0 9

Differential Operators Generating Co-Semigroups

88

in a B a n a c h space X , chosen so t h a t A " D ( A ) C_ X --+ X does g e n e r a t e a C 0 - s e m i g r o u p of contractions. As we shall see later, in this case, A g e n e r a t e s even a C0-group of isometries. E x a m p l e 4 . 4 . 1 . (The Directional Derivative Operator.) Let us consider X -- LP(]~n), w i t h 1 < p < + c o a n d let a C R n. We define the o p e r a t o r A in X by

D ( A ) - {u C X ; a . V u Au - -a . Vu = -E

EouXI ai-~x i

z:l

for u E D ( A ) 1. 4 . 4 . 1 . The operator A, defined as above, is the generator of a Co-group of isometries. Moreover, this group, G(t) 9X --+ X , t C JR, is given by a ( t ) f (x) - f (x - ta) Theorem

for each f c X , t E IR, and x C IRn. Lemma

4 . 4 . 1 . Let ;~ > O, and 1 < p < ec. If u C LP(~ n) satisfies

u +)~a. Vu - 0 in 9 (lRn ) , then u - O a. e. x 6 IRn . P r o o f . Let ~ a mollifier, let (Ck)k6N be a sequence decreasing to 0 a n d let Uk -- u~ k the ek-mollified of u. See Definition 1.3.1. T h a n k s to T h e o r e m 1.3.3, we have Un E C c~(I~ n) N L ~ (IRn). In addition, since -

-

d z

-

u(y)cp~k (x -- y) dy

for each x C IRn, we have

Uk + )~a. VUk -- 0 for each k C N, a n d x C IRn. Let x C IRa, a n d let us define h " IR --+ IR by h(t) - e t u k ( x + Ma). Clearly h is differentiable on IR, a n d

h'(t) - e t ( u k ( x + Ma) + Aa. V u k ( x + Ma)) -- O. T h e r e f o r e h is c o n s t a n t on ~. L e t t i n g t --+ - e c a n d t a k i n g into a c c o u n t t h a t u is b o u n d e d , we deduce t h a t h - 0 on IR, which implies Uk(X) -- O. 1We notice that here a. Vu is the directional derivative of u along the direction a in the sense of distributions over 9 i.e. a- Vu = (a, Vu), where Vu is the gradient of u in the sense of distributions over 9 n).

The Directional Derivative

89

LP(]~n)

(1 < p < +oc), the proof V]

Since x is arbitrary and limk-+cr Uk -- u in is complete.

Lemma

4.4.2. If )~ > O, 1 < p < +c~ and f C Lp(Rn), then the function

if0

L f (x) - -f

e-X f (x - sa) ds

satisfies L f + )~a. V ( L f ) - f

(4.4.1)

IIL f]]Lp(~ ~) <_ I]fIILp(R,).

(4.4.2)

in 9 (]~n ) , and P r o o f . Let f E Cub(I~n), and let us define

M f (x) - -s1 f0 c~ e--X f (x + sa) ds. Let cp C 9 From Fubini theorem (see Dunford and Schwartz [49], Theorem 9, p. 190), it follows that

(L f, r

- ~ n f M99 dx.

On the other hand

M()~a. V~)(x) - f0 c~ e ~a. V~(x + sa)ds sd x -~s (99(x + sa)) ds - - ~ ( x ) + M99(x).

e

--

~0 (X)

~ _

Accordingly

(Lf,~) - [ JR

n

fM~dx

- [

J~xn

f M ( A a . V ~ ) d x +
= (LI, Aa. V~} + ( f , ~ ) - ( - A a . V ( L f ) + f , ~ ) , equality which proves (4.4.1). Let l < p < + c c , a n d ~1+ 1 _ 1. From HSlder's inequality, we have

1 fcr ~ [L f (x)l < -~ ]o e rq e zp lf (x _ sa)lds <_ A- /P

lf (x - sa)lPds

9

Taking the pth power of both sides in this inequality, integrating on I~n the inequality thus obtained, and using once again Fubini theorem, we deduce 1 j l cr

90

Differential Operators Generating Co-Semigroups

from where it follows (4.4.2), in the case 1 < p < +oc. Since for p = 1, D (4.4.2) follows similarly, this achieves the proof. R e m a r k 4.4.1. Analogous considerations show that both Lemmas 4.4.1 and 4.4.2 can be extended to the case A < 0. We may now proceed to the proof of Theorem 4.4.1. P r o o f . Since D(A) includes C~(]l~n) and the latter is dense in X, it follows that D(A) is dense in X. So A satisfies (i) from Theorem 3.5.2. From (4.4.2) and Remarks 3.1.1 and 4.4.1, one may easily deduce that A satisfies (iii) in Theorem 3.5.2 as well. The proof is complete. 7-1 4.5. T h e S c h r h d i n g e r Operator The aim of this section is to present an example of differential operator arising in the mathematical modelling of some phenomena in Quantum Physics, which generates a C0-group of unitary operators. More precisely, let us consider the well-known Schrhdinger equation in the domain ~ in R 3

ut - ~ [ A ~ - V(x)~] -0 ~(0, x) - ~ 0 ( x )

(t,x) e QR (t, x) e r ~ 9 e a,

(s~)

where Q~ = R x ~t and ER = R x F. In this case too, by an appropriate choice of the Hilbert space H, the partial differential equation (8~) can be rewritten as a first-order ordinary differential equation of the form

u'-Au ~(o)

(0 9

-

where A : D(A) C_ H ~ H generates a C0-group of unitary operators. In this section we consider only the case of a vanishing potential, i.e. (V - 0), the general case making the object of Section 5.5. For the sake of simplicity, we shall also assume that m = 1/2. E x a m p l e 4.5.1. (The Schrhdinger Operator). 1. The H-l(f~; C) Setting. Let f~ be a nonempty and open subset in ]~n. We consider the space H - H-l(f~; C) - H-~(~t)+iH-~(f~) endowed with its real Hilbert space structure, and we define the operator A in H by:

D(A) - H~(f~; C) Au - iAu, for each u E D(A). We notice that, H~(f~; C) is equipped with the usual norm of the space HI(f~; C), i.e. Ilu+ivlIH~(~;C) -- (llull2Hl(fl)+llvl121(fl)) 1/2.

The Wave Operator

91

T h e o r e m 4.5.1. The operator A, defined as above, is the generator of a Co-group of unitary operators. P r o o f . We apply Theorem 3.9.1 combined with Lemma 1.6.2.

V]

2. The L2(~);C) Setting. Let ~t a nonempty and open subset in IRn, let H - L2(~;C) - L2(~t) + iL2(~t), endowed with its real Hilbert structure, and let us consider the operator A in H, defined by: D(A) - {u e H~(f~;C); Au e L2(~t;C)} Au - iAu, for each u E D(A).

T h e o r e m 4.5.2. The operator A, defined as above, is the generator of a Co-group of unitary operators. In addition, D(A) is continuously imbedded in Hl(fl; C). Moreover, if ~t is bounded, then D(A) is compactly imbedded in L2(gt; C). P r o o f . We apply Theorem 4.1.2, Lemma 1.6.2 and Theorem 3.9.1.

D

R e m a r k 4.5.1. If gt is bounded and has a C 2 boundary, or either ~ - R~ or ~t is bounded and convex, we have D(A) - H ~ ( g t ; C ) N H2(gt;C). See Remark 4.1.1. 4 . 6 . T h e W a v e Operator Let us consider the equation of wave propagation in a homogeneous and isotropic body fl in IR3 with C 1 boundary F

I utt -- Au u--O u(O, x) - uo,

ut(o, x) = vo

(t, ~) c OR (t,x) c BR xE~),

(w~)

where QR = IR x ~t and ER = IR x F, and let us observe that this can be formally rewritten as a first-order (in t) system ~ vtu~(o,

~ au 0 ~) - ~o (x), v(0, x) - vo(x)

(t, ~) c Q~ (t, x) c Q~ (t,x) C ER x e ~.

Setting w = (u, v), A(u, v) = (v, Au), and ~ = (u0, v0), this system can be rewritten in the abstract form W t --

Aw

~ ( o ) - {,

(o~)~)

in a Hilbert space H, suitably chosen so that A : D(A) C_ H -~ H generate a C0-group of unitary operators on H. In what follows, we present two

92

Differential Operators Generating Co-Semigroups

such possible choices of the function space H satisfying all the requirements mentioned above. E x a m p l e 4.6.1. (The Wave Operator). 1. The L2(fl) • H - I ( ~ ) Setting. Let ~t be a nonempty and open subset in I~n, let H - L2(fl) x H - l ( f l ) and let A ' D ( A ) C H -+ H be defined by: D(A) - H I ( ~ ) x L2(~) A(u, v ) - (v, Au), for each (u, v) e D(A). In this example, the space H is endowed with the inner product (., "/, defined by ((u, v), (f , g)} - (u, f)L2(~) + (v, g)u-~(~), where (-, .)S-i(~) is the inner product in H-1 (f~) corresponding to the inner product (-, .)H~ (~t) defined by

for each u, v C H~ (~t), i.e. the inner product defining the gradient norm I]" I[0 in Theorem 1.5.7. T h e o r e m 4.6.1. The operator A, defined as above, is the generator of a

Co-group of unitary operators on H. In addition, the graph-norm of A is equivalent with [[ 9 [[sl(~)• P r o o f . Obviously D(A) is dense in H. Let (u, v) C D(A). We have

(A(u, v), (u, v)) - (v, U)L2(~) + (Au, V)u-~(fl). Let w C H~ (f~) be the unique solution of the problem - A w -- v. By virtue of Theorem 1.5.8 we have (--~. U, V).--I(~) -- (U,

W)ul(a),

and accordingly (A(u, v), (u, v)) - (v, U)L2(~) + (u, --w),~(~)

-- (V, U)L2(Ft) -Jr- /O V u V w dw = (v, U)L2(~ ) + (u,--/kW)(H~(fl),H-l(~))

-- (V, U)L2(Ft) Jr- (--V, U)HI(~),H-I(~ ) -- O. See Theorem 1.5.8. So, A is skew-symmetric. To complete the proof, it suffices to show that (I + A) -1 E L(H). See Lemma 1.6.1 and Stone

The Wave Operator

93

Theorem 3.9.1. To this aim, let us observe first that, for each (f, g) C H, the equation ( I - A)(u, v) = (f, g) is equivalent with the system u-v+ f - A t - g - v. The latter can be rewritten under the form u-Au-

f +g

V --?J,-- f.

Since f + g E H-l(gt), from Theorem 4.1.1, we deduce that the first equation in the above system has a unique solution u E H~ (~t). Hence v, given by the second equation, belongs to L2(f~), and therefore I - A is bijective. Since [[VlIL2(Ft)+ and the norm of H~(f~) x L2(~) is given by [l(u, v)ll the latter, by Theorem 1.5.8, is equivalent with IlVllL2(~) + [ [ A u - Ui[H-l(~), which in its turn is nothing else than the graph-norm of A, it follows that ( I - A) -1 C L(H). Similarly, we show that (I + A) -1 E L ( H ) and this achieves the proof. [:] -

2. The H I ( ~ ) • L2(~) Setting. Let gt be a nonempty and open subset in I~n, let us consider the space H - H~(f~) x L2(f~) and let us define A : D(A) C H --+ H by: D(A) - {(u, v) e H; Au e L2(~t), v e H](~t)} A(u, v) - (v, A t ) , for each (u, v) e D(A). In this example, we endow H with (., .) : H x H --+ I~, defined by

((u,,), (e, for each (u, v), (~, ~) C H, in respect to which this is a real Hilbert space. See Theorem 1.5.8. T h e o r e m 4.6.2. The operator A, defined as above, is the generator of a Co-group of unitary operators on H. In addition, D(A) endowed with the graph-norm, is continuously imbedded in H~(~) x L2(~t). P r o o f . Clearly D(A) is dense in H. In addition, using Green's formula, we deduce that

94

Differential Operators Generating Co-Semigroups

for each (u, v) e D(A). From Remark 1.6.3, it follows that A is skewsymmetric. By Lemma 1.6.1 we conclude that A is skew-adjoint, while from Stone Theorem 3.9.1, it follows that A generates a C0-group of unitary operators in H. The proof is complete. D

Example 4.6.2. (The Klein-Gordon Operator) 1. The L2(f~) x H-l(f~) Setting. Let A -inf{llVUllL2(~); u e H~(~t), Ilul]52(~)

1} 2,

let m > - ~ and H - L2(~) x H - I ( ~ ) . From Corollary 1.5.1, it follows that, for each u C H-l(f~), the equation m c p ~ - Acp~ - u has a unique solution CPu C H~(f~). Then the mapping (., .)" H x H ~ R, defined by

for each (u, v), (~, ~) E H, is an inner product on S equivalent with the initial one, in the sense that the corresponding two norms are equivalent. We endow H with this inner product and we define A" D(A) C H --+ H by D ( A ) = U~(f~) x L2(~t) A(u, v) = (v, Au - mu) for each (u,v) C D(A). This is the Klein-Gordon operator in the space L2(f~) x H - I ( ~ ) . Reasoning as in the proof of Theorem 4.6.2, we obtain: T h e o r e m 4.6.3. The operator A, defined as above, is the generator of a group of isometries on H. In addition, D(A) endowed with the graph-norm, is continuously imbedded in L2(f~) • H - l ( g t ) .

2. The H~(~t) x L2(~) Setting. Let A be as above, let us consider the space H - H~(~t) x L2(~) and let us define A ' D ( A ) C H -+ H by: D ( A ) - {(u,v) e H ; A u e L2(f~), v e H~(gt)} A(u, v) - (v, Au - mu), for each (u, v) C D(A). This is the Klein-Gordon operator in the space H~(f~) • L2(f~). From the definition of ~ and the choice of m, it follows that (., .} : H • H --+ ~, defined by - (w,

+

+ iv,

for each (u, v), (5, ~) C H, is an inner product on S which is equivalent with the original one, in the sense that the corresponding two norms are equivalent. Using similar arguments as in the proof of Theorem 4.6.2, we get: 2If fl is bounded then A is the first eigenvalue of - A on H -1 (~) and A > 0.

The Airy Operator Theorem

95

4.6.4. The operator A, defined as above, is the generator of a

group of isometries on H. In addition, D(A) endowed with the graph-norm, is continuously imbedded in Hl(~t) x L2(gt). Remark

4.6.1. We notice that the Klein-Gordon equation, i.e.

l a

utt

- A u - k2u

in IR+ x onIR+ xF on

u--0 u(0) - ~

{ ut + Uxx~ - 0 ~(0, x) -~0(x)

(t,x) e R • R

(AS)

x e

and we will show that it can be rewritten under the abstract form

u ' - Au

((99

-

E x a m p l e 4.7.1. (The Airy Operator). Let H - L2(IR) and let us consider the operator A on H, defined by:

I D(A)-H3(I~) Au

-

-uzxx

=

d3u dx 3 ,

for each u C D(A). 4.7.1. The operator A, defined as above, is the generator of a Co-group of unitary operators on H.

Theorem

P r o o f . Obviously A is densely defined. Let u, v C C~(IR). Inasmuch as

l [2uxxU (ux) 2 -

it follows

R Uzxxit dx - 0

}

which shows that (An, u) - 0 for each u e C ~ ( R ) . Since C~(IR) is dense in D(A), from Remark 1.6.3, it follows that A is skew-symmetric. One

Differential Operators Generating Co-Semigroups

96

may easily verify that +i C p(A), and therefore, by Theorem 1.6.1, it follows that A is skew-adjoint. From Stone Theorem 3.9.1, we conclude that A generates C0-group of unitary operators on H, and this achieves the proof. [5

4.8. The Equations of Linear Thermoelasticity In this section, we show how the previously developed theory can be used to obtain accurate information on a system of partial differential equations modelling the evolution of three fields: the displacement, the momentum and the temperature of a thermoelastic body.

Example 4.8.1. (The Equations of Linear Therraoelasticity). Let C be a homogeneous body having the referential configuration a nonempty, open and bounded subset ~t in ]~n (n = 1, 2 or 3) whose boundary F is of class C 1. The state at the time t C I~+ is characterized by two vector fields: the displacement u(t, x), and the momentum v(t, x), and a scalar field: the temperature O(t,x). The system of equations describing the evolution of these three fields is ut -

p-iv

~

~v. w

(t, x) c Q~ + (~ + ~ ) v v

9~ + . ~ v 0

0 t -- c - l k A O -~- c - l p - l ~ ) ? T t V . v ~(t, ~) - o, o(t, x) - o ~(o, x) - ~o(x) v(o, x) - vo(x)

(t, ~) ~ Q ~

(t,x) C Qoc 0(0, x) - eo(x)

(4.8.1)

(t, ~) e r ~ 9 e ~,

where Q ~ = Ii~+ x ~ and E ~ = I~+ x F. Here p > 0 is the density of the body, O > 0 the referential temperature, and c > 0, k > 0, m, )~, and # are constants which characterize the thermoelastic properties of the body. We assume that A and # satisfy the strong ellipticity conditions # > 0 and A + 2# > 0. We begin by rewriting the system (4.8.1) under the form of an abstract Cauchy problem in a suitably chosen Hilbert space. More precisely, let H - [HI(~)] n [L2(~)] n L2(~)which, endowed with the inner product (-,-), defined by N

((~, v, 0), (~, ~, e))

is a real Hilbert space. Here n i,j--1

OUi OUj

o -U O

The Equations of Linear Thermoelasticity

97

We define the operator A : D(A) C_ H ~ H by

D(A) - [H2(gt) N H~(a)] n x [Hi(a)] n x [H2(gt) M H~(~t)] , and

A(u,v,O) -- (p-Iv,

#V" Vlt Jr- (/~ -~- # ) V V 9It -Jr-mY, c-lkAO -}- c - l p - l O T n V , v)

for each (u,v, 0) C D(A). Let us observe that the system (4.8.1) can be rewritten in the space H as

{ z~-Az z(O)

-

where A is as above, z = (u, v, 0) and ~ = (u0, v0, 00). T h e o r e m 4.8.1. The operator A, defined as above, is the generator of a Co-semigroup of contractions in H. P r o o f . Since the set [C~(f~)] n x [C~(gt)] n • C~(f~) is included in D(A) and dense in H, A is densely defined. We shall prove next that, for each > 0, there exists ( A I - A) -1 e L(H), and

I[(AI-A)-IIIL(H)

1 ~_ --~,

(4.8.2)

from where it will follow, on one hand that A is closed, and on the other hand, that it satisfies the conditions (i) and (ii) in Theorem 3.1.1. Let us observe that ()~I - A) -1 (~, ~, 0) - (u, v, 0) if and only if )~ u -- p - l v - - "u

x C ~t

s # V . Vu - (~ + # ) V V 9u - m V 0 - ~ x e gt )~0 - c - l k A O - c - l p - l O m V . v = "d x C gt u-O, 0-0 xCF.

(4.8.3)

From the strong ellipticity conditions, i.e. p > 0 and A + 2# > 0, and from the general existence results concerning systems of elliptic equations, it follows that, for each (~, ~, 0) C H, the system (4.8.3) has a unique solution (u, v, 0). On, the other hand, a simple computational argument shows that, for each (u, v, 0) C D(A), we have

(A(u, v, 0), (u, v, 0)) - -kO -1 L IIVOII2 dx <_ O, which proves that ) ~ I - A is a bijection from D(A) to H, and satisfies (4.8.2). The proof is complete. [--1

98

Differential Operators Generating Co-Semigroups

Viscoelasticity

4.9. T h e E q u a t i o n s of L i n e a r

In this section, we shall use the abstract results in Chapter 2 in order to get information concerning a system of integro-partial differential equations arising in the modelling of the evolution of the displacement, the history of displacement, and the momentum of a viscoelastic body. E x a m p l e 4.9.1. (The Equations of Linear Viscoelasticity). Let us consider an one-dimensional homogeneous body of density p > 0, and having the constitutive equation g(t - ~-)Ux(~',x) d~-,

a(t, x) - CUx(t, x) CO

where g" ~+ --+ It( is a positive, nonincreasing C 1 function satisfying a-

c-

g(s) ds > 0.

(4.9.1)

The referential configuration is the interval [ 0, 7r ]. The state of the body at the time t E ~ is characterized by the displacement u(t, x), the momentum v ( t , x ) and the history of the displacement w(t, s,x) which is defined by w(t, s, x) = u ( t - s, x) for s E IR+. Assuming that the displacement vanishes at the two endpoints of the interval [0, ~r], and taking into account that the equation of motion is pu" = ax, we obtain the system -1

~tt =

p

vt -

CUxx -

W t =

(t, x) c Q~

V

g(S)Wxx

ds

(t, s, x) c R ~

--W s

u(t, O) = u(t, 7r) = 0 ~(o, x) = ~o(x) ~(o, x) = ~o(x)

x)

(t, ~) ~ Q~

x)

(4.9.2)

tER+ z ~ (0, ~)

(~, x) c Q~,

where Qcr - II~+ x (0, 7r) and Rer = ~+ x Q ~ . We shall show next how the system above can be rewritten as a Cauchy problem for an ordinary differential equation in a suitably chosen Hilbert space. To this aim, let 2 H - H~(0, 7r) • L2(0,, ~-) x Lg(R+; Hol (0, 7r)),

where Lg2 (IR+; Ho1(0, ~)) is the 9-weighted space L 2 (]~+; 9dt; X ) . On H we consider the inner product (., .), defined by ((u, v, w), (~, ~, ~)) -

aUx~x + p - l v ~ +

g(s)[Ux - Wx][~x - ~x] ds

dx,

The Equations of Linear Viscoelasticity

99

with respect to which H is a real Hilbert space. Let us define the operator A : D ( A ) C_ H --+ H by D ( A ) = {(u, v, w) e H ; (u, v, w) satisfies (4.9.3)} v E H~(O, 7r), Ws C Lg CUxz -

fo

(::X3

~(0, .)

;

~(~+ Ho~(0, ~))' g(S)Wxz ds e L 2 (0, 7r)

(4.9.3)

and A(u, v, w) -

(

/o

p - i v , CUxx -

g(s)Wxx ds, - W s

)

for each (u, v, w) e D ( A ) . At this point, let us observe that (4.9.2) can be equivalently rewritten under the abstract form z'-Az

~,

z(O) -

where A is as above, z = (u, v, w) and ~ = (uo, vo, wo). T h e o r e m 4.9.1. The operator A, defined as above, is the generator of a Co-semigroup of contractions on H. P r o o f . Since the set

{ (~, v, ~) c c ~ (o, ~) • c ~ ( o , ~) • r~~(~+ ; c~(0,

~)); ~(0,

)

- }

is included in D ( A ) and dense in H, A is densely defined. We shall show next that, for each A > 0, there exists ( ) ~ I - A) -1 C L ( H ) and 1 II(~I - A) -1 ]]~(H) _< ~,

(4.9.4)

from where it will follow, on one hand that A is closed, and on the other hand, that it satisfies the conditions (i) and (ii) in Theorem 3.1.1. Let us observe that ()~I - A) -1(~, ~, ~) = (u, v, w) if and only if /~u- p-Iv -

~ - ~xx + Aw + w~ ~(o,.) - ~ .

/o

g(~)~xx

a~ - ~

(4.9.5)

From the last two conditions, we deduce w(s, x) - u ( x ) e -~s +

~0Se-~(s-r

-, x) d~-.

(4.9.6)

100

Differential Operators Generating Co-Semigroups

If u E H~ (0, 7r), then, again from the last two relations in the system (4.9.5), we get ,~wx - - w ~ z + ~ and therefore we successively have 2

1 z

and

1/o /o

2 dx ds - 2g(O)

~ 2 dx + ~

/oofo

g(s)wz~zdxds.

+

g~ ( ~ ) ~2 dx ds

From this equality, using Cauchy-Schwarz inequality, in the form )~a2

,---1 x/,ka~b

ab-

4-2 <-

2

b2

J

2)~

'

to estimate the last term on the right-hand side, we obtain f0~ f07r

g(~)~

2 dx ds < g(O) f07r 2 dx + 1 f0~ f0~ g ( s ) - 2 dx ds

~

-

-2

~

'

2 which proves that w E Lg(R+;H~o(O, Tr)).__ Coming back to the last two 2 relations in (4.9.5), we deduce that w~ E Lg(R+;H~(0, Tr)). Substituting v from the first equation of (4.9.5) and w given by (4.9.6) in the second equation of (4.9.5), we get

X2pu-(C-fo

~

e-XS g(s) ds

)

Uxx

_

N v

+ )~p~ -

/0

e ~r W x ~

d~ ds.

(4.9.7)

Let us observe that the right-hand side of the above equality belongs to H - l ( 0 , 7r). On the other hand, from (4.9.1) and the fact that ,~ > 0, we deduce c-

/0

e -~sg(s) ds > O.

Accordingly, (4.9.7) has a unique solution u C H~(0, 7r). Consequently, the system (4.9.5) has a unique solution, which shows that ) ~ I - A is bijective from D ( A ) to H. Finally, recalling the definition of the inner product in H and using the fact that g is C 1, and nonincreasing (which implies g~ <_ 0), we obtain 1 {A(u, v, w), (u, v, w) ) - -~

/0/0 ~

9'(s)luz - wz[ 2 dx ds < O.

From this inequality, it follows that ( M - A ) -1 C L ( H ) , and satisfies (4.9.4). This achieves the proof. K]

Problems

101

Problems Problem 4.1. Let X - C([0, 1 ]) endowed with the usual sup norm and let us consider the operator A ' D ( A ) C_ X -+ X defined by D(A) - {f e C2((0, 1)); lims+t s(1 - s)f"(s) - 0 for t - 0 or t - 1} A f ( s ) - s(1 - s)f"(s). Show that A generates a C0-semigroup of contractions on X. and Nagel et al [51], P a r a g r a p h II 3.30.(iii), p. 94.

Problem

See Engel

4.2. Let X - L2(0, 7~), t E I~+, and let S(t) 9X --+ X be defined

by

[S(t)~](x) -- ~-~ ak(~)e-

k2t

sinkx

k=0 for each ~ C L2(0, 7r), where ak(~) are the Fourier coefficients of ~ with respect to the orthogonal system {sin x, sin 2 x , . . . , sin k x , . . . }, i.e.

ak(~) --

~(y)sinkydy.

(i) Prove that {S(t); t _> O} is a C0-semigroup of contractions on

L2(0, 7r). (ii) Find its infinitesimal generator. (iii) Prove that, for each ~ C g2(0, 7r), u "[0, +oc) -+ L2(0, 7r), defined by u(t,x) - [S(t)~](x), satisfies

I u t - Uxx for (t,x) E R+ x (0,~) u(t, 0) = u(rr, 0) - 0 for t e IR+ x) =

9 e (0,

in the sense of distributions over (0, 7~).

Problem

4.3. Let p C [1, +c~), X - LB(I~) and t C I~+. Let us define

S(t)" X --+ X by S(0) - I and [S(t)f](x) - ~

1 f_+c~ (~_y)2 0r C 4t f(y)dy

for t > 0. Show that {S(t) ; t >_ 0} is a C0-semigroup of contractions on X. This is the Gauss-Weierstrass semigroup. See McBride [89], Example 1.8, p. 15.

102

Differential Operators Generating Co-Semigroups

P r o b l e m 4.4. Find the generator of the semigroup in Problem 4.3. Show that, for each f e LP(R), the function u : [0, +co) --+ LP(R), defined by u(t, x) = [S(t) f ](x), satisfies

ut - Uxx u(O, x) - f (x)

on R+ x R onR,

in the sense of distributions. See McBride [89], Example 1.8, p. 15. P r o b l e m 4.5. Let p C [1, +oc), X = LP(R), and let t C R+. Let us define S ( t ) : X --+ X by S(0) = I, and

t ff

I(y)

[ S ( t ) f ] ( x ) - 7r

t 2 + ( x - y ) 2 dy

t > 0. Show that {S(t) ; t _> 0} is a C0-semigroup of contractions on X. This is the Poisson semigroup. See McBride [89], Example 1.10, p. 20. P r o b l e m 4.6. Let {S(t); t > 0} the semigroup in Problem 4.5. Show that, for each f C LP(R), the function u : [0, +co) -+ LP(R), defined by u(t, x) = [S(t) f](x), satisfies Utt nc u x x

-- 0

x) = f (x)

onR+ xR on R,

in the sense of distributions. See McBride [89], Example 1.10, p. 20. P r o b l e m 4.7. Let p E [1, +c~), X = LP(0, 1) and t C R+. Let us define S ( t ) : X --~ X by S(0) = I and

1foX

r(t)

(x - y)t-l f ( y ) d y - (Jt f)(x)

for t > 0. Show that {S(t) ; t > 0} is a C0-semigroup of contractions on X. The integral j t f is the fractional Riemann-Liouville integral of exponent t of f. See McBride [89], Example 1.12, p. 23. P r o b l e m 4.8. Find the duals of the semigroups in Problems 4.2 and 4.3, in the particular case p - 2. P r o b l e m 4.9. Let H - L2(R) x L2(R) which, endowed with the inner product

<(Ul, Vl), (~2,

- -
-'~
103

Notes

for each (ui, vi) E H, i - 1, 2, is a real Hilbert space. Let t C I~, and let us define G(t) " H ~ H by ~ ( x + t) + ~ ( x - t)

2

[G(t) (u, v)](x) -

v ( x + t) - ~(~ - t)

+

2

v(x + t) + v(x - t) + ~ ( ~ + t) - ~ ( ~ - t) 2 2 for each (u, v) c H, where B ~ denotes the transposed of the matrix B. Prove that {G(t) ; t C I~} is a group of unitary operators on H and find its infinitesimal generator. P r o b l e m 4.10. Let X be a Banach space, A 9 D ( A ) C_ X ~ X the infinitesimal generator of a C0-semigroup in X, let a C LI(~), and let ~9 E L 1 ( - ~ , 0; X). Prove that the integro-differential equation with infinite delay x'(t) -

a(t - s ) A x ( s ) ds (X)

for s C ( - c ~ , 0 ]

x(s)-~(s)

can be equivalently rewritten in the space 3: - L l(-cx~, 0; X) under the form { u'-Au ~(o)

-

~,

with A " D ( A ) C_ X ~ ~ defined by D(,A) -

{

1,1 t ~ e WD(A) (--oc, O; X ) ; (0) --

/0

a ( ' r ) A ~ ( - ' r ) d'r

}

and ~'(0) [~](o)

if 0 < 0

-

o

a ( T ) A ~ ( - T ) d7

i f 0 - - 0.

1,1 Here I/V ,, D(A)(--~, 0; X) is the space of all ~ C W l'l(-(:x~, 0; X) satisfying ~(s) C D ( A ) a.e. for s E ( - c o , 0 ], and A~ C L l ( - c ~ , 0; X).

N o t e s . Except for Section 4.2, which follows an idea in Richtmyer [110], and for the last two ones, which are essentially based on Dafermos [44], the chapter is adapted from Brezis and Cazenave [31]. We note that the results on the Laplace operator with Dirichlet boundary condition are classical and can be found in Barbu [21], Brezis [29], Grisvard [64], Pazy [101] and

104

Differential Operators Generating Co-Semigroups

Taylor [119]. As concerns the Laplace operator with Neumann boundary condition, we followed Taylor [119]. More details on the Maxwell operator can be found in Taylor [119]. Moreover, we add that the wave operator in ]~n i.e. A ' D ( A ) C_ H --+ H defined by H - HI(]~ n) x L2(]~n) and D(A) = H2(]~ n) x Hi(JR n) A(u, v) = (v, At) for (u, v) E D(A), generates a C0-group {G(t) ; t C IR}, satisfying

IiG(t)l]~(H) <_ e 2It for each t C IRn. For a direct proof of this generation theorem see Pazy [101], Theorem 4.5, p. 222. An alternate proof is to show that A satisfies all the hypotheses of Theorem of Lit [86] mentioned in Notes to Chapter 3, with 70 = 2. The idea of treating linear integro-partial differential equations (as those of linear viscoelasticity in Section 8), by using specific semigroup techniques, belongs to Dafermos [44]. An extension of this method to the fully nonlinear case is due to Barbu [19]. Subsequent developments are due to Aizicovici [4], Crandall, Londen and Nohel [42], and Hrusa [71]. For other examples in population dynamics, which can be analyzed with the tools of the C0-semigroup theory, see Anita [7], Engel and Nagel [51], Thieme [121], Webb [132] and the references therein. The problems in this chapter are adapted from Engel and Nagel et al [51] and McBride [89].

CHAPTER 5

Approximation Problems and Applications

In this chapter we study the continuity properties of the semigroup as a function of its infinitesimal generator, properties which allow, on one hand, to approximate the solutions of a certain differential equation by the solutions of some other more simple equations, and, on the other hand, to give a rigorous meaning to some formulae in Theoretical Physics established by a more or less intuitive manner. We also study the behavior as t tends to +co of the solutions of a homogeneous first-order differential equation governed by the infinitesimal generator of a Cosemigroup, and we show how the celebrated central limit theorem in Probability Theory can be obtained via C0-semigroup techniques.

5 . 1 . T h e C o n t i n u i t y o f A ~ e tA Let X be a real or complex Banach space and let us denote by f~p(X) the space L ( X ) endowed with the pointwise convergence topology. Let us denote also by 9 ( X ) the set of all infinitesimal generators of C0-semigroups of contractions defined on X. In this section we present some continuity properties of the application A ~ S, defined on ~(X), and taking values in C([ 0, +co); L p ( X ) ) , where S is the C0-semigroup of contractions generated by A. We note that all the results here included can be suitably restated in order to handle C0-semigroups of type ( M , w ) as well. We begin with the following fundamental results. T h e o r e m 5.1.1. (Trotter, Neveu, Kato) Let {An; n C N} C ~ ( X ) , and let us denote by {Sn(t); t > 0} the Co-semigroup of contractions generated by An, for n = O, 1, . . . . Then we have: (i) if there exists

lim Sn(t)x = So(t)x for each t E I~+ and each u-+c<)

x C X , then there exists lim ()~I - A n ) - l x - ()~I - A o ) - ~x for n----~cx:)

each x C X , and uniformly for ~ in compact subsets in (0, + o c ) ; 105

Approximation Problems

106

(ii) if there exists lim ( A I - A n ) - l x -

(AI-

A0)-lx

for each x E X

n---+oo

and A E (0, +oo), then there exists lim Sn(t)x = So(t)x for each n---+oo

x E X , and uniformly for t in compact subsets in (0, +cxD). P r o o f . Let us recall that, for each n C H* and each A > O, we have A E p(An) and

/0

The item (i) follows from the preceding equalities, by observing that, by virtue of the fact that all the semigroups in the family are of contractions, the hypotheses of the Lebesgue dominated convergence theorem are satisfied. To prove (ii), we shall use Theorem 3.1.1 applied to a suitably defined linear operator on the space of all convergent sequences with elements in X. More precisely, let -- {(Xn)ncH* ; ~ lira Xn}, n--+oo

which, endowed with the norm ]]. ]]~, defined by

II(Xn)n N*

= sup n>l

IIxnII,

is a Banach space. Let

D(A) = {(Xn)nEH, e ~; Xn e D(An) V n e H* and (Anxn)nEN* e ~}~ and let A : D(A) C :~ --+ :~ be defined by

J~(Xn)nCH, --(dnxn)nCH,. We emphasize that, if (Xn)nEN, is in D(A), then lim Xn = x0, lim Anxn -- Yo, xo C D(Ao) and Aoxo = Yo.

u- +c<)

n--+oc

Let A > 0 and (fn)ncN* C ~. Then, for each n E H* the equation

)~Xn -- Anxn = fn has a unique solution Xn e D(An) and, by virtue of (ii), the sequence (Xn)~cN* belongs to %. Hence (0, +oc) C p(A). By Theorem 3.1.1, we have 1 [[(AI--,,z[)-l(fn)nEN*][oo <_ sup I I ( A I - An)-lfn[[ <_ ~ [[(fn)nCl~*][. n>l

It is easy each n E sequence and only

to see that D(,A) is dense in 2:. This follows from the fact that, for H*, An is densely defined, combined with the observation that a ((Xkn)nCN.)kCN. in 2: is convergent in the norm [[. ][oo to (Xn)~cN* if if all the sequences (xk)kcN. are convergent to Xn uniformly with

The Continuity of A ~ e tA

107

respect to n E N*. From Theorem 3.1.1, it follows that A generates a C0semigroup of contractions {S(t); t _> 0}. Inasmuch as the operator A is of "diagonal" type, we conclude that ( X J - A ) -1 - ( ( M - An)-l)n~N.. Hence fl[)~ -- (Anx)nEN*. Then it follows that S(t)(Xn)nEN* -- (Sn(t)Xn)nEN*. From the definition of the space *, we deduce lim Sn (t)Xn -- So(t)xo. The proof n--+(x~

is complete.

[-1

D e f i n i t i o n 5.1.1. A center of the operator A is any subspace Y of D(A) such that graph A is the closure in X x X of the graph of the restriction of A to Y, i.e. graph A - graph Ay. One may easily see that, in general, the center of a closed operator is not unique. D e f i n i t i o n 5.1.2. The inferior limit of a sequence of sets (Dn)ncN* is the set (:x:)

liminfDnn---+ cx?

oo

~') Din.

U

n-- 1 m--n

T h e o r e m 5.1.2. Let {An; n C N} C ~(X), and let {Sn(t); t > O} be the Co-semigroup of contractions generated by An, for n - O, 1, . . . . Let us assume that there exists a center Yo of the operator Ao, such that Y0 C lim inf D(An) and

n--~c~

lim Any - Aoy

n-+c~

(5.1.1)

for each y C Yo. Then, for each x E X , lim Sn(t)x = So(t)x, n-+oo

uniformly for t in compact subsets in (0, +oc). P r o o f . By virtue of (ii) in Theorem 5.1.1, it suffices to show that, for each ~ > 0 and each x E X, we have lim ( ) ~ I - An)-Xx - ( A I - A0)-lx. n---+oo

Since Y0 is a center of the operator A0, ( ) ~ I - Ao)Yo is dense in X, and therefore it suffices to prove the equality above merely for x C ( A I - A0)Y0. Indeed, this follows from the inequality

II()~I_~ ]](,~I-

An)-lx

- (AI-

An)-lxAn)-lxell +

( / ~ I - A0)-lxl]

I]()~I-

Ao)-lx~ - (,~I-

_<

2

An)-lxE

-- ( ) ~ I -

Ao)-lxll

llx - x ll + I I ( M - A n ) - l x e - ()~I- Ao)-lxell

A0)-lxeII

108

Approximation Problems

and from the fact that, for each x E X and each e > 0, there exists x~ C ( A I - Ao)Yo with IIx - x~ll _< e. Let x - ( A I - Ao)y with y C Y0. We have ]l(,,~I- A n ) - l x -

( / ~ I - Ao)-~xll

-

I1(~I- An)-l(/~I

- Ao - AI + An)y[[

1

= I1(~I- A n ) - I ( A u - Ao)yll _ ~II(An - Ao)yl]. By (5.1.1), it follows that lim II(AI- An) - i x - ( ~ I -

Ao)-~xll- 0,

n--+oo

and this achieves the proof.

V1

L e m m a 5.1.1. Let A" D ( A ) C_ X ~ X be a densely defined, linear, closed operator, and let A C p(A). Then # E p(A) if and only if the operator I-(~-~)(,I-A) -1 i~ inv~rtibl~ ~ith th~ i n w r ~ in ~ ( X ) , ca~ in ~hich

( A I - A) -1

-

-

( p I - A) -1 [ I - (#

-

P r o o f . Let us assume that B - I - ( p in addition, 13-1 C L(X). Then

(AI- A)(#I- A)-IB -I

A ) ( # I - A)-I] -1 .

(5.1.2)

A ) ( p I - A ) -1 is invertible and,

- [(/~ - #)I -{- ( # I - A)] ( # I - A)-IB -I

= [(A- #)(#I-

A) -1 + I]

B -1

--

I.

Since ( A I - A) commutes with both ( p I - A ) -1 and B -1, by the preceding equality, we deduce that (#I- A)-IB-I(AI-

A) - I D(A).

Therefore p E p(A), and (5.1.2) holds. Conversely, if # C p(A), then (AI- A)(#I-

A) -1 - X ( # I - A) -1 + [ ( # I - A ) - # I ] ( # I - A) -1 - B,

from where it follows that B is invertible. As B -1 - ( # - A ) ( X I - A) -1 + I, it follows that B -1 C L ( X ) , and (5.1.2) holds. The proof is complete.

FB

T h e o r e m 5.1.3. (Trotter, Neveu, Kato) Let {An; n C N} C ~(X), and let {Sn(t); t >_ 0} be the Co-semigroup of contractions generated by A~, for n - O, 1, . . . . Let us assume that, for some fixed A > 0, there exists an injective operator R~ C L ( X ) , with dense range in X , and such that lim ( A I - A ~ ) - l x - R~x

(5.1.3)

n---+ o o

for x C X . Then there exists Ao C 9 ( X ) so that Rx - ( A I - A0) -~, and lim S n ( t ) x n---~oo

So(t)x

The Continuity of A

~

109

e tA

for each x C X , and uniformly for t in compact subsets in (0, +oo).

P r o o f . Let ,~ > 0, It > 0 and n C N*. Inasmuch as ,~,# C p(An), from Lemma 5.1.1 we deduce OO

(~I-

An)-~x - (~I-

A~) -1 Z ( ~

- .)k(~Z-

An)-k~

k=0

for each x C X. For fixed ,~ > 0 and c C (0, ,~), the series above converges uniformly for I,~-#I -< "~- c. From this remark, and from (5.1.3), it follows that, for each It > 0 with I,~ - #I < "~, there exists R~ E L ( X ) such that (5.1.4)

lim ( # I - A ~ ) - l x - R~x n---+ o o

for each x C X. We shall show next that the set S, of all numbers # in I~_ for which (5.1.4) holds, coincides with ]E~_. To this aim, we shall prove that S is both open and closed in IR~. Clearly S is open. Let # C ]E~_ \ S and let ~ > 0 be such that I # - ~I < k# 1 3#). Since 2 , or equivalently )~ E (~#~ I~ - ~1 < 12. < )~, it follows that )~ E I~+. \ S, because otherwise # C S thereby contradicting the assumption # C I~_ \ S. So

Consequently R~_ \ S is open and therefore S - I~_. From this remark it follows that all the operators Rp in (5.1.4) are well-defined for each # > 0. In addition, from the resolvent equation (1.7.3), and from (5.1.4), it follows by passing to the limit that R~ - R~ - ( # - A)R~R~,

and RARp - RpRA for each A,# C IR~_. Let us denote by N and ~ the kernel, and respectively the range of the operator R~. From the fact that R , - R A [ I - ( # - A)R~], it follows that the range of R~ includes the range of RA. Interchanging A with #, we deduce that ~ is independent of # C I~_. Analogously, one proves that :N is independent of # ~ ]~_. As R:~ is injective and with dense range, it follows that : N - {0} and ~ is dense in X. Let us define the operator Ao 9 D(Ao) C_ X ~ X by D(Ao) - :~ and A0 - A I - R~ 1, where A > 0 has the specified properties. We have (,~I - A0) R~ - R~ (,~I - A0 ) - I on D (A0). In addition, (#I-

A o ) R . - [(# - s

+ ( A I - A0)]R,

= [(# - i~)I + ( ) ~ I - A o ) ] R ~ [ I - (# - A)R,] = z + (~ - ~ ) [ ~

- ~. -(~

- ~)~R~)]

- z.

110

Approximation Problems

Similarly, we have R ~ ( # I -

Ao) = I on :~. Consequently

Rp - ( p I - A0) -1 for each p > 0. Finally, [[(#I-

1

A0)-lxl]- l i r n [ [ ( # I - An)-Xx[[ ~ ~[[x[[

for each # > 0 and each x C X, showing that A0 satisfies the hypotheses of Theorem 3.1.1. So, A0 generates a C0-sernigroup of contractions. Since the conclusion follows from Theorem 5.1.1, the proof is complete. [3 5.2.

The Chernoff and Lie-Trotter Formulae

The aim of this section is to establish two approximation formulae for C0semigroups of contractions which will prove very useful in applications. We begin with the following fundamental result known as the Chernoff product formula T h e o r e m 5.2.1. (Chernoff) Let X be a Banach space, real or complex, and let (V(t); t > 0} be a family of contractions on X with V(O) - I. Let us assume that there exists A 9 D ( A ) C X ~ X which generates a Co-semigroup of contractions (S(t); t > 0} and, in addition, for each x C D ( A ) , there exists

lim 1 (V (h) - I ) x - Ax. h~O -h Then, for each x C X , we have

lim v n ( t / n ) x -

(5.2.1)

S(t)x,

n--+ c~

uniformly for t in compact subsets in I~+.

The proof of Theorem 5.2.1 is essentially based on the next lemma. Lemma

5.2.1. If L C L ( X ) satisfies IILIl~(x)< 1, then Ilen(L-I)x _ Lnxll ~_ x/~IILx - xll

for each n C N*, and each x E X .

P r o o f . From the definition of the operator en(L-I),

Ilen(L-l)x--nnxl[ --e -n

-~. ( n k - n n ) x k=0

<-

Zcc ynk lln x- L n xll . k=0

it follows that

The Chernoff and Lie-Trotter Formulae

111

Since m

IILkx - L n x l l <_

IlL k - n

X --

xll

and

L m -

I - E(Li-Li-1), i=1

we have m

From these relations and the Cauchy-Schwarz inequality, we deduce

II

II

e n ( L - I ) x -- L n x

~- e - n E

~lk

- nlllLx -

nk

xll

k--0

{ c~ nk

} 1/2 IILx - x II.

k=0

By the relation between the mean and dispersion of a Poisson distributed r a n d o m variable (see for instance Laha and Rohatgi [81], Example 1.2.2, p. 31), we know that oc nk _i_(. (k

k=0

_

_

ff]

from where we get the conclusion. We may now proceed to the proof of Theorem 5.2.1. P r o o f . Let

(an)nEI~*

be a sequence in R+ convergent to 0, and let

1 IV(an) An

--

I]

if

an>

0

an

A

if an - 0.

Then, by Theorem 5.1.2, we have that, for each x C X, lim

I]e_.tAnx -- S(t)x]l - O,

uniformly for t in compact subsets in IR+. Let t > 0 be fixed, and let an : t / n for each n C N*. From L e m m a 5.2.1 with L = V(t/n), we obtain

_ -~t ~[v(t/~)~-~]11' which shows the pointwise convergence in (5.2.1). To conclude the proof, we have merely to check that, in (5.2.1), the convergence is uniform on

112

Approximation Problems

compact subsets in IR+. But this follows from the simple observation that, for each [a, b] C R+, we have

-- tC[ a,b ]

-~

-[

inequality whose right-hand side tends to 0 for n tending to oc.

5

The next two fundamental results are simple consequences of Theorem 5.2.1. T h e o r e m 5.2.2. Let {S(t); t > 0} be a Co-semigroup of contractions and let A 9 D(A) C X --+ X be its infinitesimal generator. Then, for each x E X , we have lim

n--+c~

I-

tA

_

rt

x-

S(t)x,

1

uniformly for t in compact subsets in JR+.

P r o o f . One applies Theorem 5.2.1 to the family V ( t ) - ( I - tA) -1 for t>0. [2 D e f i n i t i o n 5.2.1. An operator L " D(L) C_ X -+ X is called closable, if the closure of its graph in X x X is the graph of a linear operator. If L is a closable operator, its closure, denoted by L, is the operator whose graph is the closure of the graph of L. T h e o r e m 5.2.3. Let A " D(A) C_ X --+ X and B " D ( B ) C_ X -+ X be the generators of two Co-semigroups of contractions {S(t); t > 0}, and respectively {T(t); t >_ 0}. Let us assume that A + B is closable and its closure, A + B " D ( A + B) C X ~ X , is the infinitesimal generator of a Co-semigroup of contractions {U(t); t > 0}. Then, for each x C X , we have lim { S ( t / n ) T ( t / n ) } n

X

--

U(t)x, 2

n--+cxD

uniformly for t in compact subsets in IR+.

P r o o f . One applies Theorem 5.2.1 to the family V(t) - S(t)T(t) for D t _> 0 and to the operator A + B " D ( A + B) C_ X --+ X .

1This is the famous Hille's exponential formula. 2This is the celebrated Lie-Trotter formula.

A Perturbation Result 5.3.

A Perturbation

113

Result

In this section we prove a sufficient condition in order for the sum of an infinitesimal generator of a C0-semigroup of contractions with a densely defined linear operator to also be the infinitesimal generator of a C0-semigroup of contractions. T h e o r e m 5.3.1. Let A " D(A) C_ X --+ X , B " D(B) C_ X --+ X be two linear operators with D(A) C_ D(B), and such that

IIAx- (A + tZ)xll > ~,llxll

(5.3.1)

for all A > O, t e [0, 1] and x e D(A), and

fo~ ~ach 9 c

D(A),

IIBxll < ~IIAxll +/~llxll ~ h ~ ~ ~ [0, 1) and 9 >

(5.3.2) O.

If t h ~

~i~t~ ~ in

[0, 1] such that A + sB is closed, and (0, +ec) C_ p(A + sB), then we have (0, +co) C_ p(A + tB) for each t C[0, 1 ]. In this case, for each t E [0, 1 ], A + tB is the infinitesimal generator of a Co-semigroup of contractions. P r o o f . Let s C [0, 1] be such that, for each A > O, A I - (A + sB) is invertible, and ( A I - (A + sB)) -1 e L ( X ) . To complete the proof, it suffices to show that there exists 5 > 0 such that, for each t C [0, 1] with I t - s I _< 5, we have (0, +co) C_ p(A + tB). From (5.3.1), it follows that [[R(A; A + sB)]l~(x ) <_ ~. We shall show next that BR(A; A + sB) e L ( X ) . From (5.3.2), it follows that

IIBxll _< ~llnxll +/311xll <_ ~ll(n + ~B)x[[ + ~sllBxll +/~llxll ~II(A + ~B)xll + ~[[Bxll + ~[Ixll for each x C D(A). Accordingly

IIBxll -< 1 - ~c ~ II(A+,B)xlI+ 1 -~c ~ Ilxll, Since R(A; A + sB) " X --+ D(A) and

(A + sB)R(A; A + sB) - AR(A; A + s B ) - I, from the last inequality, it follows that

IIBR(ik; A + sB)xll <

OL

~[I(AR(A; -1-c~

A + s B ) - I)xll + 2Ao~ + fl

1--0l

<- A(1 - c ~ ) I1~11

IIR(A; A + sB)xll

114

Approximation Problems

for each x C X. Consequently BR(A; A + sB) C L(X), and

IIBR(A; A -I- sB)ll~(x) _<

A(1 - a ) "

At this point let us observe that

A I - (A + tB) - A I - (A + sB) + (s - t)B = (I + (s - t)BR(A; A + s B ) ) ( A I - (A + sB)). Then A I - (A + tB) is invertible if and only if I + (s - t)BR(A; A + sB) enjoys the same property. But the latter is invertible for each t C [0, 1] with It - s] < A(1 - a)(2aA + /~)-1 ~ [[BR(A; A + sB)llz(x).-1 Taking (~ < A(1-a)(2aA+fl) -1, let us observe that A I - ( A + t B ) is invertible for each t C [0, 1] with [ t - s[ <_ 6. Since [0, 1] can be covered by a finite union of intervals whose length is less that 5, the proof is complete. D 5.4. The central limit theorem

In order to illustrate the power of Chernoff Theorem 5.2.1, in this section, we will give a proof of one of the most famous result in Probability Theory and Statistics: the central limit theorem. We begin by recalling that a random variable ~ defined on a probability space (f~, E,p) has a normal distribution of mean 0 and dispersion r > 0, if its repartition, or cumulative probability, F~(t) - p ( ~ <_ t) is given by

F ~ ( t ) - F N ( o , r ) ( t ) - vzTrrl/~' mx__=

exp { s2}-2r ds Oo

for each t C ]~. One may easily see that the two parameters 0 and r are the mean, and respectively the dispersion of the variable ~. T h e o r e m 5.4.1. (The central limit theorem) Let (~n)neH* be a sequence

of random variables having normal distribution of mean 0 and dispersion 1. Then, for each x E I~, we have lira F,~(x) - FN(O,1)(x), n--+c~

where, for each n C N* and x C ~, ~n (Od) -- ~

1

n ~

k--1

~k(a)).

The central limit theorem

115

In order to prove Theorem 5.4.1, some preliminaries are needed. Let 9" be the set of all repartition functions and let X = Cub(R). For each F E 9", we define F " X -+ X by the convolution F f - f 9 dF, i.e.

f (t - s)dF(s)

Ff(t) O0

IIFII (x)

for each f E X and t E R. Obviously F E L ( X ) and Lemma

5.4.1. Let (Fn)ncN*, and F in ~.

1.

Then lim Fn(x) = F(x) at n--~oo

each point x of continuity for F if and only if lira F n f - F f for each n-+oo

f

x.

Lemma

5.4.2. For each F, G c ~, we have

F,G-FG. We may now proceed to the proof of Theorem 5.4.1. P r o o f . By virtue of L e m m a 5.4.1, in order to prove the theorem, it suffices to show that

nlimcr Fu~ f - FN (O,1)f for each f in a dense subset in X. Let G be the common repartition function of all r a n d o m variables ~n, let Gr(x) = G(x/~x ) for r > 0 and x 6 ~, and let V(t) - G~/t for t > 0, and V(0) - I. Finally, let A ' D ( A ) C_ X -+ X be defined by D(A) - {f c X; f ' , f " c X}

Au-

1_,1

for each u C D(A). One may easily prove that A generates a C0-semigroup of contractions {S(t); t _> 0} on X, and, moreover, that

[S(t)f](x)- ~

i F

oxp{ }s/x y2

y) dy,

i.e. S(t) - FN(o,t), for each t > O. We shall show next that lim 1 h~O -~[V(h)f - f]-

Af

(5.4.1)

for each f C D(A). Assuming for the moment that (5.4.1) holds, from Chernoff product formula, it follows that

liin v n ( t / n ) f - FN(o,t)f

(5.4.2)

Approximation Problems

116

for each f C D(A) and t _> 0. Lemma 5.4.2, we have

Taking t -

1, we observe that, by

vn(1/n) - a n - Hn, where Hn is the repartition function of the sum of n random independent variables, each one having the repartition function Gn, while Gn-Gn,Gn*.--*Gn. Since Gn is the repartition function of the random variable ~nn~/, we have Hn - F~, which implies the conclusion. Thus, it remains to prove (5.4.1). Let us observe that

f In addition, since

~ dG~(r)- 1. O0

Gs - F~/v~, we deduce

f?

r dG~(r) - E

~i

-0,

CO

~da.(~)

- E

-

Then, for each f C D(A) and each (f > 0, we have { ?1 [ V ( t ) f - f ] - A f } (x) - -~ l[v(t)f-f](x)-I -

~

f(x ]<5 ~r2 If" ( 0 ) -

~

r)

da:/~(~)-

f(x) + rf'(x) t

f" (x)] dG1/t (r) +

1,,

~f

r' ]>_~-~

-~f " (x)

(~)

] da:/t(r) (0)

(x)]

dG1/t (r)

= J~ + J2,

where 0 is between x and x - r. Let r > 0. Inasmuch as f " is uniformly continuous on ~ there exists 5 > 0 such that If"(O)- f"(x)l < s/2 for each 0, x C ~ with ] 0 - x ! < 5. Fix 5 > 0 with property above, and let us observe that IJ:[-< ~ alrl<5

r2dG1/t(r) <

Feynman Formula for each t > O. Since If"(O)- f"(x)I <

lJ21 _
117

211f"ll~,

we have

>z 2][f"ll~r2dal/t(r)2t

(

--IIf"ll~/r

( )

r

T

/

-IIf"ll~[

s2da(s) 9

1>_6/r From this inequality and from the fact that f _ ~ s2dG(s) - 1, we deduce that lim [ s2dG(s) - O. t$O .#is >~/v~ Therefore, for the very same c > 0, there exists t(E, f) > 0 such that, for each t E (0, t(c, f)), ]J2] < c/2. Consequently we have {lt [ V ( t ) S -

(x)

for each x E IR and each t E (0, t(c, f)). The proof is complete. 5.5. F e y n m a n F o r m u l a

Our aim here is to give a rigorous interpretation of the celebrated Feynman formula by using the Lie-Trotter formula. Let us consider a nonrelativistic particle without spin, of mass m, moving in ]R3 under the influence of a potential V. We notice that the wave function of the particle is the solution u = u(t, x) of the Cauchy problem for the Schrhdinger equation

Ut --i [ 1 /XU-- g(x)u] U(0, X) --~0(X)

(t,x) C ]~ x ]R3 X e ]~3,

where R3 II~o(X) II2 d x

--

1.

From a physical point of view, Ilu(t, x)II 2 represents the probability density of the position of the particle at the time t, and at the point x. More precisely, f~ ]lu(t, x)]] 2 dx is the probability that, at the time t, the particle lies in the Borel set ~ in R 3. Analogously, lib(t, ~)]]2 is the momentum of the probability density, where ~ is the Fourier transform of u, i.e.

1

u(t, ~) -- (27r13/2

fR e-i({'W)u(t w) dw 3

'

"

118

Approximation Problems

Let us consider the Hilbert space H - L2(IR3), and let us identify u with a function from ~ to H satisfying

(A+B)u u(0) - uo, where A " D(A) C_ H --+ H and B " D ( B ) C_ H --+ H are defined by D(A) - H2(IR 3) Au- (i/2m)Au and, respectively by D(B)-{ueL2(R3);

(5.5.1)

V(.)ueL2(R3)}

(5.5.2)

L e m m a 5.5.1. The operator A, defined by (5.5.1), is the generator of a Co-group of unitary operators {S(t) " L2(IR 3) ~ L2(IR3); t C IR}, defined by -

(2it)-3/2fR ~

m

s

exp

{ imllx-{l'2 } u(~) d~ 2t

'

(5.5 3) "

for each t > 0 and u E L2(IR3). P r o o f . We begin by proving that iA is self-adjoint. that, for each u, v C H2(R3), we have

1 Au, v ) - - ] ~ (iAu, v) - ( - 2 m - -

/1

Let us observe

1 Au~dw s 2m

s -2-ramu A v dw - (u, lAy),

which shows that iA is symmetric. Since C ~ ( R a) C_ D(iA), iA is densely defined and so, by virtue of (i) Lemma 1.6.1, in order to complete the proof, it suffices to check that ( I - iA) -~ C L ( X ) . We begin by showing that the range of I - iA is dense in H. More precisely, we will prove that C ~ ( R a) is included in the range of I - iA. Let f E C~(R8), and let us consider the problem u - iAu - f, which rewrites equivalently under the form

u + 2 - ~ A u - - f. Then u, given by

1 ~ 2 m f ( ~ ) e i(z'{) u(x) - (27r)3/2 s 2m + II~ll2 d~, is a solution of the problem above. From Stone Theorem 3.9.1 it follows that A generates a C0-group of unitary operators. Since formula (5.5.3) follows by direct computation, the proof is complete. O

Feynman Formula

119

L e m m a 5.5.2. If V C L2(R3), then B, defined by (5.5.2), is the generator of a Co-group of unitary operators, {T(t) " i2(I~ 3) --+ L2(]R3), t C R}, defined by (5.5.4) [T(t)u](x) - e-itv(X)u(x),

for each u C L2(IR3). P r o o f . First, let us observe that iB is symmetric, I - i B is invertible and ( I - iB) -1 C L ( H ) . By (i) in Lemma 1.6.1, it follows that i B is selfadjoint, while, by Stone Theorem 3.9.1, B is the infinitesimal generator of a C0-group of unitary operators, {T(t); t C ]~}, defined by (5.5.4). D Let t > 0, n E N*, and let us consider Un(t)'L2(I~ 3) -+ L2(~ 3) defined by

Un(t) = [S(t/n)T(t/n)] n . A simple mathematical induction argument shows that

Un(t)u(x) -

TtIKt

3

...

3

exp{iS(xo,xl,...,Xn;t)}U(Xn)dXl.,

dxn,

(5.5.5)

where xo = x E ]~3 and

S(X0,Xl,... , X n ; t )

-- E

mlxj - xj-l[2

2(t/n) 2

- V(xj)

(t/n).

(5.5.6)

j--1

Let Cz the set of all continuous functions w 9 R+ --+ I~3 with w(0) - x. Setting xj = w ( t j / n ) , we observe that S(xo, x l , . . . ,Xn;t) is nothing else than a Riemann sum for the integral of action

s(w; t) -

{- ll (s)ll 2- v(w(s)) } ds,

where D is derivative of w with respect to the t variable. Passing to the limit formally both sides in (5.5.5) for n -+ oc, we "obtain" the famous Feynman integral representation of the wave function, i.e. lim Un(t)u(x) - k f n-+oc

exp{iS(w; t)}u(w(t)) n w ,

JCx

where k is constant. We emphasize that, this completely formal manner of approaching the problem has many vulnerable points. First, the constant k is in fact infinite. In addition, the expression Dw - H0<s
Approximation Problems

120

the definition of the expression above, i.e. the constant k, the integrand and the measure has sense in this frame. At this point, let us observe that, in the case in which A + B is closable, and A + B generates a C0-group of unitary operators on L2(R3), the Lie-Trotter formula (see Theorem 5.2.3) allows us to give a rigorous sense of the above expression by observing that, in fact, we can pass to the limit in (5.5.5) for n --+ oc not pointwisely, but in L2(R3), and this limit is the wave function u(t, x). We conclude this section with a sufficient condition in order that A + B be closable and A + B generate a C0-group of unitary operators in L2(R3). L e m m a 5.5.3. If V C(c) > 0 such that

C LP(R 3) with p > 2, then, for each e > O, there exists

IIBUlIL2(Ra) < clIAu]]L2(X3) + C(e)IlUI[L2(X3), for each u C H2(R3), where A and B are defined by (5.5.1), and respectively by (5.5.2). P r o o f . If u e H2(R 3) then (1 + 11~112)g(~) e L2(R3). On the other hand, it is obvious that (1 + 11~[12)-1 e Lp(R3). Take q - 2p/(p+ 2) and let us observe that, from Parseval identity and Hhlder's inequality, we have

IlaLLL~(a3) -

3 Ile(~)llq d~

(~ _<

)l/q ( 1 + I1~112)-~(1 + 11~ll2)qlle(~)LI q d~

3(1 + II~II2)-Pd~

3(1 + 11~]12)211~(~)1[2 d~

< Cp(II/XUlIL~(R~)+ IlUlIL=(~3)). Since p > 2 and 1 < q < 2, by Hausdorff-Young inequality 3, we get IlUllL~(R3) < II~llLq<~) provided r and q satisfy 1/r + 1 / q - 1. Therefore

II~IIL~(~)-< Cp(IIA~IIL~<~)+II~IIL2(~)). Substituting in the above inequality u(x) by u(px), we obtain

IlUllL~(~3) <_ Cp(p2IIAUlIL=(R3)+IIUlIL~(R3)). Let e > 0. Let us choose p > 0 such that

[[UlIL~(X3)IIVIILp(x3) <_ eIIAIIL=(R3) +

C(e)llUllL~(a3).

3Hausdorff-Young inequality says that I[f*gllL~(R=) _< IIIIILa(R~)IIglIL~(R=),provided and g e L~(R~). See for instance Stein and Weiss [116], p. 178.

1 <_p,q,r <_co, I/p= 1/q+ 1/r- 1, f e Lq(R n)

The Mean Ergodic Theorem

121

IIV IIL ( )

IIVIIL < )II IIL < )

Finally, by HSlder's inequality, we have so, from the preceding inequality, we deduce

and

]]VUIIL2(R3) ~ C]]/XU]]L2(R3) + C(~)]]U]IL2(]~3), which achieves the proof.

[5

T h e o r e m 5.5.1. If V 6 L2(R3), then A + B if the generator of a Co-group of unitary operators in L2(R3). P r o o f . Lemmas 5.5.1 and 5.5.2 imply that i(A + B) is symmetric. In view of Theorem 1.6.1, in order to prove that it is self-adjoint, it suffices to show that I + (A + B) 6 L(L2(R3)). This is a consequence of Lemma 5.5.3 combined with Theorem 5.3.1. We complete the proof with the help of Stone Theorem 3.9.1. D

5.6. T h e M e a n E r g o d i c T h e o r e m We conclude with a result concerning the behavior as t tends to +oc of the 1 t means 7 fo S ( s ) d s of a C0-semigroup of contractions {S(t) ; t >_ 0}.

T h e o r e m 5.6.1. Let A " D(A) C_ X -+ X be the infinitesimal generator of a Co-semigroup of contractions, {S(t); t > 0}. /f xl 6 ker(A) and x2 6 R(A), then there exists a bounded function ~ " R+ --+ X with the property lim ~(t) = 0,

t--+c~ t and such that, for each t > O, --lfo0 t S ( T ) ( X l

t

+

X2)d~- -- Xl -~- ~(t__~). t

P r o o f . Let us observe first that, for each xl 6 ker(A) and each ~- _> 0, we have S(T)Xl -- xl. As x2 6 R(A), there exists x3 6 D(A) such that x2 - Ax3. Accordingly

-Xj~0t ~(T)(Xl + X2) d T t

lj~0t -- X l -4- -~

~(T)X2

dT,

and l

-[ =

S(~')x2 dT --

Ilf0t -[ - ~dT [ S ( T ) x 3 ] d T - -

l

-[

1

S(~-)Ax3 dT

~(S(t)xa-x3)

I _< 2,,x3], t "

So, the function ~(t) - S ( t ) x 3 - x3 has all the required properties. The proof is complete. [i]

122

Approximation Problems

Let X be a Banach space and B C_ X. We denote by B•

EX*;

(x,x*)-OforallxEB}.

Similarly, if B* C_ X*, we denote by (B*) • - {x** 6 X**; (x*,x**) - 0 for all x* 6 B*}. If X is reflexive and identified with its own bidual, then (B*) • - {x C X ;

(x,x*) - O for all x* e B * } .

L e m m a 5.6.1. Let X be reflexive, and let A 9 D ( A ) C_ X ~ X be a densely defined, linear, closed operator. Then ker(A) • - R(A*) ker(A*) • - R ( A )

R ( A ) • - ker A*, R(A*) • - ker(A).

P r o o f . We shall prove only the first equality, the proofs of the other ones being similar. Let y* C R(A*). Then there exists (Xn)n~N in D(A*) such that limn~oc A* xn* - y * 9 Thus , for each x C ker(A) we have (x, y*) -

lim (x, A*xn) n---~ (x)

lim (Ax, Xn) - O, n ---~ (N:)

which shows that y* C ker(A) • So R(A*) C_ ker(A) J-. To prove the converse inclusion, let us assume by contradiction that there exists at least one y* C ker(A) -L such that y* ~ R(A*). By the Hahn-Banach theorem, it follows that there exists x C X (we recall that X is reflexive and identified with X**) such that (x,y*) r 0 and (x,z*) - 0 for all z* C R(A*). In particular, we have (Ax, x*) - (x, A ' x * ) - 0 for each x* C D(A*). On the other hand, by Lemma 3.7.3, we know that D(A*) is dense in X* and therefore A x - O. So x c ker(A) and consequently (x, y*) - 0 which is impossible. This contradiction can be eliminated only if ker(A) • C_ R(A*) and this completes the proof. [] T h e o r e m 5.6.2. (The Mean Ergodic Theorem) Let X be reflexive and let A " D ( A ) C_ X --+ X be the generator of a Co-semigroup of contractions, {S(t); t >_ 0}. Then ker(A) and R ( A ) are closed subspaces in X . Moreover, ker(A) n R ( A ) - {0}, X - k e r ( A ) | R(A), and the projection operator P " X --+ ker(A) is well-defined, and satisfies IIPIIc(x) _< 1. In addition, each x E X has a unique decomposition x - P x + ( I - P ) x and lim -1 ~0 t t~cc t

S(T)x

d~- -

Px.

C ker(A)|

123

The Mean Ergodic Theorem

P r o o f . Theorem 5.6.1 implies that k e r ( A ) n R ( A ) - {0}. Indeed, let y C ker A n R ( A ) . Then there exists (Yn)ncN in R ( A ) such that

lim

Yn -

Y.

n---+(~3

Since y E ker(A) and Yn E R ( A ) for each n C N, from Theorem 5.6.1, we have lim -1 j~0t S ( T ) y d T - y ,

t-+c~ t

and lim -1/0 t S(T)yn dT -- 0 t-+co t for each n C N. On the other hand

1/0t

S(7)y

l~0t

d T -- -t

S(7)yndT

_<

1/0t

Ily -

Ynll

-

Ily -

Ynll,

which implies lim -1 fo t S(T)y dT -- O.

t-+co t

So, y - 0, proving that k e r ( d ) N R ( A ) Let x E ker(A)| R ( A ) . We have

{0}.

IlPxll - tl 9m 71/0 t S( )x

< I1 11,

which shows that ]]Pllc(x)<-1. To complete the proof, we have to check that ker(A)| R ( A ) is dense in X. So, let us assume by contradiction that there exist x C X \ ker(A) | R ( A ) and x* e X*, satisfying both x* C (ker(A) + R ( A ) ) • and (x,x*) - 1. Clearly x* C ker(A) • and x* C R ( A ) • By Lemma 5.6.1, we deduce that x* C ker(A*)n R(A*). On the other hand, since X is reflexive, from Theorem 3.5.1, we know that A* is the generator of the C0-semigroup of contractions {S* (t); t > 0} in X*. From the considerations above, we have ker(A*) n R(A*) - {0}, and therefore x* - 0. This contradiction can be eliminated only if ker(A) | R ( A ) - X , and this achieves the proof. D In order to get a variant of Theorem 5.6.2 valid in general Banach spaces, we need the following lemma. L e m m a 5.6.2. Let u " I~+ -+ X be a bounded and continuous function. Then lim -1 ] i t u(7-) dT- - lim)~ /0 cr e-~Su(s) ds

t-+oo t

~$0

'

124

Approximation Problems

in the sense that, either both l i m i t s exist and they are equal, or both do n o t exist.

P r o o f . Let us assume for the beginning that there exists lim -1 fot u(r) dr - g,

t+ao

t

and let us observe that

f0

~-~*~(~) d~ - ~

~-~*s

9

(1s s -

Then

fo ~

~(~) a~

8

e - X S u ( s ) ds - g < ~2

)

ds.

1/oS

e-xss

-

~0~176

u(r) dr-g

ds.

s

Let s > O, and let us fix a ~ = a(s) > 0 such that -l

s

j'oS u ( r )

dr-g


for each s E [($, +cx)). From the preceding inequality, it follows

),

f0 c~

~-~'~,~(~)

ds - e <_ ),2

+)~2 <_ ~2

/o

j~

f0 ~ 1 f~s

c~

~-~'~

e -xs s -

-

8lf0~

,~(,-) d~- - e ds

u('r) d r - g ds

s

e-X~s(M

+

lel)ds

+ ~=e

se -~s d~,

where M > 0 satisfies ]]u(t)]] _< M for each t C R+. Therefore, a simple computational argument yields

fo ~

e - x S u ( s ) ds - g <_ ( M § Igl)(1 - e - x a - ~,5e - x a ) § e(Aae - x a + e-Xa).

Passing to the sup-limit in this inequality for )~ tending to 0 by positive values we obtain

(x)

j~

e - x s u ( s ) ds - g < C . lim sup )%0 Since s > 0 is arbitrary, this shows that, whenever there exists the first limit in lemma, then there exists the second one too, and both must be equal. The proof of the fact that the existence of the second limit implies the existence of the first one (and by the preceding proof, the fact that both are equal) is much more subtle and follows from a very deep Theorem of

125

The Mean Ergodic Theorem

Wiener which, for the sake of brevity, did not find its place here. However, we refer the interested reader to Widder [133], Theorem 14, p. 221. The proof is complete. [2 Theorem

5.6.3. Let X be a B a n a c h space and A 9 D ( A ) C_ X ~ X the generator of a Co-semigroup of contractions {S(t); t _> 0}. Then the following conditions are equivalent" (i)

X

-

ker A | R(A);

(ii) for each x C X there exists lira -1 t-+co t

ftJo S ( r ) x d r ;

(iii) f o r each x E X there exists l i m ) ~ ( M - A ) - l x ; Mo

(iv) f o r each x C D ( A ) there exists l i m A ( M Mo

A)-lx.

P r o o f . As in the proof of Theorem 5.6.2, one shows first that, for each x C ker(A)| R ( A ) , there exists lim -1 Ji t S ( r ) x d r . t+oo t Hence (i) implies (ii). From Lemma 5.6.2 and from the observation that, for each x C X, we have a(ai

- A)-I

-

a

we deduce that (ii) implies (iii). The fact that (iii) implies (iv) follows from the identity (AI- A)-IAx

- A(AI- A)-lx-

x - A(AI-

A)-lx,

(5.6.1)

for each x e D ( A ) . As D ( A ) is dense in X, from Lemma 5.6.2 and (5.6.1), it follows that (iv) implies (ii). In order to complete the proof we have merely to show that (ii) implies (i). Let us define P " X --+ X by Px -

lim -1 f t S ( r ) x d r t-+co t Jo

for each x C X. We have X - R ( P ) + R ( ~ - P ) and P S ( t ) x - S ( t ) P x - P x , for each x E X and each t _> 0. In addition, let us remark that p2 _ p, R ( P ) - ker(A), R ( I P ) - ker(P), and R ( A ) C_ ker(P). To check that ker(P) c_ _R(A), let x e ker(P). Then, by (5.6.1) and (iii) (which is implied by (ii)), it follows that lim A ( M - A ) - l x - P x - x - - x , ~4o

and therefore x C R ( A ) . The proof is complete.

W1

126

Approximation Problems

Problems We recall that un operator A 9 D ( A ) C_ X -+ X is called closable if the closure of its graph is the graph of a linear operator A, called the closure of the operator A. Problem

5.1. Let X - L2(0, u), and A ' D ( A )

C_ X -+ X , defined by

D(A) - (u E X ; u ' there exists a.e. and u ' A u - u' for each u E D ( A ) .

v a.e. with v E X}

Prove that A is not closable. See Richtmyer [110], Example 3~ p. 137. P r o b l e m 5.2. Which one of the operators below is closable? In the case of a closable operator, determine the domain of its closure. C X -+ X defined by"

(i) X - 1 2 , and A ' D ( A ) D ( A ) - {(Xn)nEN, (A(Xn)nEH* )kEN*

12; 3 m N*, x n (2kXk)kEH * ;

(ii) X -12, and A ' D ( A )

--

0

V n > m}

C_ X --+ X defined by"

D(A) - {(Xn)nEN. e 12; 3 m e N*, V n > m Xn -- 0} (A(Xn)nCl~*)kEl~* -- (~-2~n~1Xn, 0, 0,... ) ; (iii) X -12, and A ' D ( A )

C_ X --+ X defined by"

D(A) - ((Xn)nCN* e 12 ; 3 m e N*, V n > m (A(Xn)nEH* )kEN* -- (~-2'~n~176 (1/n)Xn~ O~ 0 , . . . ) ; (iv) X - L2(0, 1), and A ' D ( A )

X

n

--

0}

C_ X --+ X , defined by"

D(A) - C([0, 1]) ( A u ) ( x ) - u(1/2) sin ~x ; (v) X - 52(0, 1), and A ' D ( A )

C_ X -~ X , defined by"

D(A) - C([0, 1]) ( A u ) ( x ) _ ( s u ( y ) d y ] sinux.

See Richtmyer [110], Exercise 2, p. 138.

P r o b l e m 5.3. Show that if A is closable and densely defined, then A** coincides with the closure of A.

P r o b l e m 5.4. Let A ' D ( A ) C_ X --~ X be the generator a C0-semigroup of contractions on the Banach space X. Let n E N*, and let us recall that the Yosida approximation An, of A is defined by An - n R ( n ; A).

Notes

127

Using Theorem 5.1.1 prove that

S(t)x -

lime tAnx. n--+oo

See Engel and Nagel [51], 4.10, p 214. P r o b l e m 5.5. Let X - C0(IR), or X - Cub(IR) and let us consider the Co translation group [S(t)f] - f ( t + s), for each f C X and t, s CIR. For each n C N we define An

z

S(1/n)-I 1/n

Prove that" (i) An commute with S(t); (ii) An generates a uniformly continuous semigroup of contractions; (iii) limn-+oo An f = f for each f C D (A). Show that 0o tk

f ( t + s) - nlim E

~. (Akn)(f)(s)

k=O

for each f C D(A) and uniformly for s C IR, t E [0, 1 ]. Show that, for each f E X there exists a sequence (ran)nON, tending to cx~ and such that, for each k C H, ~-]mk tk k=0 ~(Akn)(f)(0) is a polynomial, and mk tk ~(A~)(f)(0).

f(t) - l i m E

k=0

This is the well-known Weierstrass approximation theorem. See Engel and Nagel [51], 4.11, p. 215. N o t e s . Theorems 5.1.1 and 5.1.3 were proved within a particular frame by Neveu [95], and generalized by Trotter [122]. The lack of an important step in the initial proof, given by Trotter, has been completed by Kato [72]. The proof of Theorem 5.1.2 in the form here presented is due to Kis:?nski [74]. Theorem 5.2.1 was established by Chernoff [37], Theorem 5.2.2 is due to Hille [69], and Theorem 5.2.3 was proved by Trotter [123]. However, the main idea in all these results goes back at the end of the ninetieth century and is due to Lie. For applications of results of this kind in optimal control theory, see Barbu [20], pp. 389-404. Theorem 5.3.1 was obtained by Pazy [100]. Although equivalent with Pazy's original result~ Theorem 5.3.1 is included here in a slightly different formulation as needed in the proof of

128

Approximation Problems

Theorem 5.5.1. We note that, under some appropriate extra-assumption, we can take c ~ - 1 in (5.3.2). Namely, we have T h e o r e m (Chernoff) Let A " D(A) C X --+ X be the infinitesimal generator of a Co-semigroup of contractions and let B " D(B) C_ X --+ X be dissipative and such that D(A) C D(B) and

IIBxil __ IBAxLI+ Zlixil for all x C D(A), where ~ >_ 0 is constant. If the adjoint of B, B* is densely defined then the closure of A + B, A + B is the infinitesimal generator of a Co-semigroup of contractions. See Pazy [101], Theorem 3.4, p. 83. The proof of Theorem 5.4.1 is due to Goldstein [60], and the rigorous approach to the Feynman formula [56] by means of the Lie-Trotter formula follows aoldstein [61] and Pazy [101]. For the physical background of this formula see Feynman [56] and also Simon [114]. The proof of Theorem 5.6.2 is due to Goldstein, Radin and Showalter [62]. As we already mentioned, within this chapter, we confined ourselves to the case of C0-semigroups of contractions although almost all the results can be appropriately restated in order to handle general Cosemigroups, i.e. of type (M, w). As concerns the results in Section 5.1, we notice that the only fact we have to add to get the general case is that all the semigroups {Sn(t); t >_ 0} there considered have to satisfy the stability condition iiS(t)[i~(x) ~_ Me ~t for each n E N*, condition which is automatically satisfied whenever the semigroups are of contractions. As concerns the problems included, they are from Engel and Nagel et al [51], Richtmyer [110] and Pazy [101].

CHAPTER 6

Some Special Classes of C0-Semigroups

In this chapter we present several remarkable classes of C0-semigroups which proved useful in the study of parabolic partial differential equations. We mean here equicontinuous, compact and differentiable C0-semigroups. Each one of these semigroups endows the solution operator f ~ u, defined by the nonhomogeneous equation u' = Au + f , u(a) = ~, by means of the variation of constants formula u(t) - S(t - a)~ +

S(t - s) f (s) ds,

with additional properties (as regularity, compactness, etc.), comparable with their finite-dimensional counterparts. All these properties will be studied in Chapters 8 and 9, and some of them will be exploited in Chapters 10, 11 and 12 in order to obtain existence results for semilinear, or even fully nonlinear differential equations in abstract Banach spaces.

6.1. Equicontinuous Semigroups The aim of this section is to study a class of C0-semigroups which, for t > 0, behave exactly as uniformly continuous semigroups. As in the preceding chapter, for the sake of simplicity, here, we confine ourselves only to the case of C0-semigroups of contractions although all the results remain valid in the general framework of C0-semigoups of type (M, w). First, we introduce:

Definition 6.1.1. A C0-semigroup {S(t) ; t _> 0} is e q u i c o n t i n u o u s if the function t ~+ S ( t ) is continuous from (0, +oc) to L ( X ) uniform operator norm II 9 II (x).

endowed with the

R e m a r k 6.1.1. Each uniformly continuous semigroup is equicontinuous, but the converse statement is not true, as we can state, from the examples analyzed in Problems 6.1 and 6.2. The next theorem provides a characterization of the infinitesimal generators of equicontinuous C0-semigroups. 129

130

Some Special Classes of Co-Semigroups

T h e o r e m 6.1.1. Let A " D ( A ) C_ X ~ X be the infinitesimal generator of a Co-semigroup of contractions, {S(t) ; t >_ 0}. Then {S(t) ; t >__ 0} is equicontinuous if and only if, for each a E (0, 1), we have lim

(

I--A

n--+ cx:)

)n

n

- S(t)

in the usual sup-norm topology of C([a, 1 / a ] ; L ( X ) ) .

The proof of Theorem 6.1.1 rests on the following convergence result. L e m m a 6.1.1. For each a e (0, 1) and each b e (1, +c~) we have n n + l ~0 a

lim

n--+oo

ni

(ve-V)ndv - 0

and

lim rt n + l ~b +c~ (ve-V)ndv - O.

n---~oo

ft!

P r o o f . Since t ~ te - t is nondecreasing on [0, 1 ], it follows that

fo

a(ve-V)ndv < a(ae-a) n.

On the other hand, ve -v < e -1 for each v > O, v r 1, and accordingly lim vn(ve -v e) n

n--+cc

--

(6.1.1)

0

for each v > O, v r 1. Observing that, from Stirling's formula n! lim =1 n--+cc x / ~ n n + l /2 e-n (see Nikolsky [96], p. 393), it follows nne-n

lim

n--+oo

= O,

ft!

(6.1.2)

we obtain nn+l

0 < n----~ limoo n!

rtne -n a(ve-V)ndv < lira an(ae-ae) n ~ =0, n-----~(~ n!

fo

which proves the first assertion. Next, let us observe that nn+l

fb +

n!

n

(ve-V)

d

--

e

-rib ~ k=O

(nb) k k!

(6.1.3)

Equicontinuous Semigroups

131

for each n C N* and b _> 0. Since b > 1, we have k

n

k! n! 1, and accordingly, the last relation implies

for k - 1, 2 , . . . , n -

nn+l n' j~b+e~(ve-V)ndv 9

< (n + 1)(be-b) n nn ~

Tt!"

Consequently, from (6.1.1) and (6.1.2), it follows

n~oclimnn+l n! ~b"+c~(ve-V)ndv

<- n~cclim(n + 1)(be-b) n-~. nn

--

0,

which achieves the proof.

[:]

L e m m a 6.1.2. Let A " D ( A ) c_ X --+ X be the generator of a Co-semigroup of contractions {S(t) ; t >_ 0}, let A > 0, and let R(A; A) - ( ) ~ I - A) -1. Then the mapping )~ ~ R()~; A) is of class C ~ on (0, +oc). In addition, for each x C X , each t > 0 and each n E N*, we have

(I _

_t A n

)-n-1 x -

S(t)x -

nn+l j~0+c~( v e - V ) n [ S ( t v ) x - S ( t ) x ] n!

dv. (6.1.4)

P r o o f . From the proof of the necessity of Theorem 3.1.1~ we know that

f0

R(s A ) x -

e-~S(s)x

ds.

(6.1.5)

In addition, by Theorems 1.7.2 and 3.1.1, it follows that ~ ~ R(s A) is even analytic from (0, +oc) to L ( X ) . Differentiating n-times both sides in (6.1.5) with respect to ~ and putting s - vt, we obtain (R()~; A))(n)x - ( - 1 ) ~ t ~+1

v n e - X t v S ( t v ) x dv

for ~ > 0 and x E X. On the other hand, (R(A; A)) (n) - (-1)nn!R(~;

A) n+l,

n [_~_ ( n ) 1 n+l x - nn+l ~0+c~( v e - V ) n S ( t v ) x n R 7; A n!

and so, substituting ~ - 7 in the relations above, we get

dv.

Taking b - 0 in (6.1.3), we conclude that

nn+l n!

~o

+CC(ve-V)ndv-

1.

(6.1.6)

for each n C N*. Therefore (6.1.4) holds and this completes the proof.

[::]

132

Some Special Classes of Co-Semigroups

We can now proceed to the proof of Theorem 6.1.1. P r o o f . Necessity. Let c~ C (0,1) and let us fix ~ E (0, c~). Since {S(t) ; t _> 0} is equicontinuous, for each c > 0 there exists (~(e) > 0 such that for each t, s C [/~, 1/~] with I t - s I _< 5(s). On the other hand, for the very same c > 0, there exists a - a(c) and b - b(e) with 0 < a < 1 < b < +oc and such that, for each t e [c~, 1/c~ ] and v e [a, b ], we have tv e [ fl, 1/fl ] and

It-tvl <_ 5(e).

So, [[S(tv) - S ( t ) l l ~ ( x ) ~ c for each t C [c~,l/c~], and v E [a,b]. From both (6.1.4) and the last relation, we deduce --n--1

I--A

-

nn+l f0a(re -v )n dv

<_ 2 n!

s(t)

n

+c

~(x)

nn+l lab (ve -~ )n d r + 2 nn+l ~b+~ (re-V) ~ dv n! n!

for each n C N*. By Lemma 6.1.1, it follows that

(

lim sup n-+oc

I-

/

t A n

- S(t)

< e L(X)

for each e > 0. Consequently, for each a E (0, 1), we have lim

= S(t)

I--A

n--+oo

(6.1.7)

n

in the norm I1" II~(x), uniformly for t E [c~, 1/c~]. In order to conclude the proof, we have merely to show that lim

(

I-

n-+o~

~ A n n t- 1

/ -n-1

= S(t)

(6.1.8)

in the norm II " IlL(x), uniformly for t e [c~, 1/c~]. From (6.1.7), we deduce that, for each sequence (an)n~N of functions from IR~_ to R~_ satisfying lim an ( t ) - t

n-+oo

uniformly on each compact subset in IR~_, we have -n-1

lim ( 1 n---~ oo

an(t) A ?Z

= S(t)

Compact Semigroups

133

in the norm [[" IlL(x), uniformly on each compact in R~_. This simply follows from the fact that, for each a C (0, 1), the set of functions -

; nCN* n

is equicontinuous on [a, l / a ] because it is relatively compact in the space C([a, l / a ] ; L(X)). See (6.1.7) and Theorem A.2.1. Taking an(t) =

nt

n§ we obtain (6.1.8), which proves the necessity. Sufficiency. By Lemma 6.1.2 it follows that t ~ ( I - tA)-n is continuous from (0, +co) to L(X) with respect to the norm [[ " [[~(x)- Since, for each a C (0, 1), we have lim n--+oc

(

I--t A

- S(t)

n

in the usual sup-norm topology of C([a, 1/a];L(X)), it follows that the semigroup is continuous from (0, +oc) to L(X), i.e. equicontinuous. The proof is complete. D R e m a r k 6.1.1. It should be emphasized that the really interesting part of Theorem 6.1.1 is the necessity, which asserts that, if a C0-semigroup is equicontinuous, then, for each a C (0, 1) and each bounded subset B in X,

the family of sequences

( ( I - n tA) -n x )

converges to S(t)x "with the nCN

very same speed" for all t C [a, l/a] and x C B. This property proves useful for the error evaluation in some approximation procedures. We conclude this section by noticing that, using a very similar proof with that of Theorem 6.1.1, we can give a direct proof of Hille's exponential formula. See Theorem 5.2.2. 6.2. C o m p a c t S e m i g r o u p s

The aim of this section is to study a class of C0-semigroups of contractions which are specific to parabolic problems in bounded domains. D e f i n i t i o n 6.2.1. A C0-semigroup {S(t) ; t > 0} is compact if for each t > O, S(t) is a compact operator. R e m a r k 6.2.1. If a C0-semigroup {S(t) ; t > 0} satisfies the condition that S(t) is compact for each t _> 0, then I is necessarily compact, and thus X is finite-dimensional. So, if X is infinite-dimensional and {S(t) ; t >_ 0} can be extended to a C0-group on X, then this semigroup cannot be compact.

134

Some Special Classes o[ Co-Semigroups

Indeed, if we assume by contradiction that the semigroup is compact, then, for t > 0, we have I = S ( t ) S ( - t ) and therefore I is compact, thereby contradicting the hypothesis that X is infinite-dimensional. We have the following characterization of the infinitesimal generators of compact C0-semigroups of contractions. T h e o r e m 6.2.1. (Pazy) Let A: D(A) C_ X --+ X be the generator of a Cosemigroup of contractions, {S(t) ; t > 0}. Then {S(t) ; t _> 0} is compact

if and only if (i) (S(t); t > 0} is equicontinuous, and (ii) for each )~ > 0 the operator ( A I - A) -1 is compact. P r o o f . Necessity. Let {S(t); t _> 0} be a compact C0-semigroup of contractions, let t > 0, and let A > 0 be such that t - )~ > 0. Then, for each s > 0, there exists a finite family {Xl,X2,... xkr in B(0, 1) such that, for each x C B(0, 1), there exists i E { 1 , 2 , . . . k(e)} with

IIs(t- ~)x- s(t-

~)~11 _< ~.

Since the family {S(-)xi; i = 1 , 2 , . . . k(e)} of continuous functions from [0, +co) to X is finite, it is equicontinuous at t. Therefore, for the very same e > 0, there exists 5(e) E (0,)~), such that

IIs(t + h)x~ - s(t)x~ll <_ for each i = 1, 2 , . . . k(e), and each h C I~ with Ihl _< 5(c). We have

IIS(t + h ) x - S(t)xll < IIS(t + h ) x - S(t + h)xill + IIS(t + h)xi - S(t)xill +llS(t)xi - S(t)xll = IIS(A + h ) ( S ( t - A ) x - S ( t - A)xi)ll +llS(t + h)xi - S(t)xil] + IIS(A)(S(t- A)xi - S ( t - /k)x)l I <_ IIS(A + h)lln(x)llS(t- )~)x- S ( t - A)xill + IIS(t + h)xi - S(t)xill

+lls(~)ll~(x)lls(t- ~)x~ - s ( t -

~)xll <__3~

for each x C B(0, 1), and each h C I~ with Ihl _
IIs(t +

h) -

s(t)ll~(x) <_ 3e

for e~ch h e R with Ihl <_ ~(~), which proves (i). To check (ii), let us recall that, for each )~ > 0 and x E X, we have

(~I - A ) - l x - R(X; A)x -

/o

~-~S(~)x e~.

Let e > 0 and let us define R~()~; A) : X ~ X by R~(A; A)x - ~ ~ e-sXS(s)xds

Compact Semigroups

135

for each x C X. We shall show next that Re()~; A) is a compact operator and that, in addition, lim IIR(~; A) - R~()~; A)II~(x) - 0.

(6.2.1)

,k--+cx)

Indeed, let us observe that R,(A; A)x - S(c) [--o 0 e-S~S(s - e)x ds Je --e

-~s(~) Ji ~

e-~XS(T)x &- -- e-X~S(e)R(A; A ) x

for each x E X. Since R()~; A) is linear continuous and e-~sS(e) is compact, it follows that R~()~; A) is compact. On the other hand,

/0

IIR()~; A) - R~(A; A)[lc(x) <

eatlls(t)llc(x)d t < 1 - e -~e

for each e > 0, which proves (6.2.1). From Corollary A.1.2 it follows that R()~; A) is compact and this completes the proof of the necessity. Sufficiency. Let {S(t) ; t _> 0} be a C0-semigroup of contractions satisfying (i) and (ii). For t > 0 and A > 0, we define S~(t): X --+ X by

S~(t)x = AR(A; A ) S ( t ) x for each x E X. We shall prove that S~(t), which obviously is compact, satisfies lim IIS:,(t) - s ( t ) I I ~ ( x ) - o. (6.2.2) ,k-+cxz

Indeed o~

II&(t) - s(t)ll~<x>

<_ ~

/0

-

~

L

e-~r

~-~(s(t

Jr ~-) - s ( t ) ) d~~(x)

+ T) -- S ( t ) l l ~ ( x ) &-.

(6.2.3)

Since the semigroup {S(t) ; t >_ O} is equicontinuous, for each e > 0 there exists 6 - 5(e) > 0 such that, for each ~- C (0, 5), we have IIS(t + ~) - S(t)ll~(x)

<_ c.

From (6.2.3) and this inequality, it follows

liSa(t) - S(t)ll~(x)

+),

<_,~

e-~lls(t

+ ~) - s(t)l[~(x) dT

e-~(lls(t + ~)ll~(x) + IIs(t)ll~(x))& _< (1 - e - ~ ) c +

2e - ~ .

136

Some Special Classes of Co-Semigroups

Passing to the sup-limit for A tending to +c~ in this inequality, we obtain lim sup IIS~ (t) - S(t)IlL(x) <- c A--+c~ for each c > 0. But this relation implies (6.2.2), and the latter, along with Corollary A.1.2, ensures the compactness of the semigroup {S(t) ; t _> 0}. The proof is complete. [3 The next simple theorem gives a very precise limitation of the class of Banach spaces on which may exist compact C0-semigroups. T h e o r e m 6.2.2. If A" D(A) C X ~ X is the infinitesimal generator of a Co-semigroup of contractions for which there exists A > 0 such that R(A; A) is compact, then X is separable. In particular, if the semigroup generated by A is compact, then X is separable. P r o o f . First, let us observe that, by virtue of the resolvent equation (1.7.3), we deduce that, for each A > 0, R(A; A) is compact. Let ()~n)nEN* be a sequence of numbers, strictly decreasing to 0. Let n C N* be arbitrary. Inasmuch as R(An; A) is compact, there exists a finite family Dn in B(0, n) such that, for each x C B(0, n), there exists Xn E Dn with

II/~nR(/~n; A)x -/~nR(/~n; A)xnl] ~_ /~n.

(6.2.4)

Let x C X and c > 0. In view of (3.2.1), there exists n E N* such that Ilxll <_ n II)~nR()~n; A)x - xll <_ ~. Taking Xn C D~ satisfying (6.2.4), we deduce IIX -- /~nR(~n; d)xnll ~_ IIx - )~nR(/~n; A)xll

+IIAnR(An; A)x -/~nR(/~n; d)xnl[ ~_ 2c. This inequality shows that the set

D - Un/~nR()~n; A)Dn is dense in X. Since this set is a countable union of finite sets, it is countable too, and therefore X is separable. Finally, if the semigroup generated by A is compact, thanks to Pazy's Theorem 6.2.1, there exists A > 0 such that R(A; A) is compact, and this completes the proof. KI We conclude this section with another characterization of the infinitesimal generators of compact C0-semigroups.

Differentiable Semigroups

137

T h e o r e m 6.2.3. Let A : D(A) C X --+ X be the infinitesimal generator of a Co-semigroup of contractions {S(t); t > 0}. Then {S(t); t > 0} is compact if and only if (i) for each a e (0, 1) we have

lim n--+o~

I--t A

- S(t)

n

(ii) for each ~ > 0 the operator ( , ~ I - A) -1 is compact. P r o o f . The conclusion follows from Theorems 5.1.1 and 6.2.1.

[-7

6.3. Differentiable Semigroups

Here, we shall study a remarkable class of C0-semigroups of contractions, (S(t) ; t _> 0}, having the property that S(t)~ C D(A) for each ~ C X and each t > 0. D e f i n i t i o n 6.3.1. i C0-semigroup (S(t) ; t _> 0} is called"

(i) differentiable at ~" _> 0, if, for each x C X, the function t ~-~ S(t)x is differentiable at r ; (ii) differentiable, if it is differentiable at each r C (0, +oc). (iii) eventually differentiable, if there exists 0 > 0 such that t ~+ S(t)x is differentiable at each ~- C (0~ +oc). R e m a r k 6.3.1. From the definition of the infinitesimal generator, it follows that, whenever (S(t) ; t >_ 0} is differentiable at ~- _> 0, we necessarily have S(T)X C_ D(A). So, if {S(t) ; t _> 0} is differentiable at r - 0, we have D(A) - X and, inasmuch as A is closed, thanks to the closed graph theorem (see Dunford and Schwartz [49], Theorem 4, p. 57), we easily deduce that A C L ( X ) . Hence, in this case, {S(t); t _> 0} is uniformly continuous. Also, if {S(t) ; t _> 0} is differentiable at ~-, then it is differentiable at each t>~-. R e m a r k 6.3.2. If {S(t) ; t _> 0} is eventually differentiable, an inductive argument based on (iii) in Theorem 2.3.2 shows that, for each x G X, the function t ~ S(t)x is of class C ~ on (0, +oc), where 0 is given by (iii) in Definition 6.3.1. D e f i n i t i o n 6.3.2. A C0-semigroup {S(t) ; t _> 0} is called-

(i) uniformly differentiable at r _> 0, if the function t ~+ S(t), from (0, +oc) to L ( X ) , is differentiable at r ; (ii) uniformly differentiable if it is uniformly differentiable at each point r G (0, +oc).

Some Special Classes oS Co-Semigroups

138

(iii) eventually uniformly differentiable, if there exists 0 > 0 such that it is uniformly differentiable at each ~- E (0, +co). R e m a r k 6.3.3. Each C0-semigroup which is uniformly differentiable at ~- is differentiable at ~-. Moreover, if a C0-semigroup is uniformly differentiable at T, then it is uniformly differentiable at each t _> ~-. We will prove latter that each differentiable C0-semigroup is uniformly differentiable. R e m a r k 6.3.4. If {S(t) ; t > 0} is uniformly differentiable, the function t ~-+ S(t) is of class C cc from (0, +c~) to L ( X ) . In particular, each uniformly differentiable C0-semigroup is equicontinuous. L e m m a 6.3.1. Let {S(t) ; t > 0} be a Co-semigroup differentiable at each t > O, A ' D ( A ) C X -+ X be its infinitesimal generator, and let n C I~*. Then 9

(i) for t > nO, S ( t ) X C D ( A n) and s(n)(t) - A n s ( t ) is a linear bounded operator; (ii) the mapping t ~ s(n-1)(t) is continuous from (n0, § to L ( X ) in the uniform operator topology. P r o o f . Let n - 1. Since, for each x C X , t ~ S(t)x is differentiable at each t > 0, we have S(t)x e D(A) and S'(t)x - A S ( t ) x for each x E X and t > 0. Inasmuch as A is closed and S(t) is bounded it follows that AS(t) is closed and thus, by the closed graph theorem (see Dunford and Schwartz [49], Theorem 4, p. 57), we conclude that it is bounded, and this completes the proof of (i) for n - 1. To prove (ii), first let us observe that there exists M >_ 1 such that IIS(t)l[~(x) <_ M for each t E [0, 1 ]. Then, for each 0 < t _< s < 0 + 1, we have

S(s)x - S(t)x -

AS(~-)x dr -

S(~- - t ) A S ( t ) x dr,

and thus

s(t)II (x)

- tIM[IAS(t) II~(x)Ilxll. So, t ~ S(t) is continuous from (0, +c~) to L ( X ) in the uniform operator topology, and this proves (ii) for n - 1. Next, we proceed by induction on n. Assume that both (i) and (ii) hold for n and let t > (n + 1)0. Choose s > nO such that t - s > 0. Then, for each x C X, we have IIs(

) -

S (n) ( t ) x -

<__

S(t - 8)AnS(s)x.

Clearly the right-hand side is differentiable at t, and thus t ~ S(t)x is (n + 1)-times differentiable, and s(n+l)(t)x - An+Is(t)x. From now on, the proof follows exactly the same lines as in the case n - 1. [:]

Diff erentiable Semigroups

139

C o r o l l a r y 6.3.1. Let {S(t); t >_ 0} be a Co-semigroup. If there exists 0 >_ 0 such that t ~-~ S ( t ) i s differentiable at every t > O, then, for each n E N*, t ~ S(t) is n-times uniformly differentiable from ((n + 1)0, +c~) to L ( X ) in the uniform operator topology. In particular, if { S ( t ) ; t >_ O} is differentiable, then t ~+ S(t) is infinitely many times differentiable from (0, +co) to L ( X ) in the uniform operator topology. Two characterizations of the class of all linear operators which generate differentiable and thus, uniformly differentiable, C0-semigroups are listed below. T h e o r e m 6.3.1. Let A : D(A) C_ X --+ X be the infinitesimal generator of a Co-semigroup of contractions {S(t); t >_ 0}. Then {S(t); t _> O} is differentiable (and thus uniformly differentiable) if and only if, for each x C X and each a E (0, 1), there exists lim A

(

I - t A

n--+c~

)n

x,

rt

uniformly for t C [ ~, 1/~ ]. T h e o r e m 6.3.2. Let A" D(A) C X --+ X be the infinitesimal generator of a Co-semigroup of contractions {S(t); t > 0}. Then {S(t); t _> 0} is differentiable (and thus uniformly differentiable) if and only if, for each (~ C (0, 1), there exists

()n

lim A I n--+oc

-tA n

uniformly for t C [c~, 1/c~]in the norm topology of L ( X ) ) . Proof. Sufficiency. Let x C X and c~ C (0, 1). From Hille's exponential formula (see Theorem 15.2.2), we have lim

I-

A

x-

S(t)x

n--+c~

uniformly for t C [a, 1/a ]. As A is a closed operator, from this remark, and from hypothesis, it follows that lim A

I-

A

x-

AS(t)x

u-+c<)

uniformly for t E [c~, 1/a ]. But this means that S(t)x c D(A) for each x C X and each t > 0, which completes the proof of the sufficiency.

Some Special Classes of Co-Semigroups

140

Necessity. Using once again the closedness of A, Hille's Theorem 1.2.2 and L e m m a 6.1.2, we get

--n--1 x - AS(t)x

nn+l -

n!

-

f.L+cr

(ve-V

)n[AS(tv)x

-

AS(t)x] dv

(6.3.1)

for each x C X and each t > 0. Let c~ C (0~1), and fix fl C (0, c~). As {S(t) ; t >_ 0} is differentiable, it follows that, for each x C X, the mapping t ~ AS(t)x is continuous on (0, +co). Then, for each c > 0 there exists 5(c) > 0 such that IIAS(t)~ - AS(~)xll _<

for each t,s E [/~, 1//~] with I t - s I _< 5(c). Moreover, for the very same c > 0, there exist a = a(c) and b = b(c) with 0 < a < 1 < b < +c~ such that, for each t C[c~, 1/c~ ] and v C [ a , b], we have

tv e [/~, 1//~] and I t - t v l

<_

5(c).

As a consequence (6.3.2)

-

for each t e[c~, 1/c~ ] and v E [ a , b]. From (6.3.1), we deduce

5

A (I - t A) -n-l x -

(6.3.3)

for each n E N* and each t E [c~, 1/c~ ], where

nn+l ~0a(ve-V)n j~(t)

-

ld ~ ( S ( t v ) ) x dv,

nn+l ~0a (ve-V)nAS(t)xdv, n!

nn+l ~ab (ve-V)~(AS(tv)x - AS(t)x) dv,

n n+l ~b+cx~(ve_v)nl d (S(tv)) x dv, n

J5 (t) -

nn+l ~b+c~(ve-V)nAS(t)xdv, n!

141

Differentiable Semigroups

for each n E N* and each t E [ a , 1 / a ]. Next, we shall evaluate each of the five terms on the right hand side of (6.3.3). To this aim, let us observe that, for each p > 0 and each ~- E [p, +oc), we have

liAS(T)xll ~ IIAS(p)xI[.

(6.3.4)

Indeed, since the semigroup is differentiable, the mapping ~- ~+ A S ( T ) x is a Cl-solution of the equation u ~ - A u on the interval (0, +oc) and then, since the semigroup generated by A is of contractions, we deduce (6.3.4). We begin to evaluate :J~(t). Integrating by parts, observing that, for each v E (0, 1), le - v - v e - V l _< 1 and taking into account that the mapping v ~ v e - ~ is nondecreasing on (0, 1), we deduce

IIJ?(t)ll

nn+ln! ~0a ( v e _ V ) n -1i ~d( s ( t ~ ) )

-

x dv

rtn+ 1 n!

nn+l /i a n!

<_

n(ve

(ae-a)nS(ta)x--

-v)~-~(~-v _ ~ - v ) s ( t ~ ) x d~

Ilxll [n n+l(ae_a)n

L

~!

+

rtn+l

------------~'(ae-a)n-1 ( ~ - 1)

]

for each n E N* and t E [a, 1 / a ]. We easily see that

nn+ 1 rt!

nn+l (~

-

nn (ae-a) n-

1)---------~(ae-a)n-l = .

(n--l) n-1 (n-

and

---~.e-n . n ( a e l - a ) n

1)!

e-

(n-l)n2 // n ~ n - 1 9

\ n-

1

]

In view of Stirling's formula (see Nikolsky [96], p. 393), we have n n

lim -~. e - n - 0 . ~_~ In addition, since a E (0, 1), we have ae 1-a < 1. Therefore, we deduce both

nn+l n---~c~ n! lim

( a e - a ) n - 0 and

nn+l n-+cx~(n -- l)! lim - - ( a e - a )

So, we conclude that lim IIJ~ (t)II - - 0 n---+oo

uniformly with respect to t E [a, l / a ] .

n - 1 - O.

(6.3.5)

142

Some Special Classes of Co-Semigroups

Regarding 3; (t), from (6.3.4), we have

for each n E N*, and t E [a,l / a ] . From this inequality and Lemma 6.1.1, we get lim 113~(t)Il= 0, n+cc

uniformly for t E [ a ,l / a ] . In order to evaluate 3; (t), let us observe that, from (6.3.2) and Lemma 6.1.2, we have

for each n E N* and each t E [ a, l/a 1. Consequently lim sup 113; (t)11 5

E.

n+cc

In order to evaluate 5'z(t), let us observe that, in view of (6.3.4), we have for each t E [ a , ] and each v E (b, +GO). As a consequence

for each n E N* and each t E [ a ,l l a ] . From this relation, (6.3.5) and from Lemma 6.1.1, we get lim 113y(t)ll = 0. n+cc

Finally, from (6.3.4), we have

for each n E N*, and each t E [ a, l/a ] and accordingly lim Il3:(t)ll = 0.

n+cc

Differentiable Semigroups Therefore limsup

(

A

I-

n--+oo

-t A

x- AS(t)x

143

<_ e

n

for each e > 0. Consequently, for each c~ C (0, 1), we have

(

lim A

I _ _t A

n--+oo

x-

AS(t)x

(6.3.6)

n

uniformly for t C [o~, 1/o~ ]. To complete the proof, we have merely to show that

(

lim A I - ~ t A x- AS(t)x (6.3.7) n~ n+ 1 uniformly for t C [a, 1/c~]. To this aim, let us observe that, from (6.3.6) and the Arzels Theorem A.2.1, it follows that, for each sequence (an)nEH of functions from I~_ in IR~_ satisfying lim an (t) -- t, n---+(x)

(

)n

uniformly on every compact subset in ]~_, we have limA

I

an(t) A

n--+ (x~

x - AS(t)x

n

uniformly on every compact subset in ]R~_. Indeed, this follows from the remark that, for each x E X and c~ C (0, 1), the family of functions

{ ( )--n--1 } t~A

I--tA

x; h E N *

n

is relatively compact in C([ c~, 1/c~ ]; X), and thus afortiori equicontinuous on [a, 1/c~]. See (6.3.6) and Arzels Theorem A.2.1. To complete the proof, we observe that the choice an(t) - n nt for n C H* leads to +l~ (6.3.7). [7 Since the proof of Theorem 6.3.2 follows exactly the same lines as that of Theorem 6.3.1, we do not give details. A useful sufficient condition in order that an operator A generate a compact C0-semigroup of contractions is: C o r o l l a r y 6.3.2. Let A : D(A) C X -~ X be the infinitesimal generator of a differentiable, i.e. uniformly differentiable Co-semigroup of contractions {S(t) ; t _> 0}. If for each )~ > O, ( A I - A ) -1 is compact, then the semigroup generated by A is compact. Equivalently, if D(A) endowed with the graph norm is compactly imbedded in X , then the semigroup generated by A is compact.

144

Some Special Classes of Co-Semigroups

P r o o f . The first part of the conclusion follows from Theorem 6.2.2 and Remark 6.3.4. The second one is a simple consequence of the fact that, for each A > O, u ~+ IIAu- Au]l , defined on D(A), is a norm on D(A), which is equivalent with the graph-norm. D

6.4. Semigroups with Symmetric Generators In this section we shall prove that all C0-semigroups of contractions which are generated by symmetric operators in Hilbert spaces are differentiable. So, let H be a Hilbert space. The next two lemmas will prove useful later.

L e m m a 6.4.1. Let A " D ( A ) C_ H ~ H be the infinitesimal generator of a Co-semigroup of contractions. Let (Xn)nEN be a sequence in D ( A ) with the property that l i m n ~ Xn -- x, and ( A X n ) n E N i8 bounded. Then x E D ( A ) and limn-~cr A X n - A x in the weak topology of H. P r o o f . Let us observe that, by virtue of Theorem 2.4.1, A is closed. Since graph A is a closed linear subspace in H x H, it is sequentially strongly x weakly closed in H x H. Therefore, if (AXnk)kEN is a weakly convergent subsequence of (Axn)~EN, then x E D(A), and the weak limit of (AXnk)kEN is y - Ax. Since (Axn)nEN is bounded, and thus sequentially weakly compact, it follows that the set of all its weak sequential limit points is nonempty and, by the argument above, contains only one element y. The proof is complete. D

L e m m a 6.4.2. Let A " D ( A ) C_ H --+ H be the infinitesimal generator of a Co-semigroup of contractions. Then, for each x E D(A), we have (Ax, x) <_ O. P r o o f . This is an easy consequence of Lumer-Phillips Theorem 3.4.2 combined with Remark 3.4.1. V] A remarkable class of uniformly differentiable semigroups is given below.

Theorem 6.4.1. Let A " D ( A ) C_ H --+ H be the infinitesimal generator of a Co-semigroup of contractions, {S(t) ; t _> 0}, let ~ E H, and u(t) - S(t)~ for t >_ O. If A is self-adjoint, then (i) u e C ( [ 0 , + o c ) ; H ) n C ( ( O , + o c ) ; D ( A ) ) n cl((0,+cxD);H), and u is the unique solution of the Cauchy problem u' - Au -

in this space; 1 (ii) IIAu(t)ll < 7---~11~]1 for each t > 0; t V "z

Semigroups with Symmetric Generators

145

(iii) the function t ~ x/~llAu(t)]l belongs to L2(0, +oc) and

fo ~ sllAu(s)ll 2 ds < -4Jill 1 12 ; 1

(iv) /f ~ e D(A), then IIAu(t)]] 2 <_ ~ ( - A ( , ~ ) (v) t ~

for each t > O, and

Au(t) C L2(O, +co), and foot IIAu(s)ll 2 ds << -~(-A~,~). 1

P r o o f . Let ~ C D(A2). By virtue of Corollary 2.3.1, it follows that the problem

{ u ~ - Au ~(o) - {

has a unique solution u of class C 2 on IR+ which satisfies u" (t) - At' (t) for each t C IR+. Taking the inner product of both sides in the above equality by ut(t), integrating from s to t, with 0 _< s _< t, and using Lemma 6.4.2, we deduce -21 ilu,(t)ll2 _

1 ~11~'(~)11 2-

fs t (Au'(T), u' (T))dT <_ O,

which shows that function t ~+ I]u'(t)]] is nonincreasing. We also have d

d-vllu(t)ll 2 - 2(At(t), u(t)),

(6.4.1)

and d

dt (At(t), u(t)) - 2(At(t), u'(t)) - 211u'(t)ll 2 _> 0.

(6.4.2)

Accordingly, t ~-+ (At(t), u(t)) is nondecreasing. Integrating (6.4.1) from 0 to t, we successively obtain

Ilu(t)l[ 2 -I1~112 -

2

/o

(At(s), u(s)> ds <_ 2t(Au(t), u(t)>

and

-t(Au(t), u(t)) _< -

f0 t (At(s),

u(s)) ds

1 1 12 1 211u(t)l12 + ~11~1 ~ ~11~112-

Integrating (6.4.2) from 0 to t and recalling that the function t ~ Ilu'(t)ll is nonincreasing, we obtain {Au(t),u(t)}-

(A~,~) - 2

f0 ~ Ilu'(s)l I~ ds

_ 2tll ~, (t)l 129

146

Some Special Classes of Co-Semigroups

Since (Au(t), u(t)) <_ 0, it follows that

2tl[u'(t) II2 < (-A~, ~).

(6.4.3)

Multiplying both sides in (6.4.2) by t, and integrating, we get

2?]lu'(t)ll 2 _<

/0

s(Au(s), u'(s)) ds -

= t(Au(t), u(t)) -

/0

/o t (Au(s),

s ~ ( A u ( s ) , u(s)) ds u(s))ds.

Since t(Au(t), u(t)) <_ O, from (6.4.1) and the last inequality, we deduce

2t2llu'(t)ll 2 _< I1~112. Accordingly,

1 for each ~ e D(A2), and t > 0. Since D ( A 2) is dense in H and the mapping ~ ~-+ u(t) - S(t)~ is nonexpansive, by the inequality above and Lemma 6.4.1, we obtain (ii). Since (i) and (iii) follow from (ii), while (iv) and (v) are consequences of (6.4.3), this achieves the proof. [--1 R e m a r k 6.4.1. All the items in conclusion of Theorem 6.4.1 hold also if A is symmetric and generates a C0-semigroup of contractions on H. Indeed, this follows from the simple remark that, in this case, A is self-adjoint. See Theorem 3.1.1 and Lemma 1.6.1. C o r o l l a r y 6.4.1. Let A" D ( A ) C_ H --+ H be a self-adjoint and negatively defined operator, let {S(t); t _> 0} be the Co-semigroup of contractions generated by A, let ~ C H, and u(t) - S(t)~ for t >__ O. Let (Xn)neN be the sequence of spaces defined as in Remark 3.5.1. Then, for each n C N*, U E CC~((O,-+-oo); X n ) , and

I[A~u(t)]l <

()n ?'t

~-~

llxll

for each t > O.

P r o o f . Let X1 be the Banach space in Theorem 3.5.1, endowed with the graph-norm l" [D(A). Then X1 is a Hilbert space with respect to the inner product (., ")1 defined by (x, y)x - (x - Ax, y - Ay}

for each x, y E X1. In addition. A(1) is self-adjoint on X1. Indeed, we have (A(1)x, y)-~ -- (Ax - A2x, y - Ay)

The Linear Delay Equation

147

= (Ax, y} - (Ax, Ay} - (A2x, y}+}A2x, Ay} = (x, Ay) - (x, A2y} - (Ax, Ay} + (Ax, A2y} - (x, A(1)y}I for each x, y 6 D(A(1)). Thus A(1) is symmetric. Since it generates a C0-semigroup of contractions on X1, it follows that ( I - A(~)) -~ E L(X1). See Theorem 3.1.1. By Lemma 1.6.1, it follows that A(1) is self-adjoint. A simple inductive argument shows that, for n 6 H*, Xn is a Hilbert space, and An is self-adjoint. Let t > 0 and n C N*. From (ii) in Theorem 6.4.1, we deduce that n

IIA (t/ ) II _

Ilxll.

Inasmuch as u ( 2 t / n ) = S ( t / n ) u ( t / n ) x 6 X2, applying once again ( i i ) i n Theorem 6.4.1, this time to the operator A(1), we obtain

I]A2u(2t/n)ll _<

n

[Iz]l.

By induction, we conclude that ]]Anu(t)ll < -

n

-~

Ilxll 9

To complete the proof, we have to show that, for each n 6 H*, u belongs to Ccr +oc); Xn). But this follows from the simple remark that, for each x C H and t > 0, we have u(t) 6 Nn>0Xn, combined with Corollary 2.4.1. D

6.5. The Linear Delay Equation A remarkable example of a differentiable C0-semigroup is given below. E x a m p l e 6.5.1. Let r > 0 and let X - C ( [ - r , 0]; R ~) which, endowed with the sup-norm, is a Banach space. If x 9 [-r, +co) --+ IRn is continuous, then, for each t _> 0, the function xt" [-r, 0] --+ R ~, defined by x (e) - x(t + e),

for each 0 E [-r, 0 ], belongs to X. Let L - X --+ It(n be a linear continuous operator, let ~ 6 X and let us consider the Catchy problem for the linear delay equation x'(t)-Lxt t>_O x(s) - ~(s) s 6 [-r, 0]. (6.5.1) By a solution of this problem we mean a function x 6 C ( [ - r , + ~ ) ; ]Rn) with the property Xl[0,+~ ) 6 C1([ 0, +co); IRn) and satisfying x(s) - ~(s), for each s 6 I-r, 0 ], and x'(t) - Lxt, for each t _> 0.

Some Special Classes of Co-Semigroups

148

P r o p o s i t i o n 6.5.1. For each ~ C X, the problem (6.5.1) has a unique solution x : I-r, +co) -+ I~n. P r o o f . Let us observe that this problem is equivalent with the delay integral equation x(t)

-

qp(t) f0 ~(0) + Lxs ds

f~ for t E (0, +co).

(6.5.2)

Therefore, to complete the proof, it suffices to show that, for each ~ C X and each T > 0, the equation (6.5.2) has a unique solution x : [-r, T] --+ R n. In order to show this, let Y = C ( [ - r , T ]; I~n) which, endowed with the supnorm I1" IIY, is a Banach space and let Q : Y ~ Y be defined by / ~(t) f0 t (Qy)(t) ~(0) + Lys ds

for t C [-r,O l for t e (0, T ].

Let us observe that x C Y is a solution of the delay integral equation (6.5.2) if and only if x is a fixed point of Q. So, in that follows, we shall prove that Q has a unique fixed point. A simple inductive argument shows that, for each k E N*, and each t E [-r, T ], we have

]l(Qky)(t) - (Qkz)(t)[[ < ILL[[ktk _

-

yllv.

Clearly, this inequality implies

IIQky - Qkzll Y < IILIIkTk _

k---y-IIx-yllY.

Therefore, for sufficiently large k, Qk is a strict contraction, which, by the Banach fixed point theorem, has a unique fixed point x C Y. Inasmuch as

I I Q x - x l l - IIQkQx- Qkx]] <_ qllQx - xll with q E (0, 1), it follows that x is a fixed point of Q. In addition, this is the unique because each one of its fixed points is necessarily a fixed point of Qk. The proof is complete. U] Let t _> 0, and let S(t)" X --+ X be defined by S ( t ) ~ - xt, where x is the unique solution of the problem (6.5.1) corresponding to the "initial datum" ~. By Proposition 6.5.1, from the linearity and continuity of the operator L, it follows that, for each t >__0, S(t) is linear continuous from X to X. More than this, we have"

149

Problems

T h e o r e m 6.5.1. The family {S(t); t >_ 0}, defined as above, is a semigroup of class Co in X whose generator, A " D ( A ) C X --+ X , is given by D ( A ) - {~ e c l ( [ - r , 0]; I~n); ~ t ( 0 ) - Lop}, and (Aw)(O) -

{ ~'(0) L~

for 0 E [-r, 0) for 0 - 0.

In addition, .for each t >_ r, S(t) is a compact operator and the semigroup is differentiable o n [ r , +oc).

P r o o f . Let us observe that, for each t _> 0, and each 0 E [-r, 0 ], we have t+o

for t + 0 _ < 0

L ( S ( T ) ~ ) d7 for t + 0 > 0. Jo From this representation formula, after a simple calculation, we deduce the expression of A. From this, and Theorem A.2.1, it follows that S(t) is compact for each t > r and the semigroup is differentiable on [r, + ~ ) . [2] [S(t)~](O) -

~a(O)+

Problems P r o b l e m 6.1. Let p C [1, + ~ ) , X = lp, (an)nON* a sequence of positive real numbers, and let {S(t) ; t > 0} the semigroup in Problem 2.1. Prove that this semigroup is equicontinuous. Using this, and Problem 2.1 show that, if a n - - rt 2 for each n C H*, then {S(t) ; t > 0}, although equicontinuous, is not uniformly continuous. P r o b l e m 6.2. Let X - co, (an)hEN* a sequence of real positive numbers, and {S(t) ; t > 0} the semigroup in Problem 2.2. Prove that this semigroup is equicontinuous. Using this, and Problem 2.2, show that, if a n - - n 2 for each n C H*, then {S(t) ; t > 0}, although equicontinuous, is not uniformly continuous. P r o b l e m 6.3. Let p E [1, + ~ ) , X - lp, ( a n ) n C N * & sequence of positive real numbers and let {S(t) ; t > 0} the semigroup in Problem 2.1. Prove that this semigroup is compact if and only if l i m n ~ a n - - cx). P r o b l e m 6.4. Let X - c o , (an)nCN* a sequence of real positive numbers and {S(t) ; t _> 0} the semigroup in Problem 2.2. Prove that this semigroup is compact if and only if limn--,oo an - co. P r o b l e m 6.5. Let H be the complexification of the space 12 (we notice that H is considered as a real Hilbert space- see Section 1.6), t _> 0, and let

150

Some Special Classes of Co-Semigroups

S(t) "12 -+ 12 be defined by (S(t)(xn)n~N*)kCN, =

e

_k2t+iek4txk )

kCN*

for

each (xn)n~N* C U. Prove that {S(t) ; t _~ 0} is a compact C0-semigroup of contractions. Find its infinitesimal generator, and show that this semigroup is not differentiable. This is Example 4.4.2, p. 181 in Balakrishnan [12]. Let H be a separable Hilbert space. An operator T C L ( H ) is called a Hilbert-Schmidt operator, if there exists an orthonormal, and complete system {en ; n C N*} in H, such that ~--]~n~__lIITenll 2 < +co. A C0-semigroup of contractions on H, {S(t) ; t _> 0}~ is Hilbert-Schmidt, if, for each t > 0, S(t) is a Hilbert-Schmidt operator. P r o b l e m 6.6. Prove that each Hilbert-Schmidt C0-semigroup is compact. P r o b l e m 6.7. Prove that the semigroup in the Problem 2.1 with p = 2, and an -- n 2 for each n C N*, is Hilbert-Schmidt, and therefore compact. P r o b l e m 6.8. Let X - L2(0, ~) and let {S(t) ; t _> 0} be the semigroup in Problem 4.2. Prove that this semigroup is Hilbert-Schmidt and therefore compact. P r o b l e m 6.9. Show that a C0-group is differentiable if and only if is uniformly continuous. N o t e s . The results in Section 6.1 are due to Vrabie [126]. Theorem 6.2.2 extends a remark in Cs and Vrabie [34]. A similar result was proved by Balakrishnan [12] in the case in which A is self-adjoint. The necessity of Theorem 6.3.2 is due to Lax (see Hille and Phillips [70]~ Theorem 10.2.2, p. 304), while the sufficiency, which is by far the most interesting, to Pazy [98]. Theorem 6.2.3 was established in Vrabie [126]; Theorems 6.3.1 and 6.3.2 were proved in Vrabie [130]. We recall also T h e o r e m (Pazy) The operator A " D ( A ) C_ X --+ X is the infinitesimal generator of an eventually differentiable Co-semigroup of type (M, w) if and only if there exist a E R and b, C C ]R*+ such that {A C C; Re)~ _> a - b l o g l I m A } - E C p(A) IIR(A;A)II~(x ) <_ ClIm)~ I for A E E, ReA <_ w.

and

See Pazy [101], Theorem 4.7, p. 54. Theorem 6.4.1 is from Brezis and Cazenave, while Example 6.5.1 is due to Hale [67]. For other significant examples of functional equations treated by semigroup methods we refer to Corduneanu [40]~ pp. 245-261. For applications to control theory see Ahmed [3] and C~rj~ [33]. The problems are from Balakrishnan [12] and Pazy [101].

CHAPTER 7

Analytic Semigroups In this chapter we introduce a class of differentiable semigroups on a complex Banach space X which can be extended as analytic functions acting from a certain sector in the complex plane including the resolvent set of the generator to L(X). After analyzing several remarkable examples as the heat equation, the Stokes equation, some parabolic and elliptic problems with dynamic boundary conditions, we define and study the fractional powers of some closed operators, with main emphasis on the negative generators of analytic semigroups.

7.1. Definition and Characterizations Let X be a complex Banach space, and let A 9 D(A) C_ X --+ X be a C-linear operator generating a C0-semigroup of contractions {S(t) ; t _> 0}. For 0 < 0 <_ 7r, we define the sector

Co-{zEC;

-O<argz
Clearly, -/9 _< arg z <_ O} U {O}.

Definition 7.1.1. We say that the C0-semigroup {S(t) ; t >_ 0} is analytic, if there exists 0 < 0 _< 7r, and a mapping S ' C o -+ L ( X ) such that" (i) S(t) - S(t) for each t _> 0; (ii) S(z + w) - S ( z ) S ( w ) for each z, w C0; (iii) _lim S ( z ) x - x f o r e a c h x E X ; N

zECe, z-+0

(iv) the mapping z ~ S(z) is analytic from C0 to L ( X ) . If, in addition, for each 0 < ~ < 0, the mapping z ~ S(z) is bounded from C5 to L ( X ) , the C0-semigroup {S(t); t >_ 0} is called analytic and uniformly bounded.

Theorem 7.1.1. Let A " D(A) C_ X ~ X be a C-linear operator which generates a Co-semigroup of contractions {S(t) ; t _> 0}. If 0 E p(A), then, the following conditions are equivalent" 151

152

Analytic Semigroups

(i) the semigroup {S(t) ; t ~_ 0} is analytic and uniformly bounded; (ii) {,X C C; Re ), > 0} C_ p(A), and there exists C > 0 such that, for each ik C C with R e ~ > 0 and I m ~ ~ 0, we have C IIR(~; A)II~(x) _~ iXm ~1 ;

(iii) there exist ~ C (0,-~) and M > 0, such that C~+5 _c p(A) and, for each ~ E C~+5, we have M IIR(A;A)[[~(x) <__ IA[;

(iv) the semigroup {S(t) ; t > 0} is uniformly differentiable for t > 0 and there exists C > 0 so that C

lls'(t) II~r

< T

for each t > O.

P r o o f . We begin by showing that (i) implies (ii). Let ~i > 0 for which there exists C1 > 0 such that N

IIS(z)ll~(x) <_ Cl for each z E C with [arg(z)[ < 5. By virtue of Remark 3.1.2, for each x E X, a > 0, and T C IR, we have R(cr + it; A)x -

/0

e-(~+~-)ts(t)x dt.

7r Let 0 C (0,-~). Since the semigroup is analytic, for 7- > O, we can shift the path of integration from (0, +co) to the ray {pe-~~ 0 < p < +co} oriented from 0 to +co. We obtain

IIR(~ + i~-; A)zll __

/0 e-P(~~

dp

C1 6' = o- cos 0 + ~-sin 0 I1~11 ___~ Ilzll. Analogously, for ~- < O, shifting the path of integration from (0, +oc) to the ray {pei~ 0 < p < +ec}, oriented from 0 to +ec, we obtain [[R(a + i7; A)xll < C~

e-P(aC~176 C

cose- ~sinellxll < ~11x11, which proves (ii).

dp

Definition and Characterizations

153

To prove that (ii) implies (iii), let us observe that, thanks to Remark 3.1.2, for each )~ C C with Re ), > 0, we have 1 ]]R(A; A)]]~(x) -< Re ~" On the other hand, by (ii), we know that, for each ~ E C satisfying Re ~ > 0, and Im ~ ~ 0, we have C

IIR(A;A)II (x) <_ IImAl"

But, from these two inequalities, it follows that there exists M > 0 such that M

IIR( ;A)II (x)_

for each A C C with Re A > 0. Let a > 0 and 7- E R. The Taylor expansion of the resolvent function around a + i~- is oo

R(s A) - ~ - ~ ( - 1 ) ~ R ( a + iT; A)~+I (a + iT - A)n n--0

This series is convergent in IIn(~ §

L(X)

for each A C C satisfying

A)IIn(x)]a + iT - s -< k < 1.

Taking ~ - Re )~§ iT in the above series and using the inequality in (ii), we observe that it is convergent in L(X) for [a - Re/k] _< k]'r[/C. Since both a > 0 and k C (0, 1) are arbitrary, it follows that p(A) includes all complex numbers ~ with Re )~ _< 0 satisfying IRe)~l/iIm)~l < 1/C. In particular, we have C~+~~ c C ; ]arg~] <-~+5 C_p(A), where 5 - k arctan(1/C), k C (0, 1). Moreover, on C~+5, we have

IIR( ; A)II

C

(x)<- l - k -

x/'C 2 + 1 1

< (l-k)

M

= ]s [s

from where it follows that A satisfies (iii). To prove that (iii) implies (iv) first let us remark that, if (iii) holds, for each t > 0, we have a(tA) C_ {A C C; Re)~ < 0}. Then, for a fixed ~ < 0 < ~+(f,~ the Dunford integral of the analytic function ez calculated at tA is well-defined, and

etA = 2~il fr et'tR(#; A) d#,

(7.1.1)

where F is the path consisting of the two rays {pe-ie; 0 < p < + ~ } , and {pei~ 0 < p < + ~ } , oriented in the sense of the increase of the imaginary

154

Analytic Semigroups

part of )~. We emphasize that, from (iii) and from the condition 0 E p(A), it follows that F is entirely contained in the domain of analyticity of the resolvent function. We shall prove next that, for each t _> 0, (7.1.2)

e tA - S(t),

where e tA is defined by (7.1.1). First, let us observe that, from (i) in Theorem 1.8.1, it follows that {etA; t > 0} U {I} is a semigroup. Therefore, to check (7.1.2), it suffices to show that, for each A > 0, we have R(A; A) - ~0 +cxDe -)~te tA dt. Let A > 0. Let us multiply (7.1.1) by e -)~t, and let us integrate from 0 to b. From Fubini theorem and the Residues theorem, we deduce i b e - ~ t e tA dt - 27ri 1 f r p -1 A ( e (t'-~)b - 1)R(p; A) dp

1 fF ~e('-~)b = R(A; A ) + ~-~ #R_( # ; A ) d # .

As lim fF e(~-X) b R(#; A) d# - 0

b--++c~

# - /~

from the above equality, we get (7.1.2). At this point, let us observe that the integral on the right-hand side in (7.1.1) can be differentiated with respect to the parameter t > 0, and we can interchange the integration with the differentiation, because the integral ~1 fr/ke~tR(/~; A) d)~ is convergent for t > 0 in the uniform operator norm of the space L ( X ) . This last assertion follows from the simple observation that, for each t > 0, we have ~i

#etLtR(#; A) d#

< _ - 7r

e_pcosOtdp _

1 1 7r cos 0 t

Differentiating both sides in (7.1.1) with respect to t > 0, using (7.1.2) and 1 the above inequality, we deduce (iv) with C = ~cos0" To prove that (iv) implies (i), let us observe that, for each t > 0 and each n E H*, we have S(n)(t)-Is'

(t)]

n.

(7.1.3)

This follows by observing that, if the semigroup {S(t); t _> 0} is uniformly differentiable, then, for each x C X, each t > 0 and each n C N*, we have

Definition and Characterizations

155

S(t)x E D(An). From Corollary 2.4.1 and the fact that the semigroup commutes with its infinitesimal generator, we have s(n)(t)-AnS(t)-AnS

(t)]n-

(t)n-[AS

[~ t (t)]n ~

which proves (7.1.3). From (7.1.3) and the inequality n n ~_ n!e n, we deduce

1

n~.lls(n)(t)ll~(x ) _~

(_C~) n

,

(7.1.4)

for each t > 0, and each n C N*. Let us consider now the power series (X)

S(z)

S(t) -~-E (z t)n s(n) (t)

(7 1.5)

n!

n=l

which, by virtue of (7.1.4), is uniformly convergent in the norm of L ( X ) for Iz - t I _~ k(~ee), for each k C (0, 1). Obviously, the family of linear operators {S(z) ; z C Co}, with 0 - arctan(~ee), extends {S(t) ; t _> 0} to Co and therefore it satisfies (i) in Definition 1.5.1. By (i) and (ii) in Theorem 1.8.1, it follows that {S(z) ; z c Ce} satisfies (ii) in Definition 7.1.1. By (7.1.5), we conclude that {S(z) ; z c Ce} satisfies (iii) and (iv) in Definition 7.1.1. Since, by virtue of (7.1.4), the uniform boundedness condition is obviously satisfied, this achieves the proof. D N

From Theorem 7.1.1 and Theorem 6.3.2 it follows:

7.1.2. Let A " D ( A ) C_ X -+ X be a C-linear operator generating a Co-semigroup of contractions {S(t); t >__ 0}. Then {S(t); t >_ 0} is analytic if and only if for each ~ C (0, 1) there exists

Theorem

lim A

n--+oc

(

I-tA

)-n

rt

-T(t)

in the usual sup-norm topology of C([(~, 1/c~];L(X)), and there exists C > 0 such that C IIT(t)ll~(x) ~_ -~ for each t > O. If H is a complex Hilbert space, we have: C o r o l l a r y 7.1.1. If A " D(A) C_ H -+ H is self-adjoint and generates a Co-semigroup of contractions {S(t) ; t _> 0}, then {S(t) ; t >_ 0} is analytic.

156

Analytic Semigroups

Proof. First, let us observe that A has a C-linear self-adjoint canonical extension. On the other hand, by (ii) in Theorem 6.4.1, it follows that (iv) in Theprem 7.1.1 holds. The proof is complete. K]

7.2. The Heat Equation

Using the results established in Chapter 6 and in the preceding section, we can obtain additional information concerning the regularity of solutions of many partial differential equations of parabolic nature. We begin with the heat equation. More precisely, let ~ a nonempty and open subset in IRn whose boundary is denoted by F, let Q ~ = IR+ • ~, E ~ = IR+ • F, and let us consider the equation

l ut-/Xu u- o x)

(t, x) c (t, x) c -

(7.2.1)

x c

known as the heat, or diffusion equation. T h e o r e m 7.2.1. For each ~ E H - ~ ( ~ ) , the problem (7.2.1) has a unique continuous solution u E C(R+; H-I(~)) which satisfies:

(i) (ii) (iii) (iv) (v) (vi)

u e C(IR~_;H~(t2))n CI(I~_;H-I(t~)) ; for each m e N, we have Ainu e CC~(IR~_;H i ( u ) ) ; u if if if It

e ~ ~ ~ E

C~(IR~_ x f~); C L2(~t), then u E C(IR+; L2(~t)) ; e Hl(t~), then u e C(IR+; HI(ft)) Cl CI(IR+; H-I(f~)) ; C H I ( ~ ) and A~ C L2(~), then A u E C(IR+; L2(t~)) and c l ( ] ~ + ; L2(t2)).

Proof. The item (i) follows from both Theorem 4.1.1 combined with (i) in Theorem 6.4.1. To check (ii), let us observe that, in view of Theorem 4.1.1 and Corollary 7.1.1, the operator A with the Dirichlet boundary condition is self-adjoint, and generates a C0-semigroup in H - I ( ~ ) . So, we are in the hypotheses of Corollary 6.4.1, from where it follows that, for each m C N, we have u e C~((0, +c~);Xm). Since X m - - D ( A m ) , and u (m) - A m u (see Corollary 2.4.1), it follows that (ii) holds. As Nm>oXm C_ CCC(~), from (ii) we get (iii). The items (iv), (v) and (vi) follow from (vi) in Corollary 3.5.1, observing that the C0-semigroup generated by the Laplace operator with the Dirichlet boundary condition in L2(~) coincides with the L2(~)-restriction of the C0-semigroup generated by the homonymous operator in H -1 (~). The proof is complete. [5

The Heat Equation

157

In the case in which ft is bounded and F sufficiently smooth, the unique solution u of the problem (7.2.1) enjoys additional regularity properties. More precisely we have" T h e o r e m 7.2.2. If ft is bounded and ~ E H~(ft) n H2(Vt), then" (i) i f F is C 2, then u E C([O,+oo);H2(f~)) NCl([o,+oo);Lg(f~)) ; (ii) if F is C 2m and m e N*, then u e C~176 +oc); H2m(f~)) ; (iii) if F is C ~ , then u e C~ +co) x f~) for each s > 0; (iv) if F is C ~ , ~ E C~176 and, in addition, ~ satisfies the usual compatibility conditions ~ - A~ . . . . - Am~ = ... = 0 on F, then u e CC~([0, +oo) x ft). P r o o f . The conclusion follows from Theorem 7.2.1 and Remark 4.1.1. [3 T h e o r e m 7.2.3. Let ~ E L2(ft), and let u be the unique continuous solution of the problem (7.2.1). Then: 1 (i) II~(t)llL~(a) < ~-~ll~llL2(a) for each t > O;

(ii) fo sll~(s) II~(a)ds < ~1 11~112~(a); _

1 (iii) IIVu(t)]]L2(n) < ~][~llL2(n) for each t > 0;

(iv)

IlVu(s)ll~=(a) ds <

1

IlZX~(t)IIL~<~)<_ ~IIV~IIL~(a)

(v) if ~ e H l ( f t ) , (vi) if ~ E H~(a)

1

~ll~ll~=(a/;

,

for t > 0;

fo sllA~(s)ll2L=(a)ds< ~IIV~II~= 1 -

(a)"

Proof. The conclusion follows from Theorem 6.4.1, taking into account both Theorem 4.1.1 and Remark 6.4.1. Kl T h e o r e m 7.2.4. If ~ is bounded, and F is of class C 1, then the Laplace operator with the Dirichlet boundary condition generates a compact Cosemigroup of contractions in H - I ( f / ) . Proof. We apply Theorem 4.1.1 and Corollary 6.3.2, observing that, by the Sobolev-Rellich-Kondrachov Theorem 1.5.4, the operator ( A I - A ) -1 is compact in L(H-I(t2)). KI T h e o r e m 7.2.5. If ft is bounded and F is of class C 1, then the Laplace operator with the Dirichlet boundary condition generates a compact Cosemigroup of contractions in L2(ft).

158

Analytic Semigroups

P r o o f . We apply Theorem 4.1.2 and Corollary 6.3.2 observing that, from the Sobolev-Rellich-Kondrachov Theorem 1.5.4, ( A I - A ) - 1 is compact from L2(f~)to L2(~t). D We conclude this section with some useful regularity results concerning the semigroup generated by the Laplace operator on Lq(f~) for q _> 1. First we need the following simple lemma. L e m m a 7.2.1. Let ~ be a nonempty, bounded and open subset in R n whose boundary is of class C 1 and let p E [ 1, +c~ ]. Let Ap be the Laplace operator subjected to the Dirichlet boundary condition on LB(~), with the convention that, for p - ~ , LP(~t) is replaced with Co(~t). Let {Sp(t); t > 0} be the Co-semigroup generated by Ap on LP(~). Then, for each p,q E [1, +co], each ~ E Co(~), and each t > O, we have Sp(t)~ - Sq(t)~. P r o o f . One may easily see that, for each p E [ 1, +oc ], Co(f~) C L p (~t), graph(Ac~) C graph(Ap) and D(Acc) C D(Ap), all the inclusions being dense. This completes the proof. D Lemma 7.2.1 allows us to denote the C0-semigroup generated by the Laplace operator subjected to the Dirichlet boundary condition on any of the spaces LP(~), by the very same symbol {S(t) ; t _> 0}. T h e o r e m 7.2.6. Let f~ be a nonempty, bounded and open subset in Rn whose boundary is of class C 1. Let q >_ 1 and let {S(t); t _> 0} be the Co-semigroup generated by the Laplace operator subjected to the Dirichlet boundary condition on Lq(f~). Then, for each 1 <_ q <_ p <_ +co, each E Lq(f~) and each t > O, we have

IIS(t)~llLp(~) <_ (47rt) - } ( 88

}l~[[L~(fl).

(7.2.2)

We shall prove Theorem 7.2.6 with the help of the next lemmas. L a m i n a 7.2.2. Let K " (0, +c~) x R n -+ R be defined by n Ilxll2 K(t) (x) - (47rt) ~ e 4t

.

If r E C(I~ n) has compact support, then v(t,x) - K(t)(x) , r

satisfies

(i) v E C([ 0, +co); Cb(]l~n)) n C((0,-~-cx)); C~(]~n)) 1 (ii) if 1 < p _ + ~ , v E C([ 0, +co); LP(]Rn)) n C~((0, +c<)); LP(]~n)) (iii) vt - Av for t > 0 and v(O) - ~2 1ck(pJ~) denotes the space of all real-valued functions defined on R'~ which are bounded and of class C k.

The Heat Equation n

1

159

1

(iv) [[v(t)[[Lp(a)<_ (47rt) -~(q-~)llr and t > O.

for each 1 ~_ q <_ p <_ +co

P r o o f . The regularity properties ( i ) a n d (ii), as well as (iii) follow from direct calculations. See also Problems 4.3 and 4.4. As concerns (iv), it follows from Hausdorff-Young inequality, by observing that o

_o

IIK(t)llL,(R~) _< P ~" (47rt)

1

-~(1-p) __~(47rt)

~

for 1 _< p <_ +oc and t > 0. This completes the proof.

D

L e m m a 7.2.3. Let ~ E L2(~) be nonnegative, i.e. ~ >_ 0 a.e. in ~ and let {S(t) ; t _> 0} be the Co-semigroup generated by the Laplace operator subjected to Dirichlet boundary condition in L2(~). Then, for each t > 0, we have S(t)~ >_ 0 a.e. in ft. P r o o f . The conclusion follows by a simple density argument observing that, by virtue of (v) in Theorem 7.2.1, for each ~ E C0(~) n H~(~t) with A u c Co(-~), we have u C C([0,+cc);H~(ft)). Therefore u - - min{u, 0} satisfies u - C C([0, + ~ ) ; H01(~t)), and thus

d L(~-)~d~--/~,~-d~--f~-/~dx

dt

Since u-(O) - 0, this inequality shows that f a ( u - ) 2 dx <_ 0 and hence u(t) >__0 for each t _> 0 and a.e. in f~. The proof is complete. D We may now pass to the proof of Theorem 7.2.6. P r o o f . By a density argument, we may confine ourselves only to the case in which ~ C C 0 ( ~ ) n H~(~). Denote by ~7 - [ ~ l and let us observe that, by virtue of Lemma 7.2.3, we have

-s(t)~ <_s(t)~ <_s(t)~, a.e. in ~. Thus Let us define r

IiS(t)~ilL,(~) <_ IIS(t)~TllL,(~). ~n __+ R by r

-

r](x)

for x E ft

0

for 9 e ~

\ ~.

Clearly r C C(R n) and has compact support. Define both

v(t,x) - K ( t ) ( x ) , ~(x) ~(t) - v(t)l~

- s(t)~.

(7.2.3)

160

Analytic Semigroups

By Lemma 7.2.3 and Theorem 7.2.1 we deduce that u e C([0, +c~); C0(~)) n C((0, +co); H i ( a ) ) n C1((0,-~-OO); L2(a)) and Au e C((0, +co); L2(a)). In addition, u(t) - v(t) >> 0 a.e. o n F , u t Hence

A u for t > 0 a n d u ( 0 ) - 0.

d ffl(u-)2dx--/utu-dx--fu-Audx _

_ _ /o

o

and thus u(t) <_ 0 a.e. in a for each t >_ 0. From of r we get

(7.2.3)

and the definition

IlS(t)9 llL ( )_ IIv(t)llL (a), and the conclusion follows from Lemma 7.2.2.

D

T h e o r e m 7.2.7. If ~ is a nonempty, bounded and open subset in I~n with C 1 boundary F, then the Laplace operator subjected to Dirichlet boundary condition generates an analytic and compact Co-semigroup of contractions in L 1(~). P r o o f . Taking q = 1 and p = 2 in (7.2.2) we get

IIS(t) IIL ( ) (_ (4~t)-411{l[L1(~) for each { C L l ( a ) and t ) 0. So, S(t){ C L2(a). As the semigroup generated by the Laplace operator subjected to the Dirichlet boundary condition in L2(t~) is analytic and compact and, by Lemma 7.2.1, it coincides with the homonymous semigroup in LI(~), the conclusion follows from Theorems 7.2.3 and 7.2.5. The next theorem exhibits some asymptotic properties of the semigroup {S(t) ; t _> 0} on each of the spaces LP(~), for p C [1, +co ]. We recall that the first eigenvalue IX of the -Laplace operator in H - I ( ~ ) is given by -inf{llVull~2(a) ; u C H i ( a ) , ]lUIIL2(a) - 1},

(7.2.4)

and that A > O. T h e o r e m 7.2.8. Let ~ be a nonempty, bounded and open subset in ]~n and let )~ > 0 be given by (7.2.4). Then -)~t

for each t >_ O.

The Heat Equation

161

P r o o f . Let ~ E 9 let u(t) - S(t)~, and let us define the function f ' I R + ~ I~+ by f(t) - (e~tilS(t)~liL2(a)) ~. A simple calculation involving Green's formula and the definition of )~, shows that

e-2~t f~(t) - 2)~ /

u(t)2dx + 2 / /-

/.

-

u(t)u~(t) dx

[o

2

Ilw(t)li

dx < 0

Hence ]]S(t)~ilL2(a ) <_ e-~tL[~l]L2(~ ) for each ~ C 9 conclusion follows by a simple density argument.

and t > 0. The D

R e m a r k 7.2.1. Using mainly the same arguments one may prove that, if A is symmetric and generates a C0-semigroup of contractions in a Hilbert space H, {S(t) ; t > 0}, and - i n f { { A ~ , ~ ) ; ~ C D(A), [i~II- 1} > 0, then

IiS(t)il (.) _<

-)~t

for each t > 0. T h e o r e m 7.2.9. Let ~t be a nonempty, bounded and open subset in ]~n whose Lebesgue measure is I~tl and let M - e~ia]2/~/(47~), where A > 0 is given by (7.2.4). Then, for each t >_ O, we have

iiS(t)ii5(L~(a)) < M e -At. In addition, for each p E[1, +cr

(7.2.5)

there exists Mp > 1 such that

[[S(t)ii~(Lp(a)) <_ Mpe -xt

(7.2.6)

for each t > O. P r o o f . Let ~ C 9

and let T > 0. If 0 _< t _< T, we clearly have

If T < t, from (7.2.2) and Theorem 7.2.8, we get

]]S(t)~l]L~(a) <_ ( 4 ~ T ) - 4 i[S(t - T)~IIL~(a ) 1 <_ (47~T)-2e-)~te-)~TIl~l]L2(a) <_ I~tl~(41rr) - 2 e_~te_~T liCitLy(a).

Analytic Semigroups

162

Taking T - I~tl~/(47r) in this inequality, we deduce (7.2.5). To prove (7.2.6), let ~ 6 9 let 1 < 2 < p < +co and let us observe that, by virtue of (7.2.2) and Theorem 7.2.8, for each t > 1, we have

]lS(t)~llLp(a) <_ ( 4 7 r ) - ~ ( 1 - 1 ) I I s ( t _ 1)~l]L2(n )

- 'ts

_< (4~-)-~ ( 89

).

Since for t 6 [0, 1 ], we clearly have

IIS(t){/llL ( ) <_ this completes the proof in the case of 2 <_ p _< +co. The case 1 _< p < 2 follows from the preceding one with the help of (7.2.2) and this achieves the proof. D R e m a r k 7.2.2. One may prove that in the inequalities (7.2.5) and (7.2.6) the constants M and Mp cannot be 1. On the other hand, if p 6 [2, +co), then 4At

_< e and this inequality is the best in the sense that d

d-T

=

4A p2"

See Cazenave and Haraux [36], Remark, p. 46. 7.3. T h e S t o k e s E q u a t i o n In this section, by using the abstract results proved in this chapter, we shall study the existence and regularity properties of one of the most important partial differential equation in Fluid Mechanics, the Stokes equation. This equation describes the flow of "moderate speed" of a viscous incompressible fluid within a domain gt in R 3. We begin with the simplest case ~ - R 3.

Example 7.3.1. (The Stokes operator in R3). Let H - {u C [L2(]R3)] 3 ; ~ 7 . u - 0}, where the condition V . u - 0 is understood in the sense of distributions over R a. One can easily see that, endowed with the standard inner product of the space [L2(R3)] a, defined by 3

i--1

The Stokes Equation

163

H is a real Hilbert space. We define A : D(A) C_ H --+ H, called the Stokes operator on I~3, by D ( A ) - {u C [H2(I~3)] 3 A H ; Au C H} An-An for u c D(A), where Au = (Aum,Au2, Au3) in the sense of distributions over I~3. T h e o r e m 7.3.1. The operator A, defined as above, is the infinitesimal generator of an analytic Co-semigroup of contractions in H. P r o o f . Since C~(I~3; I~3) - {u E [9 3 ; V . u - 0} is both dense in H and included in D(A), it follows that A is densely defined. In addition, from Theorem 4.1.2, we deduce that A is closed and symmetric. On the other hand, for each )~ > 0 and each f E H, the equation A u - Au = f has a unique solution u e [H2(]~3)]3. Let us denote by v - V . u , which clearly belongs to HI(IR3). Since V . f - 0 in H-~(]~3), it follows that I v - Av - 0 in H -1(R3), which implies v - 0. So, u C D(A) and therefore (0, +oc) C_ p(A). More than this, for each A > 0, we have 1

ilR(; ; A)II (x) <_ -S" In view of Theorem 3.1.1, we know that A is the generator of a C0-semigroup of contractions on H. From Lemma 1.6.1, we have that A is self-adjoint, while by Corollary 7.1.1, we deduce that the semigroup generated by A is analytic. The proof is complete. E] Let us consider next the Stokes equation in ~3, ut-

Au

- 0 ~(0, X) -- ~(X)

(t,x) C R ~ (t, x) e X E ]~3,

(7.3.1)

where R ~ = I~+ • ]t~3 and u(t, x) represents the instantaneous velocity of the particle occupying the point of coordinates x at the time t. T h e o r e m 7.3.2. For each ~ E H the problem (7.3.1) has a unique solution u C C([ 0, +co); H) which satisfies: (i) u e C((0, + ~ ) ; [H2(I~3)] 3) A c l ( ( 0 , + e o ) ; H )

;

(ii) for each m e N, Ainu e C~((O, +co); [H2(I~3)] 3) ~ H ) ; (iii) if~ e D(A), then u e C([0, +oc); [H2(R3)] 3) N C1([0, + o c ) ; H ) . P r o o f . We apply Theorem 7.3.1, and Corollary 2.4.1.

if]

Analytic Semigroups

164

T h e o r e m 7.3.3. Let ~ E H and let u the unique continuous solution of the problem (7.3.1). Then: 1 (i) IlZXu(t)lln fo~ each t > 0;

_<;TN[I~IIH t,V~

(ii) fo

s[[Au(s)[[2HdS < --

3

~ll~ll 1 ~ H;

1

IlVui(t)llL2(~3) <_ ~ll~lln

(iii) ~

for each t > O;

i=1

OC 3

(iv)

fo

1

E IIW~(s)II~=(R~)ds < ~11~112 --

H;

i=1

(v) if { c D(A), then

IlZXu(t)lln <_~

1

Vat,

3

~ IIV~IIL~.(R~)foreach t

> O,

i=1

and

(vi) foocc ~ll/x~(,)ll~& -<- ~1 y~ 3 IIV~ll2L 2 ( R a )

9

i-1

P r o o f . The conclusion follows from Theorems 6.4.1 and 7.3.1.

89

E x a m p l e 7.3.2. (The Stokes operator in ~t). Let ~t be a bounded domain in R 3 with boundary F of class C 2 and let u(x) be the outward normal to F at the point x C F. Let Y -- {u C [~)(~)]3

; ~7.u

__ 0

in Ft, u . u -- 0 on F}

and let H be the closure of Y in [L2(~t)] 3. Let us observe that, endowed with the usual inner product of [L2(~t)] 3, H is a Hilbert space.

Let us

denote by P " [L2(~)] 3 --+ H the orthogonal projection on H and let us define the Stokes operator on ~, A : D(A) C_ H --+ H, by D(A) - [H2(~) A H~(gt)] 3 N H A u - P ( A u ) for u C D(A). T h e o r e m 7.3.4. (Fujita-Kato) The operator A, defined as above, is the generator of a compact and analytic Co-semigroup of contractions in H. P r o o f . Since C ~ ( ~ ; N 3) - {u E [ 9 V - u - 0}, which is dense in H, is included in D(A), it follows that A is densely defined. From Theorem 4.1.2, we have that A closed and symmetric and thus self-adjoint. See Lemma 1.6.1. On the other hand, for each A > 0 and each f C H, the equation Au- P(Au) = f

The Stokes Equation

165

has a unique solution u C H which, in addition, satisfies 1

Ilull _< ~llfll. and IIR(~;A)II~(H) <_ ~. By

Therefore (0,+c~) C_ p(A) virtue of HilleYosida Theorem 3.1.1, it follows that A generates a C0-semigroup of contractions on H. Since A is self-adjoint, by Corollary 7.1.1, the semigroup is analytic. Finally, thanks to Sobolev-Rellich-Kondrachov Theorem 1.5.4, for each ~ > 0, R(~; A) is compact. An appeal to Corollary 6.3.2 shows that the semigroup is compact and this achieves the proof. K] Let us consider next the Stokes equation in ~- p(/x)~ (t, x) e Q ~ v. u - 0 (t, x) c Q ~ - 0 (t, x) e r ~ ~(0, x) ~(x) x e ~, where Qce = I~+ • f~ and E ~ = ]~+ • F.

(7.3.2)

T h e o r e m 7.3.5. For each ~ E H the problem (7.3.2) has a unique solution u E C([ 0, + ~ ) ; H) which satisfies: (i) u e C((0, +co); [H2(a)] 3) A C1((0, +oo); H); (ii) for each m e N, [P(A)]mu e CCe((O, + ~ ) ; [H2(ft)] 3) N H); (iii) if ~ e D(d), then u e C([0, +c~); [H2(~)] 3) f-) C1([0, +cx~); H). Proof. We apply Theorem 7.3.4 and Corollary 2.4.1.

[:]

T h e o r e m 7.3.6. Let ~ C H, and let u be the unique continuous solution of the problem (7.3.2). Then: (i) IIP(A)u(t)lln <

1

~---~ll~llnfo~ t,

V

(ii) fo sllP(A)u(s)ll~ds

< --

(iii) ( - P ( A ) u ( t ) , u(t))/2(a) < (iv)

fo

each t > 0;

z..~

~11~112 1 H; 1

~11~11~ fo~ ~ach t > 0;

(-P(A)u(s) ~(~)>L~(a)d~ -<- ~11~112 1 H;

1 (V) if ~ e D(A) then IIP(A)u(t)ll2H <_ ~ ( - P ( A ) ~ , ~ ) L 2 ( f ~ ) f o r each t > O, and (iv) fo ~ slfp(A)u(s)ll~ds <_ ~(-p(A)~, 1

~)L~(a)-

Proof. The conclusion follows from Theorems 6.4.1 and 7.3.4.

V]

166

Analytic Semigroups

7.4. A P a r a b o l i c P r o b l e m w i t h D y n a m i c B o u n d a r y

Conditions

Let f~ be a bounded domain in ]~n whose boundary F is of class C 2. Let QT -- [0, T] x ~, ET -- [0, T] x F, u0fl e L2(f~), u0r e L2(F), and let us consider the parabolic problem subjected to dynamic boundary conditions

u t - /x ut + u, - 0 ~(0, ~) -~0~(~) ~(0, x) - ~0~(x)

(t, z) ~ QT (t, x) C ET x e x e r,

(7.4.1)

where u~ is the outward normal derivative of u on F. For n = 3, this problem represents the mathematical model which describes the evolution of the heat distribution u(t, x) in a homogenous and isotropic solid body (represented by f~), imbedded in a moving fluid. The body has at the initial time a heat distribution on the boundary F, distribution which, at that time, might be different from that of the corresponding boundary of the moving fluid. We begin with a simple lemma which we need in the sequel. L e m m a 7.4.1. Let f~ be a bounded domain in ]~n with C 2 boundary F and l e t # >_ O, )~ > O. Then, for each f C L2(fl) and g C L2(F), the elliptic

problem I pu- Au- f )~v + u, - g

(7.4.2)

u F --v

has a unique solution u C H3/2(f~) with Au C L2(gt), u r C

Hi(r)

and

u, C L2(F). P r o o f . By the classical theory of variational problems based on the Lax-Milgram theorem (see for instance Brezis [29], IX.5., pp. 175-181), (7.4.2) has a unique solution u E g l ( g t ) with u r c n2(r) and Au C L2(gt). Then it follows that u, C H-~/2(F) and so, since u r , g c L2(F), it follows that u~ E L2(r). But then, u E H3/2(f~) and ur c H~(r). The proof is complete. D T h e o r e m 7.4.1. The operator

A ' D ( A ) C_ L2(~t) • L2(F) ~ L2(gt) • L2(F), defined by D(A) - {(u, v) e L 2 (f~) • L 2 (F) ; Au e L 2 (~), u~ e L 2 (F), u r -- v} A(u, v ) = ( A n , - u ~ ) for each (u, v) e D(A),

A Parabolic Problem with Dynamic Boundary Conditions

167

is the infinitesimal generator of a compact and analytic Co-semigroup of contractions. P r o o f . Clearly D(A) is dense in L2(~t) x L2(F), because each function in L2(F) can be approximated with functions of class C 2 on F, and each function in L2(gt) can be approximated with functions of class C2(~) whose restrictions to the boundary are preassigned C 2 functions on F. Moreover, let us observe that, for each (u, v), (~, r e D(A), we have

(A(u, v), (~, r

- (An, ~) - (u~, r

-- - ( V u , V~)

= (u, AV) - (u, ~-)L:(r) -- ((u, v), A(~, r Consequently A is symmetric. In addition, for each A > 0, the equation (M- A)(u, v) = (f, g) rewrites equivalently under the form

I

u- Au- f )~v + uv - g UF=V.

From Lemma 7.4.1, we know that, for each (f,g) E L2(~) • n2(r), the problem above has a unique solution u satisfying u C H 3/2 (gt), Au C L 2 (gt), ulr c Hi(r) and u. E L2(r). Taking the L2(~)-inner product both sides of the first two equations in the system above by u and respectively by v and taking into account the third equality, we deduce

~llvll~cr ) + (~,

~)L2(r) --

(g, v)i=(r).

Adding side by side the two equalities and using Cauchy-Schwarz inequality, we obtain

All (~, v)II~ (~)•

<- II(f , g)IIL~(~)•

II(~ , v)IIL~(~)• L~(r).

From this, we deduce, on one hand that (0, +co) C p(A) and, on the other hand, that, for each )~ > 0, we have [](AI- A)-I]IL(x) < ~. By virtue of Theorem 3.1.1, it follows that A generates a C0-semigroup of contractions, and from Lemma 1.6.1, we conclude that A is self-adjoint. Since D(A), endowed with the graph-norm, is compactly imbedded in L2(gt) • L2(F), by Corollary 6.3.2, it follows that the semigroup generated by A is compact. The proof is complete. [5 T h e o r e m 7.4.2. For each U~o e L2(~) and each Uro e L2(F), the problem (7.4.1) has a unique continuous solution u" I~+ --~ L2(~) satisfying (i) u e C([ 0, +co); L2(D)) N el((0, +co); L2(D)) ; (ii) Ur e C([0,+cc);n2(r)) n cl((0,+cr ;

168

Analytic Semigroups (iii) A u e C((0, +oc); L2(ft)), and uu e C((0, +oc); L2(r)).

I/, in addition, AUao e L2(~), u~ r e L2(r) and U~o r - u~, then (iv) u e C1([ 0, +co); L2(ft)) ;

(v) ulr e C1([0, +oc);n2(r)) ; (vi) Au e C([0,+oc); L2(ft)), and Ulr e C([0,+oc);n2(r)). P r o o f . The problem (7.4.1) can be rewritten as an ordinary differential equation, in the space H - L2(f~) x L2(r), of the form { u ~ - Au

u(O) - to, where A is defined as in Theorem 7.4.1 and u0 - ( u 0 ~, u0r). The conclusion follows from Theorem 7.4.1 combined with Theorem 6.4.1, and this achieves the proof. D

7.5. An Elliptic Problem with Dynamic Boundary Conditions Let f~ be a bounded domain in R n whose boundary F is of class C 2. Let QT -- [0, T] x f~, ET -- [0, T] x F, Uoa C L2(ft), Uor E L2(F), and let us consider the elliptic problem subjected to dynamic boundary conditions

/

-zx~ - 0

(t, ~) c Q~

ut + u~ - 0

(t,x) e ET

~(O,x) - ~ o ( ~ )

x e r,

(7.5.1)

where u~ is outward normal derivative of u on F. Let y C H1/2(F) and let us consider the nonhomogeneous elliptic problem -At-0 u-y

inft onF.

(7.5 2)

It is known that this problem has a unique solution u E Hl(f~). See Brezis [29], Exemple 2, p. 176. Let us denote this solution by u - g(y).

Theorem 7.5.1. The operator A" D(A) C L2(F) --+ L2(F), defined by D(A) - {y e H1/2(F);for which u A y - - u , for y C D(A),

g(y) satisfies u, e L2(F)}

is the infinitesimal generator of an analytic and compact Co-semigroup of contractions in L 2 (F). P r o o f . Let us observe that, for each y, z C D(A), we have (Ay, Z)L2(r) -- --(Vu, VV)L2(~) -- ( A t ,

V)[HI(Ft)].,HI(f~)

(y, Az)L2(r) -- - ( V u , Vv)n2(~) -- (Av,

U)[HI(f~)].,HI(f~),

and

An Elliptic Problem with Dynamic Boundary Conditions

169

where u satisfies (7.5.2), while v is the solution of a similar equation with y substituted by z. Here (', ")[HI(~)]. HI(~) is the usual duality between H I ( ~ ) and its topological dual [Hl(~)] *. Since An -- Av -- 0, it follows that {Ay, Z}L2(r) -- --(Vu, Vv}n2(a) -- (y, Az}n2(r) for each y, z E D ( A ) . So, A is symmetric. We shall prove next that A is self-adjoint in H - L2(F) and satisfies the conditions in Theorem 3.1.1. To this aim, we shall show that, for each )~ > 0, ( A I - A) -1 E L ( H ) , and I I ( A I - d ) - l i l ~ ( g ) < ~. One may easily see that A y - d y - f if and only if y - Ulp , where u is the unique H I ( ~ ) solution of the elliptic problem -Au - 0 Au + u . -

in ~ f

(7.5.3)

onF.

So, ( A I - A) is surjective if and only if for each f E solution in H I ( ~ ) of (7.5.3) satisfies

L2(F), the unique

u E L2(F) and u, E L2(F).

(7.5.4)

But this follows from Lemma 7.4.1. Finally, we have merely to prove that, for each A > 0, we have I I ( A I - A ) - l i i f ~ ( , ) < 1/~" -

-

(7.5.5)

To this aim, let us multiply both sides of the equation - A u = 0 in (7.5.3) by u. Integrating over ~t yields

IlWll L~(~)

- L~(r) -- 0,

from where, taking into account the boundary condition in (7.5.3) to substitute u, by f - Au, we deduce

From this inequality, observing that ( ) ~ I - A ) y - f if and only if y - ulp , we obtain (7.5.5). So, A satisfies all the conditions in Theorem 3.1.1, and Lemma 1.6.1. Moreover, from Corollary 7.1.1, it follows that A generates an analytic semigroup. Finally, the fact that this semigroup is compact follows from the last assertion by observing that the imbedding D ( A ) C_ LU(F) is compact. See Corollary 6.3.2. The proof is complete. [::] 7.5.2. For each uo E L2(F), the problem (7.5.1) has a unique solution u " R+ --~ H I ( ~ ) , satisfying"

Theorem

(i) u E C((0, +oc); H I ( ~ ) ) gl L2(0, +oc; H I ( ~ ) ) , and

IlWll

<- 89

L2(F) ;

170

Analytic Semigroups

(ii) tic e C([0, + ~ ) ; L2(F)) n C~((0, +co); L2(F)) ; (iii) u~ e C((0, +co); L2(F)). If, in addition, uo E H~/2(F), and the unique solution v of the problem - A v = 0 on ~t and v = uo on F, satisfies vu C L2(F), then

(iv) u e C1([0, + o c ) ; H l ( a ) ) ;

(v) u r e C~([0, +~);L2(F)) ; (vi) u, e C([ 0, +,c); L2(F)). P r o o f . The problem (7.5.1) can be rewritten as an ordinary differential equation in g - L2(f~) x L2(F), of the form u' - A u ~(o)

-

where A is defined as in Theorem 7.5.1 and ~ = u0. Except for (i) and (iv), the conclusion follows from Theorem 7.5.1 combined with Theorem 6.4.1. In order to check (i) and (iv), let us observe that, for each t, s E [0, +oc), we have - A ( u ( t ) - u ( s ) ) - O. Taking the L2(~) inner product both sides by u ( t ) - u(s), and taking into account the boundary conditions, we deduce

IIV(u(t) - u(s))ll~2(~-,t) _~ Iluu(t)

- u.(s)llL2(r)llu(t)

- u(s)llL=(r).

From this inequality, from (ii) and (iii), we get the first assertion in (i). From the same inequality, from (iv) and (v), we deduce (vi). To prove the second assertion in (i), it suffices to observe that

IIVu(t)ll~2(a) - (u~(t), u(t))L=(r) - O, or equivalently ld Integrating this equality from 0 to T, we obtain Ji T

1 IlVu(t) ll~2(~) dt -< ~ II~ll~=(r)

for each T > 0, which achieves the proof of (i). The proof is complete.

V1

7.6. F r a c t i o n a l P o w e r s of C l o s e d O p e r a t o r s In this section we shall define the fractional power of the negative of an infinitesimal generator of a C0-semigroup. First, let us recall that, for each C (0, +co), the improper integral F(a) -

/o

t~-le-tdt

171

Fractional Powers of Closed Operators

is convergent, and its values is the so-called Euler F-function. Let a > 0 and a > 0. The change of variable ta - ~" shows that

a- a _ -

1 F(a)

.~~o t a-1 _~

-ta e

dt.

Then, if A " D ( A ) C X --+ X is a linear operator with - A the infinitesimal generator of a semigroup of class Co, {S(t) ; t >_ 0}, it is quite natural to consider that, substituting a with A and e - t a with S(t) in the relation above, we obtain an operator playing the role of A - a . We recall that, at least formally, S(t) can be interpreted as S(t) - e -tA, and this suggests and explains the above analogy. More precisely, we have D e f i n i t i o n 7.6.1. The operator A - a " D ( A -~) C_ X ~ X defined by

1 F(a)

A-~x-

/o

t~_lS(t)xdt '

(7.6.1)

for each x C D ( A -~), where

D ( A -a) -

x C X;

t a - x s ( t ) x d t is convergent

is called the power of exponent - a D ( A ~ - X and A ~ - I.

of the operator A.

,

By definition,

R e m a r k 7.6.1. Clearly D ( A -a) is a linear subspace in X and A - a is a linear, closed operator. Moreover, for each 0 _< a _
D ( A -~) c_ D ( A - a ) . R e m a r k 7.6.2. If the C0-semigroup, {S(t) ; t > 0}, generated by - A , is of type ( M , - w ) , with M _> 1 and w > 0, i.e., for each t > 0, we have

IIS(t)ll cx)

Me

then, for each a >_ O, D ( A -a) - X . Indeed, this follows from the remark that, in this case f0 ~ t ~-~ I I S ( t )IId t(x)

< +oo.

In addition, since f o t a - l s ( t ) dt is convergent in the uniform operator norm, it follows that, for each a _~ 0, A - a is a linear bounded operator. E x a m p l e 7.6.1. Let H be a real Hilbert space, whose inner product is denoted by (-, .), and let A " D ( A ) C H -+ H be such that - A generates a C0-semigroup of contractions on H. Let us assume that A is self-adjoint and invertible with compact inverse A -1. Then, by a theorem of Hilbert,

172

Analytic Semigroups

there exists a sequence of positive numbers Pk > 0, ~k+l _~ #k, for k E N* and an orthonormal basis {ek; k C I~*} of H such that A-1 ek -- #kek for k C N*. Let us denote Ak -- #k -1, and let us observe that ek C D ( A ) and Aek -- )~kek for k E N*. We also have that limk~ccAk -- +co. Let a > 0 and let us define Aa " D ( A ~ ) C_ H ~ H by D(A~) -

u C H ; u-

ukek, ~ k=l

A2k"lukl 2 < +cc

k=l

cK)

A . ~ - Z ~k"~k~k for ~ e D(A.). k-1

A simple calculation shows that A~ - A a. We notice that D(Aa), endowed with the natural inner product (., "}a " D ( A a) x D ( A a) -+ R, defined by oc)

k=l

for each u , v C D(A"), u Y~'~k=lUkea and v - E ~ C _ l Vkek, is & real Hilbert space. In addition, with respect to this inner product, the family {),/"ek ; k E N*} is an orthonormal basis in D ( A " ) . i e m m a 7.6.1. If the Co-semigroup, {S(t) ; t >_ 0}, generated b y - A , is of type ( M , - w ) , with M >_ 1 and w > O, then, for each a,/3 C [0, +co), we have A-(~+~) = A - ~ A - ~ . P r o o f . Let us observe that

/o fo r(~)r(~) /o S r(~)r(~) (/o

A_~A_ ~ = _ _-

= Inasmuch as

1

r(~)r(~)

1

1

1111 r(~)r(9)

~

t a-1

t ~-~ 8/~-l S ( t ) S ( 8 ) dt ds (u - t ) ~ - l d t S ( u ) d u

ta-l(u - t)e-ldt

)

S(u) du

v a - l ( 1 -- v ) ~ - l d v ~0 c~ Ua+~-ls(u) du.

1 v c ~ - l ( 1 _ V)/3-1dv __ r(oL)r(/~) c(~+9)'

173

Fractional Powers of Closed Operators

from the equalities in above, we deduce A_~A_ ~ =

1

r(~+~)

and this achieves the proof.

/0

ua+~-lS(u) du - A -(~+~) [:3

L e m m a 7.6.2. If the Co-semigroup, {S(t) ; t _> 0}, generated b y - A ,

is of type ( M , - w ) , with M > 1 and w > O, then, for each a E (0, 1), we have

A_ a : sin 7ra f o r A-~(AI + A) -1 dA.

(7.6.2)

J0

T

P r o o f . Since - A is the infinitesimal generator of a C0-semigroup of type (M,w), by Theorem 3.3.1, we have that

II(AI +

M

lll_t}-lll ~,

~

/~ -+-co

for each A > - w , and therefore the integral on the right-hand side in (7.6.2) is convergent. In addition, as ( M + A ) -1 -

e-~ts(t)dt '

~0~176

we have sinTra f0cr

~--(~i

+ A)-~ex

-

sin 7ra If~176 )~-ae-~td)~

7r

S(t) dt

71-

sinTra ~oCCv - ~ e -v dv fo ~ t a - l s ( t ) dt - sin ~c~ r ( ~ ) r ( 1

-

a)A -a

71"

7r

Since

r(~)r(1 -~)

-

sin 7ra

this completes the proof.

D

L e m m a 7.6.3. If the Co-semigroup {S(t) ; t _> 0}, generated by - A , is of type ( M , - w ) , with M >_ 1 and w > 0, then there exists C > 0 such that, for each a E [0, 1], we have

IIA-'~II~(x) _< c. P r o o f . It suffices to prove the inequality only for a C (0, 1). From Lemma 7.6.2, we have

iiA_~ll~(x ) _<

sinTra 71-

+

sin 7ra 71"

~

flA_~(AI

+ A) -1 dA

J0

~(x)

cc,~-a(Ai + A ) - l dA ~(x)

174

Analytic Semigroups

Inasmuch as II(AI + A) -~ lin(x) < ~M with w > 0, there exist Co > 0 and C1 > 0 such that II(AI + A) -111~(x) _< Co for each A C [0, 1 ], and

IIA(A/+ A)-III~(x) ~ C1 for each A > 1. Accordingly, we have IIA-c~ II~(x) -< Co sinTr(17r(l_-a)a) -t- C1 sin~raTra < C, where C - Co + C1. The proof is complete.

[-1

L e m m a 7.6.4. If the Co-semigroup, {S(t) ; t _> 0}, generated by - A , is of type ( M , - w ) , with M >_ 1 and w > O, then, for each x C X , we have lim A - a x - x. aS0

P r o o f . Let x E D ( A ) . Since ( - w , +co) C p ( - A ) , we have 0 E p(A) and therefore there exists y E X with x - A - l y . We then have

)

A - ~ x - x - A-(~+l)y - A - l y -

F(1 + a) - 1 S ( t ) y dt.

From the growth condition IIs(t)ll~(x) _< Me -~t, we deduce

II1-~

- xll _< MllYll f0 c r ( 1 t +c~ c~) - 1

~-~tat.

On the other hand, there exists C > 0, such that, for each a C [0, 1] and each t >_ 1, we have ta -1
r(1 + ~)

-

It then follows that, for each k _> 1, we have IIA-~x

_< Mllyll

fo ~

-

~11

r(1 t~ + a) - 1 e-~tdt + CMllyll

Ji ~ te-~tdt.

Let e > 0. Let us fix a sufficiently large k _> 1, such that the second term on the right-hand side in the above inequality be less than e/2. We observe that there exists 5(e) > 0 such that, for each a C (0, 5(e)), the first term on the right-hand side in the above inequality is less than e/2. Then, for each x C D ( A ) , we have lim~s0 A - ~ x - x. Since D ( A ) is dense in X and, by virtue of Lemma 7.6.3, the family of operators {A-~; c~ C [0, 1]} is bounded in L ( X ) , the proof is complete. [3

Fractional Powers of Closed Operators

175

From Lemmas 7.6.2 and 7.6.4, it follows: C o r o l l a r y 7.6.1. If the Co-semigroup, {S(t) ; t > 0}, generated by - A , is of type ( M , - w ) , with M > 1 and w > O, then { A - t ; t > 0} is a semigroup of class Co. L e m m a 7.6.5. If the Co-semigroup, {S(t); t > 0}, generated b y - A , is of type ( M , - w ) , with M >> 1 and w > O, then, for each c~ >> O, A -~ is injective. P r o o f . Obviously A -1 is injective and thus, for each n E N*, A -n has the same property. Let c~ > 0 and x C X with A - a x - O. Let us fix n C N* with n > c~. We have A - n x - A - n + a A - a x - O. Accordingly x - 0, and therefore A -~ is injective. The proof is complete. D L e m m a 7.6.5 allows us to define the power of an arbitrary real exponent, of any operator A of the form A - - B , with B the infinitesimal generator of a C0-semigroup of type ( M , - w ) . More precisely, we have: D e f i n i t i o n 7.6.2. Let A " D ( A ) C_ X --+ X be a linear operator, with - A the infinitesimal generator of a C0-semigroup of type ( M , - w ) , with M _> 1 and w > 0, and let c~ > 0. By definition A ~ - ( A - ~ ) -1.

(7.6.3)

In all that follows, A a is defined by A a - A -~ with/~ - -c~, where A -~ is given by (7.6.1) if c~ < 0, and respectively by (7.6.3) if c~ > 0. According to the previous convention, we have A ~ - I. T h e o r e m 7.6.1. If the Co-semigroup {S(t) ; t >_ 0}, generated by - A , is of type ( M , - w ) , with M > 1 and w > O, then" (i) (ii) (iii) (iv)

for for for for we

each each each each have

c~ E IR, A ~ is a closed operator with D ( A ~) - R ( A - ~ ) ; 0 < c~ < ~, we have D ( A ~) C D(AZ) ; c~ > O, D ( A ~) is dense in X ; c~,/~ C IR and each x E D(AT) (7 - max{a,/~, c~ +/~}), A ~+~x - A ~A ~x.

P r o o f . The items (i) and (ii) follow from Remark 7.6.1, even in a more general setting when (S(t) ; t > 0} is only of class Co. The item (iii) follows from the observation that D ( A n) is dense in X and, for each c~ > n, we have D ( A n) C_ D(A~). See Theorem 2.4.3, and (ii). Finally, (iv)is a direct consequence of both L e m m a 7.6.1 and Definition 7.6.2, and this achieves the proof. [3

Analytic Semigroups

176

For 0 < a < 1 and x E D(A) precisely, we have :

C: D(Aa), we can define Aa

explicitly. More

Theorem 7.6.2. If the Co-semigroup {S(t) ; t 2 0)) generated by -A is 1 and w > 0, a E (0, I ) , and x E D(A), then: of type ( M , -w), with M

>

A'X

sinaa a

l

=-

CO

~ a - (XI l

+ A ) - ~ A Xd

~ .

Proof. By (i) and (ii) in Theorem 7.6.1, we have R(A&-') = D(A'-o) and D(A'-") C D(A). Therefore, A"-'x E D(A), for each x E X. On the other hand, by Lemma 7.6.2, we get A"-lZ

=

lrn +

-

IT

X~-'(XI

A ) - ~ xd ~ ,

(7.6.4)

while from the preceding observation, both sides of the equality above are in D(A). Applying A to both sides in (7.6.4), and using Theorem 1.2.2 and (iv) in Theorem 7.6.1, we obtain the conclusion. The proof is complete.

Theorem 7.6.3. Let A : D(A) CI X + X, where -A is the generator of a Go-semigroup of type ( M , -w), with M 2 1 and w > 0, and let a E ( 0 , l ) . Then there exists C > 0 such that, for each x E D(A) and each p > 0, we have (7.6.5) IIAaxII I C(P"IIXII + P"-'IIAxII) and (7.6.6) IIAaxIl I 2cllxl11-"IIAxIIa. Proof. By Theorem 7.6.3, since ( X I we deduce

+ A)-'Ax

=x

-

X(XI

+ A)-'x,

s i n a ( 1 - a)

sin aa a where

C = (1+ M ) sup

s i n a ( 1 - a)

aE(0,')

So, (7.6.5) holds. Since (7.6.6) is obviously true for x = 0, and follows from (7.6.5), putting p = IIAxIIIIIxlI, for x # 0, the proof is complete.

177

Further Investigations in the Analytic Case

C o r o l l a r y 7.6.2. Let A " D ( A ) C_ X --+ X , where - A is the generator of a Co-semigroup of type ( M , - w ) , with M > 1 and w > O, and let c~ C (0, 1 ]. Let B " D ( B ) C_ X --+ X be a closed operator with D ( A ~) C_ D ( B ) . Then there exists C > 0 such that, for each x C D ( A a ) , we have IIBxII <_ CIIA~xll .

(7.6.7)

Moreover, there exists C1 > 0 such that, for each p > 0 and each x C D ( A ) , we have

IIBxll _< cl(p"llxll + p~-lllAxll).

(7.6.8)

P r o o f . Since D ( A ~) C_ D ( B ) , the operator B A -~ is defined on the whole X and is closed. From the closed graph theorem (see Dunford and Schwartz, Theorem 4, p. 57), it is in L ( X ) , which proves (7.6.7). Since (7.6.8) follows from (7.6.5) and (7.6.7), the proof is complete. [~ 7.7. F u r t h e r I n v e s t i g a t i o n s in t h e A n a l y t i c C a s e L e m m a 7.7.1. Let A " D ( A ) C X -+ X , w h e r e - A is the infinitesimal generator of an analytic Co-semigroup of type ( M , - w ) , with M > 1 and w > O. Then" (i) there exists C > 0 such that, for each t > O, we have

IIAS(t)ll~(x) <_ C t - 1 ; (ii) there exist Co > 0 and a C (O,w) such that, for each t > O, we have

IIAS(t)ll~(x) <_ Cot-le-at ; (iii) for each m C N*, we have I]AmS(t)ll < (Corn)rot-me -at, where Co > 0 and a E (0, w) are the constants given by (ii).

P r o o f . The item (i) follows from Theorem 7.1.1. On the other hand, since the semigroup is of type ( M , - w ) , we have

IIAS(t) II~(x) - IIS(t/2)AS(t/2)II~(x)

<_ IIS(t/2)II~(x)IIAS(t/2)II~(x)

<_ 2MCt-~e -(~/2)t,

which proves (ii) with Co - 2 M C and a - w/2. observe that

To check (iii), let us

IIAmS(t)[I - II(AS(t/m))mll <_ IIAS(t/m)[I m <_ (Comt-le-aCt/m)) m -(Corn)rot-me-at '

and this achieves the proof.

[-1

178

Analytic Semigroups

T h e o r e m 7.7.1. Let A " D(A) C_ X -+ X , where - A is the infinitesimal generator of an analytic Co-semigroup of type ( M , - w ) , with M >_ 1 and w > O, let ~ C (0, 1), and let B ' D ( B ) C_ X ~ Xbe a closed operator with D(A) C_ D ( B ) . If there exist C > 0, ~, r (0, 1), and Po > O, such that, for each x C D(A) and each p > po,

IIBxll

C(p llxll + p -lllAxll),

(7.7.1)

then, for each ~ < ~ < 1, we have D ( A ~) C D(B). P r o o f . Let x C D ( A I - a ) . Then A - a x E D(A) C_ D ( B ) . closed, from Theorem 1.2.2, it follows that BA-~x-

1 F(a)

/o

As B is

t~_lBS(t)xd t

if the integral on the right-hand side is convergent. Let us observe that

IIBA- xll <_ 1

Pta_ 1

(/0

IIBS(t)xll dt +

L

ta_ 1

[IBS(t)xll dt

)

.

Taking ~ - p0 -1, and using (7.7.1), with p - t -1 in the first integral on the right-hand side, and respectively, with p = P0, in the second integral, and using (i) and (ii) in Lemma 7.7.1, we deduce that there exists C1 > 0 such that IIBA-axll <_ Cl[[Xll for each x C D ( A I - a ) . Since B A -~ is closed and D ( A 1-~) is dense in X, it follows that the preceding inequality holds true E] for each x C X. Hence D ( A ~) C_ D ( B ) , and this achieves the proof. T h e o r e m 7.7.2. Let A " D(A) C_ X --+ X , where - A is the infinitesimal generator of an analytic Co-semigroup of type ( M , - w ) , with M > 1 and w > O. Then" (i) for each t > 0 and ~ >_ O, we have S ( t ) X c_ D(Aa) ; (ii) for each ~ > 0 and x E D(Aa), we have S ( t ) A ~ x - A a S ( t ) x ; (iii) for each t > O, and each ~ >_ O, the operator AaS(t) is in L ( X ) . In addition, there exist Ma > 0 and a C (0, w) such that, for each t > O, we have IIA~S(t)[[r

<_ M ~ t - ~ e - a t ;

(iv) for each ~ C (0, 1 ], there exists Ca > 0 such that, for each t > 0 and each x C D(Aa), we have

IIS(t)x - xll _< C t llA xll. P r o o f . Since the semigroup generated by - A is of type ( M , - w ) , with M >__ 1 and w > 0, from Lemma 7.6.5, we deduce that, for each c~ > 0, A ~

Further Investigations in the Analytic Case

179

is well-defined. I n a s m u c h as {S(t) ; t _> O} is analytic, for each a _> 0 and each t > O, we have oO

S ( t ) X C_ N D(An) C_ D(Aa), n--O

which proves (i). Let a _> 0 and x C D(Aa). By virtue of L e m m a 7.6.4, there exists y C X such t h a t x - A - a y . We then have

S(t)x-

S(t)A- ay-

F (1a ) fo cc s a _ l s ( s ) S ( t ) y d s

= A-"S(t)y - A-~S(t)A"x, and thus we get (ii). Clearly (iii) holds for a - 0. So, let us assume that a > 0. Since A ~ is closed, for each t > 0, A~S(t) enjoys the very same property. Moreover, in view of (i), AaS(t) is defined on the whole X. By virtue of the closed graph theorem, it follows t h a t it is bounded. Let n C N* with n - 1 < a _< n. From (iii) in L e m m a 7.7.1, we deduce

IIA~S(t)ll~(x) -

IIA"-nAns(t)ll~(x)

O0

-

r(~

sn-~-l][AnS( t + ~)ll~(x)d~

~)

-

OO

< --

(c~)n r(~

-

~)

fo

s n - a - l ( t zt_ s)-ne-a(t+S)ds

(Cn) he-at ~00cc u n - a - l ( 1 4- u)-ndu -- -Ma -e t~

-at

-< r(n - ~)t~

which proves (iii). Finally, (iv) follows from the fact that, by virtue of (iii), for each a C (0, 1), we have

't A S ( s ) x ds

IIS(t)x - x l l -

fo

t A l - a S ( s ) A a x ds

foot IIAI-~s(s)II~(x)IIA~xll ds <_ M , foos~-le-a~ilA~xll ds

/o The proof is complete.

[q

Analytic Semigroups

180

Finally, we notice that we can define the fractional powers of the negative of an infinitesimal generator of an analytic C0-semigroup by using the functional calculus for unbounded operators presented in Section 1.9. More precisely, i f - A is the infinitesimal generator of an analytic C0-semigroup with 0 C p ( - A ) and c~ > 0, we can define A-~=

1 f r ),-a (A - )~I)-1 d),, 2ui

where F is a simple rectifiable curve included in p(A), which does not intersect the negative axis, curve consisting of two rays connected by an arc of circle centered at the origin and oriented from cce -i~ to cce i~ with 0 fixed in (0, ~). In addition, )~-~ is taken to be positive for A real and positive. For more details with regards to this manner of defining A - a , see Pazy [101], Section 2.6, p. 69. Problems Problem 7.1. Show that if a C0-group has an analytic extension in a sector of the complex plane, then its infinitesimal generator is bounded. See Goldstein [61], Exercise 14, p. 75. Problem

7.2. Let H - L2(0, ~) and let A " D(A) C_ H --+ H be defined

by D(A) - H 1(0, 7r) N H I (0, 7r) A u - - u " for u c D(A) 1 1]. Find D(A"). and let a C [~, P r o b l e m 7.3. L e t X - L2(I~+), and A " D(A) C_ X -+ X with - A the generator of the C0-semigroup of translations S(t) 9X -+ X, defined by

[S(t) f](s) - f (t + s) for e a c h t _ > 0, each f C X, a n d a . e , for s C ~+. Let c~ C (0,1). Show that C~(0, +co) C_ D ( A - a ) , and that, for each f C D(A-~), and a.e. for s C (0, +c~), A - a f is the Weyl fractional integral of order ~ of f, i.e.

(A-~ f)(s) - F(c~) lfs~

(u - s)a-l f (u) du - (K~ f)(s).

Prove that, for each f, g C C ~ ( 0 , +oc), we have

/0

/0

Notes

181

where J a f is the Riemann-Liouville integral, of exponent a, of f, i.e.

(J~f)(u)-

1 F(u)

(u_s)a_lf(s)ds.

See also Problem 4.7. See McBride [89], p. 103. P r o b l e m 7.4. Let {S(t) ; t >__0} be the C0-semigroup in Problem 4.3, and let A be its generator. Prove that A1/2 - - ( - A ) 1/2 is defined by means of the singular integral operator m 1 ./_+~ f ( x - y ) - f (x) [A1/2f](x) - l i h--+O -~ ~ ~ y2 + h2 dy

for each f E D(A1/2). Prove that the C0-semigroup generated by A1/2 is the Poisson semigroup in Problem 4.5. See Goldstein [61], pp. 62-63. N o t e s . Theorem 7.1.1 was proved by Hille in 1942, and Theorem 7.1.2 is due to Vrabie [130] and is somehow related to the following characterization of generators of analytic semigroups due to Crandall, Pazy and Tartar [43]. T h e o r e m (Crandall-Pazy-Tartar) Let A " D(A) C X --~ X be the generator of a Co-semigroup of type (M,w), {S(t); t _> 0}. Then {S(t) ; t _> O} is

analytic if and only if there exist C > 0 and A >_ 0 such that IIAR(A;A)

for A > hA, n -

n+lll~(x)

< --

c

?~)~n

1,2, . . . .

See Pazy [101], Theorem 5.5, p. 65. The results in Section 7.1, 7.2 and 7.3 are from Brezis and Cazenave [31] and Cazenave and Haraux [36]. For similar decay estimates, as those in Theorem 7.2.6, in the case of the C0group generated by the linear Schrhdinger operator in the whole space, see Sulem and Sulem [118], pp. 41-51. Sharper time-space estimates, for the Schrhdinger C0-group, known as Strichartz inequalities, may be found also in Sulem and Sulem [118] loc. cit., as well as in Brezis and Cazenave [31], pp. 88-94. Sections 4 and 5 are from Bejenaru, Diaz and Vrabie [23]. Several interesting examples of analytic semigroups, arising from thermoelastic plate systems subjected to various boundary conditions and requiring a much more delicate analysis, can be found in Lasiecka and Triggiani [84], Chapter 3, pp. 311-413. Sections 7.6 and 7.7 present, in a slightly different manner, the main concepts and results in Pazy [101] concerning fractional powers of closed operators, introduced by Bochner [25], Phillips [104], redefined and studied within a more general frame by Balakrishnan [10], and Komatsu [75], [76], [77], [78], [79]. For more details on this subject see Amann [5] and Pazy [101]. We note that whenever A generates an analytic

182

Analytic Semigroups

semigroup, then - ( - A ) 1/2 generates an analytic semigroup too, and the unique solution of the boundary-value problem u" = - A t , limt$0 u(t) = and u bounded is the solution of u' - - ( - A ) ~ / 2 u , t > 0, limt+0 u(t) - ~. This is in fact the definition given by Balakrishnan [11] for the square root o f - A . We note that Barbu, in [15] and [16], extended this definition to the nonlinear case as well, i.e. when A is maximal monotone operator acting in a real Hilbert space H. Further details on the fractional powers of the Stokes operator may be found in Fujita and Kato [58] and Constantin and Foia~ [38]. The problems in this chapter are from Goldstein [61] and McBride [89].

CHAPTER 8

The Nonhomogeneous Cauchy Problem

In this chapter we introduce several concepts of solution for the nonhomogeneous Cauchy problem u ' - Au + f u(a) - ~,

where A 9 D ( A ) C_ X ~ X is the infinitesimal generator of a C0-semigroup of contractions, ~ E X, and f C Ll(a, b ; X ) . Namely, we introduce the concepts of C 1, strong and respectively C~ and we analyze the relationship between these types of solutions. We then study the compactness properties of the C O. solution operator f ~ u which will prove useful later in order to obtain existence results for some classes of semilinear Cauchy problems.

8.1. The Cauchy Problem

u'-

A u + f , u(a) -

As we already have seen in the preceding sections, if A " D ( A ) C_ X --+ X is the infinitesimal generator of a C0-semigroup {S(t) ; t _> 0}, then, for each a _> 0, and ~ C D(A), the function u 9 [ a , + c c ) -+ X, defined by u(t) - S ( t - a)~ for each t _> 0, is the unique solution of the homogeneous Cauchy problem u' - A u u(a) - ~.

(8.1.1)

From this reason, it is quite natural to consider that, for each ~ C X, the function u, defined as above, is a solution for (8.1.1), in a generalized sense. The aim of this section is to consider the nonhomogeneous problem u'- Au + f u(a) - ~,

(8.1 2)

where A is as before, ~ C X, and f C L 1(a, b ; X ) . D e f i n i t i o n 8.1.1. The function u 9 [a, b] --+ X is called classical, or C 1solution of the problem (8.1.2), if u is continuous on [a,b], continuously 183

The Nonhomogeneous Cauchy Problem

184

differentiable on (a, b], u(t) E D(A) for each t E (a, b] and it satisfies u'(t) = Au(t) + f (t) for each t E [a, b] and u(a) = ~. D e f i n i t i o n 8.1.2. The function u : [a, b] --+ X is called absolutely continuous, or strong solution, of the problem (8.1.2), if u is absolutely continuous on [a, hi, u' E L l(a, b;X), u(t) E D(A) a.e. for t E (a, b), and it satisfies u'(t) = Au(t) + f (t) a.e. for t E (a, b) and u(a) = ~. R e m a r k 8.1.1. Each classical solution of (8.1.2) is a strong solution of the same problem, but not conversely. R e m a r k 8.1.2. As we already have seen, if A generates a uniformly continuous semigroup and f is continuous from [a, b] to R ~, then a function u : [a, b] ---+X is a classical solution of the nonhomogeneous problem (8.1.2) if and only if it is given by the so-called variation of constants formula

u(t) - S ( t - a)~ +

S ( t - s)I(s) ds

(8.1.3)

for each t E [a, b]. See Theorem 2.1.2. Simple examples show that, in the case in which X is infinite-dimensional and A is unbounded, i.e. it generates only a C0-semigroup, the problem (8.1.2) may fail to have any classical solution and this, no matter as regular is the datum f. See Example 8.1.1 below. The next simple result is fundamental in the study of the nonhomogeneous problem (8.1.2). T h e o r e m 8.1.1. (Duhamel Principle) Each strong solution of (8.1.2) is given by (8.1.3). In particular, each classical solution of the problem (8.1.2)

is given by (8.1.3). P r o o f . Let u be a strong solution of (8.1.2), t E (a, b] and let us define g ' [ a , t ] --~ X b y g ( s ) - S ( t - s ) u ( s ) for e a c h s E [a,t]. T h e n g i s a . e . differentiable on (a, t), its derivative belongs to Ll(a, t ; X ) , and

g'(s) - - A S ( t -

s)u(s) + S ( t - s)u'(s)

= -AS(t-

s)u(s) + S ( t - s ) A u ( s ) + S ( t - s)f(s) - S ( t - s)f(s) a.e. for s E (a, t). Since f E Ll(a, b; X), s ~-, S ( t - s)f(s) is integrable on [a, t ]. Integrating the above equality from a to t, we obtain (8.1.3).

[-1

From Theorem 8.1.1 we deduce the following uniqueness result. C o r o l l a r y 8.1.1. For each c~ E X and each f E L l ( a , b ; X ) , the problem (s.1.2) ol tion.

185

The Cauchy Problem u' - Au + f, u(a) -

The previous considerations justify why, in the case of infinite-dimensional spaces X, the variation of constants formula is promoted to the rank of definition. More precisely, we introduce" D e f i n i t i o n 8.1.3. The function u "[ a, b] ~ X, defined by (8.1.3) is called C o, or mild, solution of the problem (8.1.2). In all that follow, we focus our attention on some sufficient conditions in order that a C~ of (8.1.2) be a C 1, or strong solution of the same problem. We begin with an example which shows that, the continuity of f alone is not sufficient for the C~ given by (8.1.3) to be a strong one and, even less, a Cl-solution. E x a m p l e 8.1.1. Let A " D ( A ) C_ X --+ X be the infinitesimal generator of a C0-semigroup {S(t); t >_ 0}, for which there exists rl E X such that S(t)~ ~ D ( A ) for each t _> 0. (Such semigroup is that in Example 2.4.1 in which rl is a nowhere differentiable function). Let us define f ( s ) - S(s)r] for each s E [0, T] and let us observe that f is continuous. On the other hand, the problem (8.1.2) with ~ - 0 has no strong solution, and so it has no classical solution. Indeed, if u is a strong solution of the problem (8.1.2), in view of Remark 8.1.2, it is given by u(t) -

S(t-

s)S(s)rl ds - tS(t)rl

for each t C [0, T ], function which clearly is not a.e. differentiable on [0, T ]. T h e o r e m 8.1.2. Let A " D ( A ) C_ X ~ X be the infinitesimal generator of a Co-semigroup {S(t); t _> 0}, let f E L l ( a , b ; X ) be continuous on (a,b), and let v(t) -

/a

s(t -

f

for t C [a, b]. If at least one of the conditions below is satisfied

(i)

v

continuously

ei

ntiabl

on (a,

b);

(ii) v(t) C D ( A ) for each t C (a, b), and t ~ Av(t) is continuous on

(a,b) then, for each ~ C D ( A ) , (8.1.2) has a unique classical solution. If there exists ~ E D ( A ) such that (8.1.2) has a classical solution, then v satisfies both (i) and (ii).

P r o o f . Let us observe that, for each t C (a, b) and h > 0, we have 1 (S(h) I)v(t) v(t + h ) - v(t) ---h h

1 ft+h 1 S(t + hhjt

s ) f (s) ds.

(8.1.4)

186

The Nonhomogeneous Catchy Problem

Let us assume that (i) holds. Since f is continuous, and v is continuously differentiable on (a, b), it follows that the right-hand side of the equality above converges for h tending to 0. Hence, the left-hand side converges too, and consequently v(t) C D(A), and

v'(t) - Av(t) + f (t)

(8.1.5)

for each t C (a, b). If (ii) holds, then v is differentiable from the right on (a, b), and its right derivative is continuous on (a, b). Since v is clearly continuous, it follows that v is continuously differentiable on (a,b), and satisfies (8.1.5). Hence, in both cases (i) and (ii), v satisfies (8.1.5). Since v(a) = 0, it follows that, for each ~ C D(A), t ~ u(t) = S ( t - a)~ + v(t) for t C [a, b]is a classical solution of (8.1.2). Let us assume now that there exists ~ C D(A) such that (8.1.2) has a classical solution u which, in view of Remark 8.1.2, is given by (8.1.3). So, the function t ~ v(t) = u ( t ) - S ( t - a)~ is differentiable on (a, b), and in addition v'(t) = u ' ( t ) - S ( t - a)A~ is continuous on (a, b). Thus v satisfies (i). If ~ E D(A), we have S ( t - a)~ C D(A) for each t E [ a , b], and therefore it follows that v(t) = u(t) - S ( t - a)~ C D(A) for each t C (a, b), and t ~ Av(t) = A u ( t ) - A S ( t - a ) ~ = u ' ( t ) - f ( t ) - S ( t - a ) A ~ is continuous on (a, b). So, v satisfies (ii), and this achieves the proof. E:] C o r o l l a r y 8.1.2. If A : D(A) C_ X -+ X is the infinitesimal generator of a Co-semigroup {S(t); t _> 0}, and f is of class C 1 on [a~ b], then, fOP each E D(A), the problem (8.1.2) has a unique classical solution. P r o o f . Let us observe that

t ~+ v(t) -

fat S ( t -

s)f(s)ds -

~ot-a S ( s ) f ( t -

s)ds

is continuously differentiable on (a, b). Indeed, a simple calculation shows that

v'(t) - S(t - a)f(a) + = S(t - a)f(a) +

0t-a S ( s ) f ' ( t

- s)ds

S(t - s ) f ' ( s ) d s

for each t E (a, b). The conclusion follows from (i) in Theorem 8.1.2.

[3

C o r o l l a r y 8.1.3. If A : D(A) c_ X -+ X is the infinitesimal generator of a Co-semigroup {S(t); t _> 0}, f E L l ( a , b ; X ) is continuous on (a,b), f ( s ) E D(A) for each s C (a,b), and A f ( . ) E L l ( a , b ; X ) , then, for each E D(A), the problem (8.1.2) has a unique classical solution.

The Cauchy Problem u' = Au

+ f , u(a)= (

187

Proof. For each t E (a,b) and s E (a,t ] ,we have S ( t - s )f ( s ) E D ( A ) . Moreover, the function t I+ A S ( t - s )f ( s ) = S ( t - s ) Af ( s ) is integrable, and thus v , defined as in Theorem 8.1.2, satisfies v ( t ) E D ( A ) for each t E ( a ,b). In addition, the function

I'

t ~ + A v ( t ) = A S(t-s)f(s)ds=

I'

S(t-s)Af(s)ds

is continuous on [ a ,b ] . We conclude the proof with the help o f (ii) in Theorem 8.1.2. Theorem 8.1.3. Let A : D ( A ) & X -+ X be the infinitesimal generator of a Co-semigroup { S ( t ) ;t 2 0 ) , let f E L1(a,b ; X ) , and let v(t)=

l

S ( t - S ) f ( s )ds

for t E [ a ,b ] . If at least one of the conditions below is satisfied ( i ) v is a. e. diifferentiable on (a,b), and v' E L' (a,b ; X ) ; (ii) v ( t ) E D ( A ) a.e. fort E (a,b), and A v ( . ) E L 1 ( a , b ; X ) , then, for each J E D ( A ) , the problem (8.1.2) has a unique strong solution. If there exists J E D ( A ) such that the problem (8.1.2) has a strong solution, then v satisfies both ( i ) and (ii). As the proof of Theorem 8.1.3 is a copy o f that of Theorem 8.1.2, we do not enter into details. W e state, also without proof, two consequences o f Theorem 8.1.3 which are variants of Corollaries 8.1.2 and 8.1.3. Corollary 8.1.4. If A : D ( A ) X + X is the infinitesimal generator 0 ) , f is a.e. differentiable on ( a ,b), and of a Go-semigroup { S ( t ) ; t f' E L1(a,b ;X ) , then, for each E D ( A ) , the problem (8.1.2) has a unique strong solution.

>

Corollary 8.1.5. If A : D ( A ) C X -+ X is the infinitesimal generator of a Co-semigroup { S ( t ) ;t 2 O), f E L1(a,b ;X ) is continuous on ( a ,b), f ( s ) E D ( A ) a.e. for s E ( a ,b), and A f ( . ) E L1(a,b ;X ) , then, for each E D ( A ) , the problem (8.1.2) has a unique strong solution. Theorem 8.1.4. Let A : D ( A ) C X + X be the infinitesimal generator of a Co-semigroup { S ( t ) ; t 2 0 ) and let f E C ( [ a b, ] ; X ) .If at least one of the two conditions below is satisfied

(9 f E L1(a,b; D ( A ) ); (ii) f E W 1 > ' ( a , b ; X ) , then, for each [ E D ( A ) , the problem (8.1.2) has a unique classical solution.

The Nonhomogeneous Cauchy Problem

188

Proof. It suffices to consider only the case x = 0. So, we will prove first that, whenever f satisfies either (i), or (ii), the function

belongs to C1([a,b]; x). Let t E [ a , b) and h E (0, b - a). We assume first that f satisfies (i). We then have v(t

+ h) - v(t) = h

l

t

S(t -s)

S(h) - I f (s) ds h

lth+

+

S ( t h - S) f (s)ds.

Since IIS(h)x - xll 5 IIAxIlh for each x E D(A) and h dominated convergence theorem, it follows that S(h) - I f = Af h

lim hJ0

d+v -(t) dt

=

> 0, by the Lebesgue

/

a

t

S(t - s)Af(s)ds

+ f (t)

for each t E [a,b). Now, if we assume that f E W ' ~ ' ( U , ~ ; Xfor ) , each t E [ a , b) and h E (0, b - a ) , we have

Since lim

f(t+h

hi0

-

.) - f ( t - . ) h

= f,(.)

in L1(a, t; X ) and limS(h) hLO

($la+h

S ( t - s)f (s) ds)

= S ( t - a )f

(a),

we have

9

for each t t [ a , b). So in both cases (i) and (ii), E C([a , b); X ) . Similar E C ( ( a ,b]; X ) and thus v E C1([a,b]; x). The arguments show that proof is complete.

2

Smoothing Effect. The Hilbert Space Case

189

A Banach space X has the Radon-Nicod~)m property, if every absolutely continuous function f 9 [a, b] --+ X is almost everywhere differentiable on (a, b), f' C Ll(a, b ; X ) and

f (t) - f (a) +

f' (s) ds

for each t E [a, b]. We notice that all reflexive Banach spaces, as well as all separable duals have the Radon-Nicod#m property. See Diestel and Uhl [46], Theorem 1, p. 79 and Corollary 4, p. 82. C o r o l l a r y 8.1.6. Let A " D(A) C X --+ X be the infinitesimal generator of a Co-semigroup {S(t); t > 0} and let f " [a, b] --+ X be an absolutely continuous functions. If X has the Radon-Nico@m property then, for each e D(A), the problem (8.1.2) has a unique classical solution. P r o o f . Since f c w l ' l ( a , b ; X ) , we can apply Theorem 8.1.4 and this achieves the proof. D A specific form of Corollary 8.1.6, very useful in applications, is" C o r o l l a r y 8.1.7. Let A " D(A) C_ X --+ X be the infinitesimal generator of a Co-semigroup {S(t); t > 0} and let f 9 [a,b] --+ X be a Lipschitz continuous functions. If X is reflexive, for each ~ C D(A), the problem (8.1.2) has a unique classical solution. 8.2. Smoothing Effect. The Hilbert Space C a s e Let H be a Hilbert space. L e m m a 8.2.1. Let A " D(A) C H ~ H be a linear self-adjoint operator which generates a Co-semigroup of contractions. Let u C W 1 2 ( O , T ; H ) with u(t) e D(A) a.e. for t e [a,b], and Au e L 2 ( a , b ; H ) . Then the function t F-+ l (Au(t), u(t) ) is absolutely continuous on [a, b] and

d ( 1 ( A n ( t ) u ( t ) ) ) - (An(t) u'(t)) dt -2 ' ' "

(8.2.1)

P r o o f . Let ~ > 0 and let Ax E L ( H ) be the Yosida approximation of A. Obviously A), is a self-adjoint operator. By Theorem 1.3.4, we know that u is absolutely continuous on [0, T ], a.e. differentiable on (0, T), its derivative belongs to L2(a, b; H), and u is given by

]o u'(s)ds.

The Nonhomogeneous Catchy Problem

190 We have d(X

dt

-~(A~u(t), u(t))

)

1

1

- -~(A~u(t), u'(t)) + -~(A~u'(t), u(t)) = (A~u(t), u'(t)).

Integrating this equality from s to t, we obtain

1 -~(A~u(t), u(t)) - 1 (A~u(s), u(s)) -

(A~u(~), u'(~-)) dT.

By virtue of Lemma 3.2.1 and Lebesgue dominated convergence theorem, we can pass to the limit in this equality. We get

1(At(t) u ( t ) ) -

1 (At(s)u(s))-

(Au(~') u'(7-)) d~-

is for each a <__ s _< t _< b. Therefore, the function t ~ 89 absolutely continuous on [a, b], and (8.2.1) holds. The proof is complete. D The next theorem gives a sufficient condition in order that, for each ~ C X and each f C L2(a, b; X), the unique C~ of the problem (8.1.2) be a strong solution. 8.2.1. Let A : D(A) C_ H ~ H be the infinitesimal generator of a Co-semigroup of contractions {S(t); t >__0}. If A is self-adjoint then, for each ~ C X and f C L2(a,b;X), the unique C~ of (8.1.2) is strong. Moreover, the function t ~ (t-a)X/2u'(t) belongs to L2(a, b; H), the function t F-~ ~l(Au(t), u(t)) belongs to Ll(a, b), and, for each c C (a,b), is absolutely continuous on [c, b ]. If ~ C D(A), then, for each f G L 2 (a, b; X), the unique C~ of (8.1.2) is strong and satisfies u' G L2(a,b; H), and the function t ~-+ 1(At(t), u(t) ) is absolutely continuous on [a, b]. Theorem

P r o o f . From Theorem 6.4.1 we know that, for each ~ C X and t > 0, we have S(t)~ c D(A). This means that the function

is a.e. differentiable on [ a, b] and satisfies

u'(t) - A t ( t ) + f (t) a.e. for t C [a, b]. Let us take the inner product both sides in this equality by ( t - a ) u ' (t). Inasmuch as A is self-adjoint and generates a C0-semigroup, by virtue of Lemma 8.2.1, we obtain ld (t - a)I1~' (t)II 2 - (t - a)-~ -~ ((At(t), u(t))) + (t - a)(f (t), u' (t)).

Smoothing Eflect. The Hilbert Space Case

Integrating from a to b this inequality we get

Recalling that, by virtue of Lemma 6.4.2, (Ax, x) 5 0 for all x E D(A), we deduce and using the inequality (f (t),u1(t))5 f (t)112 ill~'(t)11~,

ill

But (Au(t17u(t)) = (ul(t) - f (t),'ll(t)) = therefore we have

+

(llu(t)112)

-

(f (t),

w,and

and

So, t ++(Au(t),~ ( t )is) in L' (a, b). Inasmuch as the semigroup generated by A is of contractions, from (8.1.3), we have

and accordingly

From this inequality we deduce the first part of the conclusion. In order to prove the second one, let us consider E D(A), and let us observe that, in this case, u' E L2(a,b ;H). Indeed, taking the inner product both sides in (8.1.1) by u', and integrating over [ a , b ] , we obtain

From this equality, the fact A is dissipative, and from the inequality

192

The Nonhomogeneous Cauchy Problem

we deduce

/a

114(t) II2 dt <_ - ( A ~ , ~) +

fa

Ill(t) II2 dt.

Then it follows that u' E L2(a, b ; H). Consequently, Au C L2(a, b ; H) too, and the conclusion follows from Lemma 8.2.1. The proof is complete. El 8.3. A n A p p r o x i m a t i o n R e s u l t

The next result shows that each C~ of the problem (8.1.2) can be approximated uniformly on [a, b] by classical solutions of certain suitably defined approximating Catchy problems. T h e o r e m 8.3.1. Let A : D(A) C_ X --+ X be the infinitesimal generator of a Co-semigroup of type (M, w), let f C Ll(a, b ; X ) and ~ E X . Then there exist a sequence (~n)ncH in D(A) and a sequence (fn)ncN in c l ( [ a , b ] ; X ) such that / l i m n ~ ~n -- ~ strongly in X strongly in Ll(a, b; X ) limn-,cc fn - f limn~cc Un(t) - u(t) uniformly on [a, b], where Un is the unique classical solution of (8.1.2) corresponding to ~n and fn, and u is the unique C~ of the problem (8.1.2). Proof. As, by Theorem 2.4.1, D(A) is dense in X, for each ~ C X there exists (~n)nEN such that limn--+c~n -- ~. Moreover, since c l ( [ a , b ] ; X ) is dense in L l ( a , b ; X ) , for each f C L l ( a , b ; X ) , there exists (fn)ncN such that limn--~oofn = f. By virtue of Corollary 8.1.2, for each n C N, the Catchy problem +fn ~t 'n - A n n ~n (a) --

~n

has a unique classical solution given by the variation of constants formula Un(t) - S ( t - a)~n +

S ( t - s)fn(s)ds.

See Remark 8.1.2. But the semigroup is of type (M, w) and thus ]]S(t)ilc(x) <_ M e wt for each t E IR+. We then have

Ilu(t)

- Un(t)ll ~_ I[S(t - a)(~ - ~n)ll-~-

_< IIS(t - a ) l l ~ ( x / l l ~

- ~nll +

[IS(t -- s ) ( f (s) -- fn(s))ll ds

IIS(t -- s ) l l ~ ( X / l l l ( s )

-- f ~ ( s ) l l d s

Compactness of the Solution Operator from LP(a, b ; X)

<--Mew(t-a)I1~ <-- Melwll(b-a)

193

- ~nll + M fat e~(t-~) IIf (s) - fn (s)II ds

(

/ab II~-~nll

+

II1(~) - fn(~)II d~) [:]

inequality which completes the proof. 8.4. C o m p a c t n e s s of t h e S o l u t i o n O p e r a t o r f r o m LP(a, b; X )

We start with a necessary and sufficient condition in order that a given set of C~ be relatively compact in C([a, b]; X). Let us consider the nonhomogeneous Catchy problem u' - Au + f u(a) - ~,

(8.4.1)

where A ' D ( A ) C_ X --+ X generates a C0-semigroup of contractions, { C X and f C Ll(a, b ; X ) . We notice that, although we confine ourselves only to the case of C0-semigroups of contractions, all the results here included hold true in general, i.e. for C0-semigroups of type (M, w). Definition 8.4.1. The operator Q" X x L l ( a , b ; X ) -+ C ( [ a , b ] ; X ) ,

defined by Q([, f) - u, where u is the unique C~ (8.4.1) corresponding to ~ E X and f E Ll(a, b; X ) , i.e. u(t) - S(t - a)~ +

of the problem

S(t - s) f (s) ds

for each t C [0, T], is called the solution operator attached to the problem (8.4.1). R e m a r k 8.4.1. The operator Q is nonexpansive and so, it maps each bounded subset in X x L 1 (a~ b ; X ) into & bounded subset in C([ a, b]; X). Indeed, we have ]](Q(~, f ) ( t ) - (Q(~, g)(t)]] <_ I I s ( t - a)~ - S ( t - a)~ll +

<-- I1~ - vii +

IIS(t - s ) ( f ( s )

- g(s))ll d s

IIS(t - s)II~(x)IIf (s) - g(s)II ds

< I1~ - VII +

/a

[If(s) - g(s)II as

194

The Nonhomogeneous Cauchy Problem

for each (~,f),(rl,9) E X x L l ( a , b ; X ) and each t E [a,b]. supremum for t E [a, b] in this inequality, we deduce

Taking the

IIQ(~, f) - Q(~7, g)Iloo _< II~ - r/ll -t-Ilf - gllLl(a,b ;X), which proves the assertion. Next, we shall concentrate our attention on some compactness properties of the operator Q. T h e o r e m 8.4.1. Let A : D(A) C_ X ~ X be the generator of a Cosernigroup of contractions {S(t); t > 0}, let 9 be a bounded subset in X , and 9: a uniformly integrable subset in L l(a, b ; X ) . Then Q(~), ~) i8 relatively compact in C([c, b]; X ) for each c E (a, b), if and only if there exists a dense subset D in [a, b] such that, for each t E D, the t-section of Q( 9 :~), i.e. Q( 9 9")(t) = {Q(~, f)(t) ; (~, f) E 9 x 9"}, is relatively compact in X . Moreover, if the latter condition is satisfied, and a E D, then Q( 9 9=) is relatively compact in C([ a, b]; X). P r o o f . By virtue of Theorem A.2.1, the necessity is obvious. In order to prove the sutticiency, we shall make use again of Theorem A.2.1. To this aim, we shall show that Q( 9 9") is equicontinuous on (a, b ]. Let e > 0. Since 9" is uniformly integrable, there exists 7(e) > 0 such that

fE iif(~)ii d~ _< C for each measurable subset E in [a, b] with #(E) _< -),(e), and uniformly for f E 9:. We recall that # is the Lebesgue measure on R. Let t E (a, b] and let us fix ,~ = ,~(e) > 0 such that t - ,~ > a, 2A _< 7(e) and t - ,~ E D. This is always possible because D is dense in [a, b ]. Inasmuch as Q( 9 9 : ) ( t - A ) is relatively compact in X, for each e > 0, there exists a finite family { (~1, fl), (~2, f 2 ) , - . . (~k(e), fk(e))} in 9 x 9: such that, for each (~, f) E 9 x 9:, there exists i E {1, 2 , . . . k(e)} with the property IIQ(5, fD(t - A) - Q(~i, fiD(t - ~)11 _ c. On the other hand, the family {Q(~I, fl), Q(~2, f2),... Q(~k(e), fk(e))} is equicontinuous at t, being a finite family of continuous functions on [a, b ]. Therefore, for the very same e > 0, there exists 5(e) E (0, ,~ ], such that

IIQ(~, f~)(t + h) - Q(~, f~)(t) II _< c for each i = 1, 2 , . . . k(e), and each h E R with Ihl _< 5(c). We then have IIQ(~, f ) ( t + h) - Q({, f)(t)]] < IIQ(~, f ) ( t + h) - Q(~, f~)(t + h)ll

+IIQ(~, f~)(t + h) - Q({~, f~)(t)ll + IIQ(~,

f~)(t) - Q ( 5 ,

f)(t) II.

Compactness of the Solution Operator from LP(a, b ; X )

195

Since for each ( 7 , g ) E D x 3 and each h E ( - S ( E ) , S ( E ) ) , we have

from the preceding inequality, we get

for each (I f, ) E D x 3 and each h E ( - S ( E ) , S ( E ) ) . From this inequality and from the manner of choice of both X > 0 and S ( E ) > 0, it follows

IIQ(t, f ) ( t + h ) - Q(t,f )(t)ll 5 7~ x 3 and each h E R with lhl < a(&). Therefore Q ( D , 3 )

for each (I,f ) E D is equicontinuous on (a,b ] , and, by virtue of Theorem A.2.1, it is relatively compact in C ( [ c , b ] ; X )for each c E (a,b). If a E D, then Q ( D , 3 ) is equicontinuous at a as well, and this achieves the proof.

The Nonhomogeneous Catchy Problem

196

R e m a r k 8.4.2. The condition of uniform integrability which circumvents the general frame of Theorem 8.4.1 cannot be relaxed as we can see from the simple example below. Let A : IR --+ I~ be defined by An - 0 for each u C I~, let 9 = {0} and let 9~ = {fn; n E H*}, where, for each n C H*, fn :[ 0, 1] ~ I~ is defined by fn(t)

-- nt n-1

for t E (0, 1). A simple calculation shows that Q(iD,9") = {un; n c H*}, where for each n C N*, un :[ 0, 1] -+ I~ is defined by ~(t)

= t~

for each t C[0, 1 ]. So, Q( 9 9:) is not relatively compact in C([ 0, 1]) even though it satisfies (ii) in Theorem 8.4.1 with D = [0, 1 ]. We emphasize that this is a consequence of the fact that 9~ lacks uniform integrability "near" t = 1". On the other hand, Q( 9 is relatively compact in LP(a,b) for each p C [1, +c~). This observation suggests the possibility to get a variant of Theorem 8.4.1 concerning the relative compactness of Q( 9 9") in LP(a, b;X), by assuming only the boundedness of 9" in L~(a, b;X). We notice that this is the topic of Section 8.6. An extremely useful consequence of Theorem 8.4.1 is the following sufficient condition of relative compactness of the set Q( 9 9"). 8.4.2. Let A : D(A) C_ X ~ X be the generator of a compact Co-semigroup of contractions, let ~ c X, 9 = {~} and let ~ be a uniformly integrable subset in L l ( a , b ; X ) . Then Q( 9 is relatively compact in C([a,b];X).

Theorem

P r o o f . By virtue of Theorem 8.4.1, it suffices to check that, for each is relatively compact in X. Let t C (a,b) and let > 0 with t - )~ > a. We have

t C (a,b), Q( 9

Q(~,f)(t) - S ( t - a ) ~ + S ( X )

fa

S(t-)~-s)f(s)ds+

S(t-s)f(s)ds. )~

Since S()~) is compact, from Remark 8.4.1, we deduce that the operator Px : Q( 9 ~ X, defined by

j~at-'~

P~Q(~, f)(t) - s ( t - a)~ + S()~)

S ( t - )~- s)f(s) ds,

maps the set Q( 9 9")(t) into a relatively compact subset in X. In addition, from the preceding relation and from the uniform integrability of 2F, it follows that lim I I P ~ C ( r f ) ( t ) - Q ( ~ , f)(t)II - 0

The Case when (AI - A)- 1 is Compact

197

uniformly for f e 9". From Lemma A.1.2 it follows that Q(D,9~)(t) is relatively compact in X for each t C [a, b]. An appeal to Theorem 8.4.1 completes the proof. KI

8.5. The Case when ( A I - A) -1 is Compact Let A " D ( A ) C_ X ~ X be the infinitesimal generator of a C0-semigroup of contractions {S(t) ; t _> 0}, ~ E X, f E Ll(a, b ; X ) and let us consider the Cauchy problem (8.5.1)

u ' - Au + f u(a) - ~.

Let Q" X • L l ( a , b ; X ) ~ to (8.5.1), i.e.

C ( [ a , b ] ; X ) be the solution operator attached

Q(~, f )(t) - S(t - a)~ +

S(t - s) f (s) ds.

See Definition 8.4.1. Our aim here is to prove a necessary and sufficient condition in order that Q(D, ~) be relatively compact in C([ a, b]; X). The main result in this section refers to the case in which, for each ~ > 0, ( A I - A) -1 is a compact operator. T h e o r e m 8.5.1. Let A " D(A) c_ X -~ X be the infinitesimal generator of a Co-semigroup of contractions with ( ) ~ I - A) -1 compact for each ~ > 0, let D be a bounded subset in X , and g: a uniformly integrable subset in L 1 (a, b ; X ) . T h e n Q(D, ~) i8 relatively compact in C([ a, b ]; X ) if and only if it is equicontinuous from the right at each t E [ a , b) P r o o f . Necessity. By virtue of Theorem A.2.1 the necessity is evident. Sufficiency. Let D be a bounded subset in X, 9" a uniformly integrable subset in Ll(a, b;X), and let us assume that Q( 9 9") is equicontinuous from the right on [a, b). According to Remarks A.5.1 and 6.3.1, Q(D, 9") is bounded in C([a, b]; X). More precisely, there exists M > 0 such that

I]Q(~, f)(t)ll _< M

(8.5.2)

for each ( ( , f ) E D • ~ a n d t E [a,b]. Let t C [a,b), and let ~ > 0 w i t h t + A < b. Let us define I ~ ' Q ( D , 5)(t) ~ X by I~Q((, f ) ( t ) - s163 A)Q(~, f ) ( t ) fo~ e~ch (~, f) ~ ~) • ~. In view of (S.5.2) ~nd of t5~ r of th~ operator R()~; A), it follows that I ~ Q ( D , 5 ) ( t ) ) is relatively compact. We shall show that

lira III~Q(5,

/~----> CX:)

f)(t)

-

Q(5, f)(t)ll- 0

(8.5.3)

The Nonhomogeneous Cauchy Problem

198

uniformly for f E 3 . Indeed,

At this point, let us observe that l l s ( ~ ) Q ( If, ) ( t ) - & ( I , f )(t)ll I l l s ( ~ ) Q ( Sf, ) ( t ) - &(I, f )(t

+ 7)II

+ l I & ( < , f ) ( t + ~ ) - Q(<,f)(t)ll (8.5.5) for each ( I , f ) E 'D x 3, and T > 0 with t T 5 b. Since the family Q ( ' D , 3 ) is equicontinuous from the right at t , for each E > 0 there exists S = S(E) > 0 such that, for each ( I , f ) E 2) x 3, and each T E ( 0 , S), we have IIQ(I, f ) ( t + 7 ) - Q ( I , f )(t)ll 5 E. As

+

I l s ( ~ ) Q ( I , f ) ( t-) Q ( I , f ) ( t +

411 5 [+'

Is@+

-

o ) l l ~ ~ (~o )~l l dl el f

for each ( [ ,f ) E 'D x 3, and each T > 0 with t + T 5 b, by virtue of the uniform integrability of 3, for the very same E > 0 , there exists y ( ~ >) 0 , with t Y ( E ) 5 b, and such that

+

l l S ( T ) Q ( I , f ) ( t ) - & ( I , f ) ( t + .)I1 5 & for each f E 3 and each T > 0 with T 5 Y ( E ) . Let us fix p p 5 m a x { S ( ~ )y, ( ~ ) ) From . (8.5.2), (8.5.4), (8.5.5), it follows

> 0 with

Passing to the limsup for X tending to +m in the inequality above, we obtain I2 ~ , limsup I l I x Q ( I , f ) ( t ) - Q ( I , f X+m

Compactness of the Solution Operator f ~ u from Ll(a, b ; X)

199

uniformly for (~, f) C 9 • 9". Since c > 0 is arbitrary, this implies (8.5.3). But (8.5.3), along with the relative compactness of the sets I~Q( 9 for each ,~ > 0 with t - ,~ > a, shows that we are in the hypotheses of Lemma A.1.2. Therefore, for each t E [a,b), Q( 9 is relatively compact. An appeal to Theorem 8.4.1 completes the proof. D 8.6. C o m p a c t n e s s of t h e S o l u t i o n O p e r a t o r f r o m L 1(a~ b ; X) The aim of this section is to prove a necessary and sufficient condition in order that a family of C~ be relatively compact in L P ( a , b ; X ) for each p E [1,+oc). More precisely, let A " D(A) C_ X --+ X be the infinitesimal generator of a C0-semigroup of contractions {S(t) ; t _> 0}, let ~ c X and f C L l ( a , b ; X ) . We denote by u - Q ( ~ , f ) the unique C~ of the nonhomogeneous Catchy problem

{

u(a) -

(8.6.1)

The main result in this section is: T h e o r e m 8.6.1. Let A" D(A) C_ X --+ X be the infinitesimal generator of a Co-semigroup of contractions {S(t) ; t _> 0} and let 9 x ~ be a bounded subset in X x L 1(a, b ; X ) . Then Q( 9 9~) is relatively compact in LP(a, b; X ) for each p E [1, +oc) if and only if for each c > 0 there exists a relatively compact subset C~ in X such that, for each (~, f) C 9 x ~ there exists a subset EE,~,f in [ a, b] whose Lebesgue measure is less that c, and such that Q(~, f)(t) c C~ for each (~, f) E 9 x 9~ and t C [a, b] \ E~,~,f. P r o o f . Necessity. By Theorem A.5.1 the necessity is obvious. Sufficiency. We shall use again Theorem A.5.1. First, let us observe that, by virtue of Lebesgue dominated convergence theorem, it suffices to show that Q( 9 9=) is relatively compact in Ll(a, b ; X ) and bounded in L~(a, b ; X ) . To this aim, let us recall that there exist m~) > 0 and rag= > 0 such that I]~II-< m~) and ]]f]]Ll(a,b;X) ~ 77~y~ (8.6.2) for each (~, f) G 9 • 5. Then, by virtue of Remark 6.3.1, we have I[Q(~, f)(t)ll-< m~ + m y

(8.6.3)

for each (~, f) E 9 • 9" and t e [ a , b]. We will prove next that Q( 9 9") is l-equiintegrable. To this aim let ~ > 0, let (~, f) C 9 • ~" and let C~ and E~,~,f be the sets with the properties mentioned in the hypotheses. We have

fab-h IIQ(5, f ) ( t

+ h) - Q(~, f)(t)I j dt

The Nonhomogeneous Cauchy Problem

200

~a b- h <+

/a

IIQ(r f ) ( t + h) - S(h)Q(r f)(t)ll dt

[IS(h)Q(~, f)(t) - Q(~, f)(t)[ I dt ~_

+[

,b]\E~,~,Z

+f

dE

~

iiS(h)Q( , f ) ( t )

-

.It

IIf(s)ll ds dt

f)(t)lld t

IIS(h)Q(~, f)(t) - Q(~, f)(t)] I dt

hllfiiLl([a,b];X ) -F [ I[S(h)Q(~, f)(t) - Q(~, f)(t)]] dt + 2M9:r J[a ,b]\E~,~,S

for each f E 9" and h E (0, b - a ], where M s - m~) + rag=. As Q (~, f) (t) E C~ for each (~, f) E 9 x 9 =, and each t E [a, b] \ E6,~,f, and Cc is relatively compact in X, there exists 5(c) E (0, b - a ] such that, for each h E (0, 5(c)]

]IS(h)Q(~, f)(t) - Q(~, f)(t)l ] _< E, uniformly for f E E and t E [ a , b ] \ EE,~,f. Then, taking into account (8.6.2), (8.6.3) and the preceding inequalities, we obtain

~a b-h ]]Q(~, f ) ( t + h) - Q(~, f)(t)l I dt < (b - a + rag: + 2M9:)c for each f E E and h E (0, 5(E)] N (0, ~]. Obviously, this relation shows that Q(9 is 1-equiintegrable. From Theorem A.5.1 it follows that it is relatively compact in L i ( a , b ; X ) . An appeal to Lebesgue dominated convergence theorem completes the proof. [::] Two remarkable consequences of Theorem 8.6.1 are listed below. C o r o l l a r y 8.6.1. Let A " D(A) C X --+ X be the infinitesimal generator of a Co-semigroup o] contractions and let 9 x 2: be a bounded subset in X x L i ( a , b ; X ) . If {Q(~,f)(t); (~,f) E 9 x ~T, t E [a,b]} is relatively compact in X , then Q( 9 ~) is relatively compact in LP(a, b; X ) , for each p E [1,+oo). C o r o l l a r y 8.6.2. Let X be a finite-dimensional Banach space, A E L ( X ) and let 9 x E be a bounded subset in X x L l ( a , b ; X ) . Then Q(9 is relatively compact in LP(a, b; X), for each p El1, +c~). The next lemma, which is closely related to Egoroff uniform asymptotic convergence theorem (see Dunford and Schwartz [49], Theorem 12, p. 149), will prove useful later.

Compactness of the Solution Operator f ~ u from Ll(a, b ; X)

201

L e m m a 8.6.1. Let U C Ll(a, b; X ) be nonempty and bounded, and let

{Qh " U --+ L l ( a , b ; X ) , h e (0, b - a]} be a family of (possible nonlinear) operators such that lim h$0

IIQhu(t)ll dt - 0

uniformly for u C U. Then, for each e > O, there exists a sequence (hn)ncN, decreasing to O, with the property that, for each u E U, there exists a subset EE,u in [a, b] whose Lebesgue measure p(Ee,u) is less than e, and lim Qh~ u(t) -- O,

n--+oc

uniformly for u C U and t C [a, b f \ EE,u. P r o o f . Let c > 0 and let us choose two sequences (hn)neN and ( a n ) h e N , both decreasing to 0, such that

y ~ an <_ c and

fb

Ja

2

(8.6.4)

IIQh~u(t)ll d t < a n

n--O

for e a c h n C N a n d u C U .

ForuCUandnC

E n - {t e [a, b f; IIQhnU(t)]l ~_ an}

N let us define and E u -

UE~. nEN

By virtue of the second relation in (8.6.4), it follows that #(E~) < an for each n E N, and therefore #(E~,u) < c. From the definition of EE,~, we deduce that IIQh~u(t)ll < an, for each n C N, u C U and t C [a, b f \ E~,~ and this completes the proof. E:] T h e o r e m 8.6.2. Let A : D(A) C X --+ X be the infinitesimal generator of a compact Co-semigroup of contractions. Then, for each bounded subset is 9 x 5: in X x L l ( a , b ; X ) , and for each p C [1, +oc), the set Q ( 9 relatively compact in LP(a, b; X ) . P r o o f . We will show that the set Q( 9 9") satisfies the hypotheses of Theorem 8.6.1. To this aim, let us observe that

fa

b IIQ(~, f)(t) - S(h)Q(~, f ) ( t - h)l I dt

+h

<_

IIf (s)l] ds dt <_ h IlfllL~(a,b;X), +h

h

The Nonhomogeneous Catchy Problem

202

Since 9" is bounded in Ll(a, b ; X ) ,

for each ({, f) e 9" and h C (0, b - a ] . we have

IIQ(~, f ) ( t ) - S(h)Q(~, f ) ( t - h)ll dt - O,

lim h$0

(8.6.5)

+h

uniformly for (~, f) E 9 x 9". Let us define U - Q( 9 9~) and Qh" U ~ LI(a, b ; X) by

(Qhu)(t) -

0

iftE[a,a+h]

u(t) - S ( h ) u ( t - h),

if t E (a + h, b]

for u C U. From (8.6.5) it follows that Lemma 8.6.1 applies, and therefore for each c > 0 there exists E~,u C_ [a,b] with #(E~,u) < c and (hn)ncI~ decreasing to 0 such that lim lit(t) - S ( h n ) u ( t -

n---+cx~

h n ) l l - O,

uniformly for u C Q( 9 and t E [ a , b ] \ E~,u. Inasmuch as S(hn) is compact for each n C N and {u(s); u E Q( 9 s c [a, b]} is bounded in X, it follows that the set Cs - {u(t); u e Q( 9 2"), t e [a, b] \ Ee,u} is relatively compact in X. Consequently we are in the hypotheses of Theorem 8.6.1. So Q( 9 is relatively compact in LP(a, b ; X ) for each p C [1, +c~), and this achieves the proof. [3 Problems P r o b l e m 8.1. Let A ' D ( A ) C_ X -+ X be the infinitesimal generator of a C0-semigroup of contractions in X, ~ C X, and f C LI(o, + ~ ; X ) . Prove that the nonhomogeneous equation

u'-

Au + f ( t ) -

can be equivalently rewritten as a homogeneous equation of the form U(O)

-

Uo

in the Banach space 2 : - X x LI(O, + ~ ; X), endowed with the norm

II(u, f)llx -Ilull + IlfllL (0,+ ;x/

Problems

203

where Uo = (~, f), U(t) = (u(t), f ( t + . ) ) , and A : D(A) C X --+ X is defined by

D ( A ) - {(~,f) E 2:; ~ E D(A), f C wl'l(0,+oc;X) ~4(~, f) - (A{ + f(0), f') for each ({, f) C D(~4). Prove that A is the infinitesimal generator of a C0-semigroup of contractions in 3:. P r o b l e m 8.2. Let a C R 3, u0 C Lp(~3), and f E LI(lt~+; Lp(R3)). Find the solution of the Cauchy problem for the nonhomogeneous transport equation ut+a'Vu-f(t,x)

(t,x) e I ~ + x I ~ 3 x e R 3.

P r o b l e m 8.3. Let A be an n x n matrix with real entries, uo C Lp(~n), and let f C LI(I~+; L p (R n)). Solve the Cauchy problem

{ ~tt + (fl[X, V~t)

f(t, X) (t, X) e ]~+

]~n

x e Rn

Let X - L ~ ( R ) be the space of a.e. equivalence classes of measurable functions from I~ to I~, periodic of principal period 2~r, and square integrable on [0, 27r ]. Endowed with the inner product and with the norm of L 2 (0, 27r), this is a real Hilbert space isometric to L2(0, 27r). P r o b l e m 8.4. Let X - L2~(]~), and let A ' D ( A ) c X -+ X be defined by D ( A ) - {u e L22~(I~);u ' e L~(]~)} A u - u' for each u C D(A). Prove that, for each A E IR+, ( A I - A) -1 is a compact operator. Let 9" = {sin(n(. + x)); n C N*}, and let Q(3") the set of all Co solutions of the problem u'-Au+f

-o, corresponding to f C 9~. Show that Q(9") is not relatively compact in C([0, 1 ]; X), although 9" is uniformly integrable in L~(0, 1; X). This shows that, in Theorem 8.4.2, the compactness of the C0-semigroup is essential. P r o b l e m 8.5. Let A : D ( A ) c_ X --+ X be the infinitesimal generator of a C0-semigroup of contractions. If ( A I - A) -1 is compact for some A > 0, then, for each ~ E X and f C L I(]~+; X), the orbit {u(t) ; t _> 0}, of the C~ u of the Cauchy problem u'is relatively compact in X.

Au + f ( t )

204

The Nonhomogeneous Cauchy Problem

N o t e s . The results in Section 8.1 are classical and were adapted mainly from Pazy [101] and Brezis and Cazenave [31]. Theorem 8.1.4 is from Cazenave and Haraux [36]. The concept of C~ known mainly under the name of "mild solution" proved very useful in the study of many problems of practical interest. For details, see Barbu [20] and Pavel [97]. Theorem 8.4.1 was proved by Vrabie [125] in the general case in which A generates a contraction semigroup, possible nonlinear, and Theorem 8.4.2 is in Baras, Hassan and Veron [14]. We notice that an L ~ variant of this result was established independently by Pazy [99] in 1975, by using a proof very similar to that in [14]. Nonlinear versions of this compactness result were obtained by Baras [13] and Vrabie [124]. Theorem 8.5.1 is a "linear" consequence of a general result in Mitidieri and Vrabie [90], Theorem 8.6.1 is also a specific case of a result in Vrabie [128], and Theorem 8.6.2 is from Baras, Hassan and Veron [14]. As concerns the problems here included, the first one is a particular case of a classical trick due to Dafermos and Slemrod, while the other ones are simple applications of the abstract results in this chapter.

CHAPTER 9

Linear Evolution

Problems

with Measures

as D a t a

In this chapter we consider the abstract nonhomogeneous Cauchy problem du - {Au}dt + dg ~(a)

-

~,

where A : D(A) c_ X --+ X is the infinitesimal generator of a C0-semigroup of contractions, ~ C XA, and g E B V ( [ a , b]; XA), while XA is a space of abstract measures including X**. Namely, we introduce the concept of L~-solution and we study the compactness properties of the L~-solution operator (~,g) ~ u, which will prove useful in obtaining existence results for some classes of semilinear Cauchy problems involving measures.

9.1.

The Problem

du - { A u } d t + dg, u(a) -

Let X be a Banach space, A " D ( A ) C_ X --+ X the infinitesimal generator of a C0-semigroup of contractions on X , ~ C X , g C B V ( [ a, b l; X), and let us consider the Cauchy problem du - { A u } d t + dg

~(a)

--

(9.1 1)

~.

In order to give a precise meaning of (9.1.1), we begin with the concept of the so-called L ~ - s o l u t i o n (see Definition 9.1.1 below), which is inspired by the variation of constants formula. To this aim, let t C (a, b ], T C T[ a, t ], ~P" a - to < tl < " " < tk -- t, let ~-/ C [ t / - 1 , t / ] , i -- 1 , 2 , . . . , k , and let us consider the Riemann-Stieltjes sum of ~" ~ S ( t - T) over [a, t] with respect to g, i.e. k

~Ia,~l(~' S' g' ~) -- Z S ( t - ~) (g(t~) -- g(t~_,)). i=1

Let { S ( t ) ~ ; t _ 0} be the sun dual semigroup (see Definition 3.8.2), and let us observe that, for each x ~ E X | and each e > 0, there exists 5(x G, c) > 0 205

Linear Evolution Problems with Measures as Data

206

such that, for each IP C [P[ a, t] with A(iP) - m a x { t / - ti-1 ; i - 1 , 2 , . . . , k } _~ (~(x~ we have

[(~[a,t](~,S,g,~) --~[a,t](~,S,g,O~),x| for each 7i,0i E [ti-1, ti ], i -

<_ C,

(9.1.2)

1 , 2 , . . . ,k. Indeed, let us remark that

I(~[a,t] (~, S, g, ~) - ff[a,t] (~, S, g, 0i), xG)I k E (g(ti) -- g(ti-1)~ ( S ( t -

7i) | - s ( t - e~)|

|

i--1

_<

sup

(g, [a, t ] ) .

{ l l ( S ( t - 7~)o _ s ( t - 0 ~ ) ~ 1 7 6

i=1,2,...,k

Since the sun dual semigroup is a C0-semigroup of contractions, it readily follows that, for each x ~ C X ~ and ~ > 0, there exists (~(x~ c) > 0 such that, for each Ti, 0i E [ a , t] with I T / - 0il _< (~(x*~ c), we have c

II(S(t - 7i)* - S ( t - O i ) ~

<_ V a r ( g , [ a , t ] ) '

and this completes the proof of (9.1.2). But (9.1.2) shows that the mapping 7 ~ S ( t - T) satisfies the usual Catchy condition of dg-integrability on [a, t] in a(X, X ~ called the weak-Q topology on X. Let X A -- ( X ~ *. We obviously have X** C_ X A and the arguments above show that there exists a unique element f ta S ( t - s)dg(s) C X A such that t

k

fa S(t - s)dg(s) -

lim E

S(t - 7i)(g(ti) - g(ti-1))

)~([P)$0 i - 1

weakly-@ and it is called the Riemann-Stieltjes integral on [a,t] of the operator-valued function ~- ~ S ( t - T) with respect to the vector-valued function g. If c~" [a, b] ~ I~ is a given function, by a similar procedure, we can define t

~a (~(s)S(t - s)dg(s) -

k

lim E (~(7i)S(t - 7i)(g(ti) - g(ti-1)), ~(~)4o i=~

(9.1.3)

of course, whenever the limit on the right-hand side exists in the weak-q) topology on X. It it easy to see that this happens, for instance, if c~ is the characteristic function of a proper subinterval o f [ a , t ].

207

The Problem du = {Au}dt + dg, u(a) =

R e m a r k 9.1.1. An elementary computational argument shows that, for each c E [a, b) and each ~ > 0, such that c+~ C [a, hi, and each t C [c+~, b ], we have s(t

-

dg(

) -

JC

-

rig(s)

JC

(9.1.4)

+ S ( t - c)(g(c + O) - g(c))~

where X(c,c+5] denotes the characteristic function of (c, c +(~ ]. Similarly, for each c E (a, b], and each 5 > 0, such that c - ~ C [a, hi, and each t c [ c , b], we have s(t

-

dg(

) -

(s)S(t

-

dg(

+ S ( t - c)(g(c) - g(c - 0)).

)

(9.1.5)

R e m a r k 9.1.2. If X is reflexive the weak-| topology on X is nothing else than the weak topology on X, and therefore X A = X . In general this is not the case as the following simple example shows. E x a m p l e 9.1.1. Let X = LI(]R) and let {S(t) ; t > 0} be the C0-group of translations, i.e. ( S ( t ) f ) ( x ) = f ( x + t) for each f E X, each t C IR, and a.e. for x CIR. It is well-known that, in this case, X ~ = Cub(R), i.e. that space of all bounded, uniformly continuous functions on IR, space endowed with the usual sup-norm. See Example 3.8.1. At this point let us observe that the weak topology on X does not coincide with the weak-| topology. More than this, X is not sequentially weakly| complete, and its sequential completion is a space of measures strictly larger that X. One may easily verify that all Dirac measures belong to X A . R e m a r k 9.1.3. If g is defined by a density, i.e. there exists a function f C L l ( a , b ; X ) such that dg(s) = f ( s ) d s , then, for each t C [a,b], we have

/a

S(t-

s)dg(s) -

/a

S(t-

s ) f (s) ds e X .

From Remarks 9.1.2 and 9.1.3, it became clear that, in order for the Riemann-Stieltjes integral defined above to belong to X, we must impose some extra-conditions on X, on g, or on A. Since in applications X is a space with quite bad geometric properties, as Ll(Ft) for instance, and thus nonreflexive, and g is not a priori known being the "weak limit" of a certain sequence of Ll-functions which excludes the situation in Remark 9.1.3, in that follows, we will mainly focus our attention only on the properties of A which may ensure the existence of a "good integral". The next theorem

208

Linear Evolution Problems with Measures as Data

gives a useful sufficient condition in this respect. The simple remark below will prove useful in that follows. R e m a r k 9.1.4. If g E B V ( [ a, b]; X), then g([ a, b]) is strongly relatively compact in X. This is an easy consequence of the fact that g is piecewise continuous on [a, b] (see Proposition 1.4.2). T h e o r e m 9.1.1. If {S(t) ; t > 0} is continuous from (0, +oc) to L ( X ) in the uniform operator topology, then, for each g E B V ( [ a , b ] ; X ) and each t E (a, b), the limit in (9.1.3) exists in the norm topology of X and consequently we have ~a t S(t - s)dg(s) E X.

(9.1.6)

P r o o f . It suffices to show that, for some fixed t~ > 0, for each t E (a, b ], and each e > 0, there exists ~?(e) > 0, such that

I[(Y[a,t] (~, ~, g, Ti) -- O'[a,t] ([P', ~, g, 7]) II ~ ~c~ provided )~([P) _< ~(e) and [P' is finer than IP, i.e. contains all the points of [P. To fix the notation, let iP E [P[a,t], [P : a = to < tl < " " < tk = t be a partition of [ a, t ], and let ~-~ be arbitrary in [ t~, t~+l ], i -- 0, 1 , . . . , k - 1. If [P' is another partition of [ a, t ] which is finer that [P, relabelling if necessary, we have T' : ti = ti,o <_ 7i,o <_ ti,1 <_ Ti,1 _<,-.., _< Ti,mi-1 <_ ti,m~ -- ti+l, i = 0, 1 , . . . , k 1. Let e > 0, and fix 6 > 0 with t - 26 E [a,b], and such that II(S(s) - I)(g(~) - g(~-'))[I <- e (9.1.7) for each s E [ 0 , 6] and each T, ~-' E [ a , t), and Var (g, [t - 26, t)) _< e.

(9.1.8)

This is possible because the semigroup is strongly continuous, g([ a, t)) is strongly relatively compact in X (see Remark 9.1.4) while, by virtue of L e m m a 1.4.1, lim550 Vat (g, [ t - 5, t)) - O. Since ~- ~ S(~-) is continuous at 7- = 5 > 0 in the uniform operator topology, there exists ~ E (0, 5] such that - s( )ll <_ ~ (9.1.9) for each s E [0,~]. Let us assume now that A([P) <__~? and let us fix p = p((~) E {1, 2 , . . . , k - 1} such that tp < t - 6 and tp+l >_ t - 6. Let us denote by

h-1 mi-1 s h -- E E (~(t - Ti) -- S ( t i=n j=O

Ti,j))(g(ti,j+l) -- g(ti,j)),

The Problem du - {Au}dt + dg, u(a) -

209

and let us observe that

(Y[a,t](~, S, g, Ti) - a[a,t](~', S, g, Ti,j) Set Sko -- T1 + T2 + T3, where T1 - S~, T2 - Skp - T3, and T3 is the last term in S0k. By (9.1.9), and the fact that ~ < t - tp, we deduce

p-1 mi-1 [[TI[[ <__E

~

[[[S(tp - Ti) - S(tp - ~-i,j)]S(t - tp)(g(ti,j+l) - g(ti,j))[[

i=0 j=0 p-1 mi-1

~-- E ~ [[S([Ti -- 7i,jl)S(5) -- S(5)[[[[g(ti,j+l) -- g(ti,j)l] i=o j=o

_< Var(g, [a, b ])c. Since tp+l - t p <_ ~l, t k - l , m k _ ~ - I view of (9.1.8), we get

tp+l <_ t -

tp+l <_ 5, and 7/ C (0, 5], in

lIT2 [[ <_ 2Var(g, It - ~7- 5, t)) < 2Var(g, It - 25, t)) < 2c. Finally, by virtue of (9.1.7), we have

lIT311 < II(S(l k-

I ) ( g ( t ) - g(tk-l,m _l- ))II <

Consequently

S,g,

Ti,j)[[ ~ ( V a r ( g , [ a , b ] ) + 3)s,

and this completes the proof.

[:]

Let ~ C X and g C B V ( [ a, b ]; X ) . D e f i n i t i o n 9.1.1. The function u ' [ a , b] -+ XA, defined by u(t) - S ( t - a)~ +

S(t - s)dg(s)

for each t C [a,b], is called the L ~ - s o l u t i o n of the problem (9.1.1) on the interval[a, b] with the initial condition u(a) - ~ if it satisfies (9.1.6). R e m a r k 9.1.5. Clearly, whenever dg is defined by means of a density f, i.e. dg - f d t with f C L l ( a , b ; X ) , the L~-solution of (9.1.1) corresponding to (~, g) is nothing else than the mild, or C~ of the Catchy problem u'-Au+f -

-

This happens, for example, whenever X has the Radon-Nicod~m property (see Diestel and Uhl [46], Definition 3, p. 61), and g is absolutely continuous on [a, b], case in which f - g' a.e. on [a, b]. Some specific but important

210

Linear Evolution Problems with Measures as Data

such instances are those in which X is either reflexive, or a separable dual. See also Diestel and Uhl [46], Theorem 1, p. 79, and Corollary 4, p. 82. The proofs of the next two propositions follow from a simple computational argument, and therefore we do not give details.

Proposition 9.1.1. Let v" [ a, c] --+ X be the Lee-solution of the problem (9.1.1) on [a, c] with the initial condition v(a) - ~, and w " [c, b] --+ X the L ~ - s o l u t i o n of the very same problem on [ c, b] with the initial condition w(c) - v(c). Then the function u "[ a, b] -+ X , defined by ifte[a,c] if t e ( c , b ] ,

u(t) - { v(t) w(t)

is the L~176 of the problem (9.1.1) on the interval [a,b] with the initial datum u ( a ) - ~. Proposition 9.1.1, to which we refer as to the concatenation principle, is equivalent to the fact that the family { U ( t , s ) ; a <_ s <_ t <_ b} in L ( X ) , defined by U(t, s)~ - S ( t - s)~ + f ts S ( t - T)dg(T) for each ~ E X, is an evolution system, i.e. U ( s , s ) - I and U ( t , s ) - U ( t , T ) U ( T , S ) for each a _< s _< 7 _< t _< b, and this follows from the simple proposition below.

Proposition 9.1.2. If u " [a, b] -+ X is the L ~ - s o l u t i o n of the problem (9.1.1), then, for each c e (a,b), we have - s(t-

+

s(t-

9.2. Regularity of L~ We begin with the following fundamental regularity result.

Theorem 9.2.1. Let ~ e X , g e B V ( [ a , b ] ; X ) ,

and let u be the L cosolution of (9.1.1) corresponding to ~ and g. Then, for each t E [a, b) and each s e (a, b ], there exist u(t + O) and u* (t - 0) = limh40 S ( h ) u ( s - h), and u(t + O) - u(t) = g(t + 0) - g(t),

(9.2.1)

u(s) - u* (s - O) = g(s) - g(s - 0).

(9.2.2)

and If, in addition, either the semigroup generated by A is continuous from (0, +ec) to L ( X ) in the uniform operator topology, or it can be imbedded into a group, then there exists u(s - O ) = u * ( s - O) and accordingly u(s) - u(s - O) = g(s) - g(s - 0).

(9.2.3)

Regularity of L ~ -Solutions

211

So, in this case, u is continuous f r o m the right (left) at t C [a, b] if and only if g is continuous f r o m the right (left) at t. Also in this specific case, u is continuous at any point at which g is continuous and thus u is piecewise continuous on [ a, b ].

P r o o f . Since, by Proposition 9.1.2, u satisfies the evolution property and by (9.1.6), for each t E [a, b] and h > 0 with t + h _< b, we have u(t + h ) -

u(t) - S ( h ) u ( t ) + / t "t+h s ( t + h - s ) d g ( ~ )

- ~(t)

t+h = S ( h ) u ( t ) - u(t) +

X(t,t+h](s)S(t + h -

+S(h)O(t

s)dg(s)

+ O) - g ( t ) ) ,

we get []u(t + h) - u(t) - g(t + O) + g(t) II I I S ( h ) ( ~ ( t ) + g ( t + O) - g ( t ) ) - ~ ( t ) - g ( t + 0) + g(t)II + Var (g, (t, t + h ]).

From the strong continuity of the semigroup and Lemma 1.4.1, we deduce (9.2.1). To check (9.2 2), let s E (a, b] and h > 0 with s - h > a and let us observe that, by using similar arguments, we obtain u(s) - S ( h ) u ( s - h) +

= S(h)~(~

- h) +

f" -h

/s

-h

S ( s - 7-) dg(~-)

XE~-h,~)(~)S(~ - ~ ) e g ( ~ - ) + g(~) - g(~ - o),

and therefore Ilu(s) - S ( h ) u ( s - h) - g(s) + g(s - 0)[I _< Var (g, [s - h, s)). An appeal to Lemma 1.4.1 completes the proof of (9.2.2). Let us assume next that the semigroup {S(t) ; t > 0} is continuous in the uniform operator topology from (0,+co) to L ( X ) . Let s C (a,b] and ~ > 0 with s - ~ >_ a and let h E (0, 5 ]. We have ]lu(s - h) - S ( h ) u ( s - h)l I

~a s-h _< IlS(h)~ - ~]] +

( S ( s - T) - S ( s - h - T)) rig(T)

.

Since the semigroup is strongly continuous at 0, to complete the proof, it suffices to show that the second term on the right-hand side tends to 0 when h tends to 0. Let e > 0. By virtue of Lemma 9.4.1, there exists ~ > 0 such that V~r (g, [~ - ~, ~)) _< ~.

212

Linear Evolution Problems with Measures as Data

So, in view of (9.1.5), for each h e (0, (~), we have h

<

fa s-~ ( s ( ~

- ~) - s ( ~ - h -

~))dg(~)

+

~ss-h ( S ( s

- T) - S ( s - h -

~'))dg(T)

<_ IIS(5) - S ( 5 - h)[]Var (g, [ a, b ]) + 2Var (g, [ s - 5, s - h)) +ll(S(h)

- I ) ( g ( s - h) - g ( s - h -

0))11 <_ IIS(~) - S ( ~ - h)llVar (g, [a,b])

+2Var (g, Is - 5, s)) + II(S(h) - I ) ( g ( s - h) - g ( s - h - 0))[[. Since the semigroup is continuous in the uniform operator topology from (0, + ~ ) to L ( X ) and, by Remark 9.1.4, g([a, b l) is relatively compact in X, for the very same c > 0, there exists ~ C (0, 5) such that, for each h C (0, ~), we have sup ec(0,~)

IIs(5)- s ( 5 - 0)11 ___

and ll(S(h)

-

I)(g(s

-

h)

-

g(s

-

h -

0))11

~

c.

Summing up, we get

~a8-h

(S(s-

~-) - S ( s - h -

~-)) dg(~-)

_< (Vat (g, [a, b]) + 2)E

which shows that limh40 Ilu(s -- h) - S ( h ) u ( s - h)l I - O. Recalling that, as we already have shown, the limit limh40 S ( h ) u ( s h) - u* ( s - 0) exists, we conclude that there exists u ( s - 0) and u ( s - 0) - u* (s - 0). This achieves the proof of (9.2.3) in the case in which the semigroup is continuous in the uniform operator topology on (0, +c~). Finally, if {S(t) ; t > 0} can be imbedded into a group, for each t > O, S ( t ) is invertible and so u(t-

h) - S ( h ) - 1 S ( h ) u ( t -

h).

Thus, by virtue of (9.2.2) and the strong continuity of both S ( h ) S(h) -1, we get (9.2.3). The proof is complete.

and Z]

R e m a r k 9.2.1. It is a simple exercise to show that, even for general C0semigroups, there exists limh40 u ( s -- h) in the weak-| topology on X , and equals u * ( s - 0). So, we have u ( s ) - u ( s - O) = g ( s ) - g ( s - 0)

(9.2.4)

A Characterization of L~-Solutions

213

for each s E (a, b l, where the one-sided limit u ( s - 0) is considered in the weak-| topology on X.

Corollary 9.2.1. Let g C BV([a, b]; X). If g is right continuous on [a, b l, then, for each ~ C X , the L ~-solution of (9.1.1), corresponding to ~ and g, is right continuous on [a, b]. If, either the semigroup generated by A is continuous from (0, +c~) to L ( X ) in the uniform operator topology, or it can be imbedded into a group and g is continuous, then for each ~ C X , the L~-solution of (9.1.1), corresponding to ~ and g, is continuous on [a, b]. R e m a r k 9.2.2. A quite n a t u r a l question we may raise is w h e t h e r or not any L ~ - s o l u t i o n is of b o u n d e d variation. The answer to this question is in the negative, as we can see from the next simple example. Let X be a reflexive Banach space and let us assume t h a t the semigroup generated by A is not differentiable, i.e. there exists x C X such t h a t S(t)x ~ D(A) for each t _> 01. Now, let g ' [ 0, 2] ~ X be defined by

g(t)-{

0 x

fortE[0,1) fortC[1,2].

Clearly, g is of b o u n d e d variation on [1, 2 ]. Moreover, one m a y easily see that the unique L ~ - s o l u t i o n u of the problem (9.1.1), corresponding to = 0 and to g, is defined by 0

u(t) -

S(t-1)x

for t C [0,1) fortE[i,2].

At this point, let us recall that, whenever X is reflexive, and u : [a, b] --+ X is of b o u n d e d variation, then u is a.e. differentiable on [a, b ]. See Bochner and Taylor [26], T h e o r e m 5.2. From this remark, it is clear t h a t u cannot be of b o u n d e d variation on [0, 2] since it is nowhere differentiable on [ 1, 2]. We conclude by noticing t h a t it should be of great interest to know, under what circumstances on X, ~, A and g, the corresponding L ~ - s o l u t i o n of (9.1.1) is of b o u n d e d variation. One may prove t h a t this is the case whenever A C L ( X ) , i.e. when the generated semigroup is uniformly continuous, but we don't know any other relevant situations.

9.3. A C h a r a c t e r i z a t i o n of L ~ - S o l u t i o n s In this section we prove a characterization of L ~ - s o l u t i o n s in the terms of the duality between X and XA. Namely, let A G : D(A G) C_ X G --+ X G 1A typical instance of this sort is: X = L 2(1~+; R) and A : D(A) C_X -+X defined by Af = f', for each f E D(A) = {f E L 2(F~+; R); f' E L 2(R+; 1~)}, which generates the well-known translation semigroup.

214

Linear Evolution Problems with Measures as Data

be the infinitesimal generator of the sun dual semigroup {S(t)G ; t > 0}. Throughout this section we denote by C([a,b];D(AG)) - { f e C([a,b];ZG) ; f(t) e D(A ~ for t e [a,b]}. I f g C BV([a,b] ;X) and ~aG C C([a,b] ; X ~ b

we denote by

k

(dg(t) ~~

-

'

lim E ( g ( t i ) - g(ti_1), ga| )~(9))$0

Definition 9 . 3 . 1 . A f u n c t i o n u : [a,b] -+ X is called a variational, or V-solution of the problem (9.1.1) if u satisfies (9.2.1) and (9.2.4) and, for each f o e C ([ a, b ]; D (AG)), we have

/a

(u(t), f|

dt +

where ~po C C([ a, b]; D ( A ~ of the | problem

/i

(@(t), ~G(t)) + (~, ~G(a)) - O,

(9.3.1)

a, b]; X e) is the unique strong solution

{ ~ G ' _ _AO~O + fG

(9.3 2)

0.

Clearly, each V-solution of (9.1.1) is weakly-(/) piecewise continuous on [a,b]. Since D(A ~ is dense in X ~ it follows that C ( [ a , b ] ; D ( A ~ is dense in C([a,b];X G) too and, accordingly, we have: T h e o r e m 9.3.1. For each ~ C X and each g C BV([ a, b]; X ) , the problem (9.1.1) has at most one V-solution defined on [a, b]. P r o o f . The conclusion follows from the simple remark that, whenever u and v are two V-solutions of (9.1.1), u - v is weakly-q) continuous on [a, b] (see (9.2.1)and (9.2.4))and b(u(t) -- v(t),

f|

(t)) dt

0

for each f E C ( [ a , b ] ; X ~

ff]

As concern V-solution, we have the following characterization theorem. T h e o r e m 9 . 3 . 2 . Let~ E X , andg C B V ( [ a , b ] ; X ) . Thenu C LCC(a,b;X) is the V-solution of the problem (9.1.1) on [a, b] if and only if u is the L ~ solution of the same problem on the same interval.

A Characterization of Lm -Solutions

215

Proof. Let u be an L"-solution of (9.1.1). Since, by Remark 9.2.1, u satisfies (9.2.4), by virtue of the uniqueness Theorem 9.3.1, it suffices to verify (9.3.1). We have

(9.3.3) b ] where Let us observe that, for each t, s E [ a ,b], we have X[ a,t (s) = ~ [ ~ ,(t), XE is the characteristic function of the subset E & [ a, b 1, and therefore

lb(I'

~ (- t4 dg(s),f Y t ) ) dt

We also have l b ( S ( t- a)[, f o ( t ) ) dt = and

(t, lb

S ( t - a)' f o ( t ) ) dt,

cp@(s) = - JQ S ( t - s)@f @ ( t )dt

(9.3.4) cp@(a)= - s;S(t

- a)"f3(t)dt.

From these relations and (9.3.3), we deduce (9.3.1). Conversely, if u is a V-solution of (9.1. l ) on [ a , b ] and f E C([a, b] ; X @ ), then cp3 satisfies (9.3.4) and a simple backward calculation shows that (9.3.1) implies @

216

L i n e a r E v o l u t i o n Problems with Measures as Data

Now the conclusion follows from the arbitrariness of f o C C ( [ a , b ] ; X ~ combined with the fact that, thanks to (9.2.1), and (9.2.4), the first factor under the integral above is weakly-| continuous on [a, b]. The proof is complete. E] 9.4. C o m p a c t n e s s

of t h e L e e - S o l u t i o n Operator

From now on we shall assume that X and A are fixed, and such that, for each (~,g) e X x BY([ a, b ]; X), the Catchy problem (9.1.1) has a unique L~-solution. See Remark 9.1.2 and Theorem 9.1.1. Our goal here is to prove a necessary and sufficient condition in order that the family of all L~176 of the semilinear Catchy problem (9.1.1), when ~ ranges in a bounded subset in X, and g ranges in a subset in B Y ( [ a , b]; X) of equibounded variation, be relatively compact in LP(a, b; X ) for each p in [ 1, +co). This condition is in fact an extension of Theorem 8.6.1 from mild, or C~ to L~-solutions, allowing of course the right-hand side in (9.1.1) to be a measure generated by a function of bounded variation. Let E X and g C BY([ a, b]; X), and let us denote by u = Q(~, g) the unique L~ of the Catchy problem (9.1.1) corresponding to ~ and g. Remark have

9.4.1. Since {S(t); t _> 0} is a semigroup of contractions, we Ilu(t) II <_ I111 + Var (g, [ a, b ]).

The next simple lemma will prove useful in what follows. L e m m a 9.4.1. For each g C BV([ a, b]; X ) and h C (0, b - a), we have

fb-. f*+"S(t+ h Ja

s)dg(s)

dt <_ hVar(g,[a,b]),

dg( )

_< h Var (g, [a, b ]).

dt

and

fab +h

/

r~

S(t--

dt

Jt-h

P r o o f . We shall prove only the first equality, the second one being obtained via very similar arguments. Since t < s _< t + h if and only if s - h __ t < s, we have )C(t,t+h](8) -- X[ s_h,s) (t) for each t E [ a , b - h] and each s E [a + h,b]. Let us denote by Vg(s) - Var(g,[a,s]). Since

f -

t+h S(t + h - s)dg(s)

X(t,t+h]S(t + h - s)dg(s) + S(h)[g(t + 0) - g(t)]

Compactness o] the L ~ -Solution Operator

217

and Iig(t + o) - g(t)[I = 0 a.e. for t e [ a, b ], we have

dt Ja

-

Jt

X(t,t+h](s)S(t + h - s)dg(s)

dt

X[~-h,s)(t) dVg(s) dt +h

-

/a (/a +h

X[s_h,s)(t)dt

)

dVg(s) <_ hVar(g,[a,b]).

The proof is complete.

[3

We recall that a subset 9 in BV([a, b]; X) is of equibounded variation if there exists m 9 > 0 such that Vat (g, [a, b]) _~ m S for each g C 9. We may now proceed to the statement of the main result in this section. T h e o r e m 9.4.1. Let A : D(A) C X --+ X be the infinitesimal generator of a Co-semigroup of contractions {S(t) ; t _> 0}, let 9 be a bounded subset in X and 9 a subset in BV([ a, b]; X) of equibounded variation. Then Q( 9 9) is relatively compact in LP(a, b; X) for each p C[1, +c~) if and only if for each c > 0 there exists a relatively compact subset Ce in X , such that, for each (~,g) E 9 x g, there exists a subset Es,(,g in [a, b] whose Lebesgue measure is less than c, and such that Q(~, g)(t) c Ce for each (~, g) E 9 9, and each t C [ a , b ] \ Ee,r P r o o f . By virtue of Theorem A.5.1, the necessity is obvious. To prove the sufficiency, we also make use of the same Theorem A.5.1. First, let us observe that, by virtue of Lebesgue dominated convergence theorem, it suffices to show that Q ( 9 ~) is relatively compact in L~(a,b;X), and bounded in L ~ (a, b ; X ) . To this aim, let us recall that there exist m~) > 0, and m S > 0, such that m~) and Var (g, [a, b]) _~ rag,

(9.4.1)

for each (~, g) C 9 x 9. Then, by virtue of Remark 9.4.1, we have

+mg for each (~,g) C 9 g a n d t C [a,b]. In order to prove that Q( 9 is l-equiintegrable, let c > 0, (~, g) c 9 x g, and let C, and E~,~,g be the sets

Linear Evolution Problems woth Measures as Data

218

having the properties mentioned by hypotheses. A simple computational argument, along with Lemma 9.4.1, shows that a b-h IIQ(~, g)(t + h) - Q(~, g)(t)ll dt

_<

~ b-h b-h ~

IIQ(r g)(t + h ) - S(h)Q(~,g)(t)[[ dt

+

~ab-h I~t t+h

<

+

IIS(h)Q(5, g)(t) - Q(~, g)(t)II dt

fJ[ a,b]\Er

s(t + h-

dt

IIS(h)Q(r g)(t) - Q(~, g)(t)l [ dt

j.h

+/_

JE e,~,g

IIS(h)Q(~, g)(t) - Q(~, g)(t)II dt

<_ hVar(g,[a,b]) + [ IIS(h)Q(~,g)(t) - Q ( ~ , g ) ( t ) l l d t + 2Mge, J[ a,b]\Ee,~,g for each (~,g) C 9 x S, and h C (0, b - a ] , where M 9 = m ~ + m 9. As Q(~, g)(t) c C~ for each (~, g) c 9 x 9, and each t C [a, b] \ E~,~,g, while C~ is relatively compact in X, there exists 5(c) E (0, b - a] such that, for each h e (0, 5(c)], IIS(h)Q(5, g)(t) - Q(r g)(t)ll _< c, uniformly for (r g) E 9 x 9, and t c [a, b] \ E~,~,g. Then, taking into account (9.4.1), (9.4.2) and the preceding inequalities, we obtain

L

b-h [IQ(~,g)(t + h)

Q(5, g)(t)l I dt <_ ( b - a + m 9 + 2Mg)e

for each (~, g) C 9 x 9, and h C (0, 5(e)] N (0, e]. Obviously, this relation shows that Q( 9 9) is l-equiintegrable. From Theorem A.5.1, it follows that D it is relatively compact in Ll(a, b;X), and this completes the proof. R e m a r k 9.4.2. If X is finite dimensional then, for each bounded subset 9 in X, and each subset 9 in BV([a,b] ;X) of equibounded variation, Q( 9 9) is relatively compact in LP(a,b;X), and thus p-equiintegrable. This follows from the observation that, by virtue of Remark 9.4.1, the set {Q(~,g)(t); (~,g) c 9 x ~, t c [a,b]} is bounded and, inasmuch as X is finite dimensional, the set above is relatively compact. So, we are in the hypotheses of Theorem 9.4.1 and the conclusion follows.

Compactness of the L~-Solution Operator

219

To see that in infinite dimensional spaces the p-equiintegrability condition is not an intrinsic property of the set Q( 9 9) with 9 and 9 bounded, and respectively of equibounded variation, let us analyze the following example. E x a m p l e 9.4.1. Let X - L22~(R) be the space of all equivalence classes, with respect to the almost everywhere equality on R, of measurable and 27r-periodic functions from R to R. Endowed with the L2(0, 27r)-norm this is a real Hilbert space. Let A 9 D(A) C_ H --+ H be the operator defined by D(A) - {u C H; u' E H} and Au - u' for each u C D(A). Obviously, A generates a C0-group of isometrics on H, i.e. the translation group. Let 9 - {0}, and ~ - {t ~-~ - n 1 cos {n(t + 9)}; n C N* }. It is easy to see that 9 is of equibounded variation on [0, 1 ]. On the other hand, in this case, Q( 9 ~) - {t ~+ t sin{n(t + .)}; n c N*}, which is not LP-equicontinuous on [0, 1 ], because it is not relatively compact in L p(0, 1 ;L 2 (0, 27r)). C o r o l l a r y 9.4.1. Let A 9 D(A) C_ X ~ X be the generator of a Cosemigroup of contractions {S(t) ; t > 0}, let 9 be a bounded subset in X , and ~ a subset in BV([a, b]; X ) of equibounded variation. If

{Q(~,g)(t); (~,g) e 9 x ~, t C [a,b]} is relatively compact in X , then, for each p C [1, +co), Q( 9 ~) is relatively compact in LP(a, b ; X ) . We conclude this section with a significant extension of Theorem 8.6.2. T h e o r e m 9.4.2. If A " D(A) C_ X --+ X is the infinitesimal generator of a compact Co-semigroup of contractions, if) is a bounded subset in X , and 9 is a subset in BV([ a, b ] ; X ) of equibounded variation, then, for each p C [ 1, +oc), the set Q( 9 ~) is relatively compact in LP(a, b ; X ) . P r o o f . We show that Q( 9 ~) satisfies the hypotheses of Theorem 9.4.1. To this aim let us observe that, by virtue of Lemma 9.4.1, we have

1

Ilu(t) - S(h)u(t - h)ll dt <_

+h

+h

f

S(t - s)dg(s)

dt

h

< hVar(g,[a,b]), for each (~, g) E 9 x 9 and h E (0, b - a ]. Inasmuch as 9 is of equibounded variation, we get lim h$0

uniformly for f C ~.

/a

+h

Ilu(t) - S(h)u(t - h)l I d t - O,

(9.4.3)

220

Linear Evolution Problems woth Measures as Data

Let us define U -

Q( 9 9) and Qh" U --+ L l ( a , b ; X ) by 0

(Qhu)(t) -

u(t) - S ( h ) u ( t - h)

i f t C [a,a + h] if t C (a + h,b].

From (9.4.3), it follows that Lemma 8.6.1 applies, and therefore, for each E > 0, there exists E~,u C_ [a, b] with #(E~,u) < ~, and (hn)ncN decreasing to 0, such that lim lit(t) - S ( h n ) u ( t - h n ) l l - 0, n---+(x)

uniformly for u E Q( 9 9), and t E [a, b] \ EE,u. Since S(hn) is compact for each n C N and {u(s); u E Q( 9 9), s c [a, b]} is bounded in X, it follows that the set CE - {u(t); u e Q( 9 ~), t e [a,b] \ E~,u} is relatively compact in X. So, we are in the hypotheses of Theorem 9.4.1, and thus Q( 9 ~) is relatively compact in LP(a, b;X) for each p C [1, +c~). This completes the proof. D

9.5. E v o l u t i o n E q u a t i o n s w i t h "Spatial" M e a s u r e s as D a t a Let X be a real Banach space and let us consider the Catchy problem (9.1.1), where A : D(A) C_ X ~ X generates a compact C0-semigroup of contractions, ~ C XA and g C BV([a,b];XA), where XA -- ( X ~ *. See Section 9.1. By the Hahn-Banach theorem (see Hille and Phillips [70], Theorem 2.1.2, p. 29) it follows that XA is a closed subspace of X**. We notice that whenever X is reflexive, and thus X ~ = X*, we have XA = X, and therefore the problem (9.1.1) can be easily treated by the previously developed theory. This is no longer true in the nonreflexive case when XA ~ X, and this explains why, throughout this section, we assume that X is nonreflexive, although all the abstract results hold (trivially) true in general. Another reason, much more subtle for doing this, is that the analysis of partial differential equations involving measures with respect to the spatial argument relies heavily on nonreflexive settings and techniques, as L 1 spaces, and vague topologies. See Example 9.5.1 below. In order to give a precise sense of (9.1.1) in this more general setting, we need the following convergence result. T h e o r e m 9.5.1. Let A " D(A) C_ X ~ X be the generator of a com-

pact Co-semigroup of contractions, let ~ c XA and g E BV([a,b]; XA), and let (~k)keN and (gk)keN be two sequences in X, and respectively in

Evolution Equations with Spatial Measures as Data

221

BV([ a, b]; X ) , such that (gk)kEN has equibounded variation, and lim ~k -- ~

in a ( X A , X ~

lim gk(t) -- g(t)

for each t E [a, b] in a(XA, XG).

k--+c<) k--+c~

Then there exists u E L ~ ( a , b ; X ) , such that, for each p E [1, +co), lim Q(~k, gk) = u

(9.5.1)

k-+cx)

strongly in LP(a, b ; X ) , and pointwise in a(XA, X| x| | we have (u(t),x e) - (~, S ( t - a)Gx G) + u(t+O)-u(t) = g(t+O)-g(t)

In addition, for each

( d g ( s ) , S ( t - s)|176

and u ( s ) - u ( s - O ) = g ( s ) - g ( s - O ) ,

(9.5.2) (9.5.3)

where the one sided limits on the left-hand sides of (9.5.3) are considered in the weak-| topology on X . P r o o f . Let us observe that, for each k, p E N, each t E [a, b ], and each x G E X ~ we have

(Q(~k,gk)(t) - Q(~p, gp)(t),x ~ -

(d(gk - gp)(s),S(t - s)Gx~

where {S(t)G ; t >_ 0} is the sun dual semigroup. See Section 3.8. Since {S(t); t >_ 0} is compact, by virtue of Schauder's theorem, p. 282 in Yosida [136], it follows that its sun dual is a compact semigroup too and therefore both X and X ~ are separable. See Theorem 6.2.2. So, we are in the hypotheses of Lemma 1.4.7, and accordingly lim k ,p --," o e

/a

(d(gk -- 9p)(s), S(t - s ) ~ ~ - 0

for each t E [a, b ], and x ~ E X ~ Furthermore, we have lim (~k -- # , S(t

-

a) ~ G)

-

0

k,p---~oc

for each t E [a, b] and x G E X ~ Summing up, we conclude that, for each t E [a,b], (Q(~k, gk)(t))kEN is a Cauchy sequence in the weak-| topology on XA. Therefore there exists u : [ a, b] ~ XA such that lim Q(~k, gk)(t) = u(t) k--~ o c

weakly-q) in XA. Fix p E [1, +co), and let us observe that, by virtue of Theorem 9.4.1, on a subsequence at least, we have lim Q ( ~ k , gk) = v

k-+cx~

Linear Evolution Problems with Measures as Data

222

strongly in LP(a,b;X). So, v coincides with u. As {Q(~a,ga); k E N} is relatively compact in LP(a, b;X), we get that (Q(~k, gk))kEN itself converges in LP(a,b;X) to u, and this proves (9.5.1). Finally, let us observe that (9.5.2) is a direct consequence of Lemma 1.4.7, while (9.5.3) follows by using the very same arguments as in Theorem 9.2.1, and this achieves the proof. [-1 R e m a r k 9.5.1. By Lemma 1.4.7, one may easily verify that the limit in (9.5.1) does not depend on the choice of the sequences (~k)kEN and (gk)kEN which approximate ~ and respectively g. Therefore, Theorem 9.5.1 allows us to extend the concept of the L~-solution to the case in which ~ E XA and g E BV([a,b];XA), whenever the latter can be approximated in the pointwise convergence weak-| topology by a sequence of functions (gk)kcN with equibounded variation and, of course, the semigroup generated by A is compact. More precisely, let ( E XA and g E BV([a, b]; XA). D e f i n i t i o n 9.5.1. A function u E L ~ ( a , b;X) satisfying (9.5.2) is called ~n c ~ - g ~ a l ~ z ~ e ~ol~tio~ of the problem ( 9 . 1 . 1 ) o n [a, b ] We also notice that, under these circumstances, the operator A has a smoothing effect on the data in the sense that, for each ~ E XA and g E BV([a,b];XA), as in Theorem 9.5.1, the L~-generalized solution u is an X-valued function and not an XA-valued one, as we might expect at a first glance. Since XA is obviously dependent of A, in all that follows, we call it the space of admissible measures for A. A prototype of the situation described in Theorem 9.5.1 is illustrated by the following suggestive example. E x a m p l e 9.5.1. Let f~ be a bounded domain in IR~ with sufficiently smooth boundary F, and let us consider the linear parabolic problem

/

~ - zx~ + ~(t - to) | ~(x - x0) u- 0 ~(0) - ~ ( x - y0)

on (0, T) • in (0, T) • r on a,

(9.5.4)

where to E (0, T) and xo, Yo E t~. To get an L~-generalized solution of (9.5.4) we proceed as follows. Let X - LI(~), and let A ' D ( A ) C_ X --+ X be defined by

D(A) - {u E Wo'l(a); Au E L l ( a ) } and A u -

Au

for each u E D(A). We recall that A generates a compact C0-semigroup of contractions {S(t) ; t _> 0} on X. See Theorem 7.2.7. Let {S(t)G ; t _> 0} be the sun dual semigroup on X G. In our case, one may easily verify that X G - C0(~), i.e. the space of all continuous functions from f~ to

Problems I~ vanishing on F.

(fk)kCN in

223

At this point, let us take two sequences (~k)k~N and

LI(~), and respectively in I]~kIILI(~-~)-

LI(O,T;LI(~)),

I]hiiil(O,T;il(~-~))

-

satisfying

1

for each k C N, and lim ~ ~oT fk(~, x)~(~, ~) d~ dx = ~(to, ~o),

k--~c~

lim ~

~k(x)r

= r

k--+ c r

m

for each ~ E C([0, T]; C0(~)) and r E C0(~). Now, let us consider the sequence of problems

I Ukt--Auk+fk uk--0

on (0, T) • in(0, T) •

uk(0) - ~k

on ~,

and let us observe that we are in the hypotheses of Theorem 9.5.1. Thus, the problem (9.5.3) has a unique Lee-generalized solution. R e m a r k 9.5.2. Many other problems can be approached by using a similar scheme. We mention here only one of the simplest, i.e. the heat equation with point control

l ut- Au+f~(x-xo) u--O ~(o) -

on (0, T) • f~ in (0, T) x F o n ~'~

where the control f C I~. Problems.

P r o b l e m 9.1. Let X = Cub(R) and let {G(t); t c I~} the translation group, i.e. [G(t)f](s) = f ( s - t) for f C X and t, s C I~. Find the sun dual group {S(t)~ t > 0}. P r o b l e m 9.2. Let X be a real Banach space, {S(t) ; t > 0} a C0-semigroup of contractions on X, x C X with x ~ 0, and let g : R --+ X be defined by -x

g(t) -

0 x

ift <0 if t - 0 i f t > 0.

Linear Evolution Problems with Measures as data

224

Prove that g E BV(IR; X) and, for every a, t E IR with a < t, the RiemannStieltjes integral f ta S ( t - s)dg(s) exists in the norm topology of X and t

/a S ( t - s) dg(s) -

0 x

2x x 0

ifa_
This example shows that, even for nonreflexive Banach spaces X, for semigroups which are not continuous in the uniform operator topology from (0, +oc) to L ( Z ) and for functions g which are not defined by densities, it may happen that the Riemann-Stieltjes integral above not only to belong to X, but to exist even in the norm topology of X. See Remarks 9.1.2, 9.1.3 and 9.1.5. P r o b l e m 9.3. Let X be a real Banach space, {S(t) ; t > 0} a C0-semigroup of contractions on X, x E X with x ~ O, h C BV(IR) and let g : I~ ~ X be defined by g(t) = h(t)x for every t C Ii~. Show that g C BY(R; X) and, for each a, t C R with a < t, the Riemann-Stieltjes integral f ta S ( t - s)dg(s) exists in the norm topology of X. P r o b l e m 9.4. Let X be a real Banach space, {S(t) ; t >__0} a C0-semigroup of contractions on X, let xi E X satisfying xi ~ 0 and hi c BV(IR) for i - 1 2, n and let g 9 IR --+ X be defined by g(t) - ~-]n hi(t)xi for every t C IR. Show that g E B V(IR;X) and, for each a , t C I R w i t h a < t, the Riemann-Stieltjes integral f: S ( t - s)dg(s) exists in the norm topology of X. "

9 "

i = 1

P r o b l e m 9.5. Let X be a real Banach space, A" D(A) C_ X --+ X the generator of a C0-semigroup of contractions, and let BVA([a, b]; X) be the space of all g C BV([a, b ] ; X ) enjoying the property that, for every t E [a, b], there exists the integral fo S ( t - s)dg(s) in the norm topology of X (and belongs to X, of course). Show that BVA([a,b] ;X) is a closed subspace in BV([a, b ] ; X ) , the later being endowed with the seminorm [[g]] - Var (g, [a, b ]). A subset 9 in BV([a, b]; X) is equi-absolutely continuous if there exists a function t~" IR+ --+ I~+, nondecreasing, continuous at 0, with t~(0) - 0 and such that Var (g, [ c, d ]) <_ ~(d - c) for all g C ~ and [ c~ d] C_ [ a, b ]. P r o b l e m 9.6. Let A 9 D(A) C_ X ~ X be the generator of a C0-semigroup of contractions {S(t) ; t >__0}, let 9 be a bounded subset in X, and 9" an equi-absolutely continuous subset in BV([a, b] ;X). Show that Q( 9 9") is

Notes

225

in C([ a, b]; X) and is relatively compact in C([ c, b]; X) for each c e (a, b), if and only if there exists a dense subset D in [ a, b ] such that, for each t C D, the t-section of Q( 9 9~), i.e. Q( 9 9")(t) = {Q(~, f)(t) ; (~, f) c 9 x 9"}, is relatively compact in X. Moreover, if the latter condition is satisfied, and a E D, then Q( 9 9~) is relatively compact in C([ a, b] ;X). P r o b l e m 9.7. Prove that if A " D(A) C_ X -+ X is the generator of a compact C0-semigroup of contractions, 9 is a bounded subset in X, and is an equi-absolutely continuous subset in BV([ a, b] ;X), then, for each c E (a, b), the set O(z),

-

g);

c 9 x

is relatively compact in C([ c, b]; X). N o t e s . The results in this chapter were proved by Vrabie [131]. Related results referring to linear evolution equations with measures as data were obtained recently by Amann [6] by assuming that A generates an analytic semigroup and dg is a Radon measure. See also Ahmed [2]. We notice that the first-order necessary conditions of optimality for optimal control problems with state constraints are of the form (9.1.1). See Barbu [18], Casas [35], Fattorini [52], Lasiecka [83], [82], Raymond [107], RaymondTrSltzsch [108], Raymond-Zidani [109] and the references therein. The main results in Section 9.4, Theorems 9.4.1 and 9.4.2, extend Theorem 8.6.1 and Baras-Hassan-Veron's Theorem 8.6.2 to the general case of distributed measures, while Theorem 9.5.1 in Section 9.5 exhibits a closedness property of the L~ operator. Concerning the problems here included we notice that all of them are new. There are many other interesting open problems from which I list only a few. 1. It would be of great interest to prove a relative compactness sufficient condition of the same form as that in Theorem 9.4.2 not only in L p (a, b; X), but also in the pointwise convergence topology. 2. A second problem is to develop an existence and regularity theory for another type of solution, more general than that of L~ Namely, eliminating the condition (9.1.6) imposed in Definition 9.1.1, we get another concept of solution, weaker than that of L~-solution. More precisely, we say that u : [a, b] --+ XA is an XA-valued solution of (9.1.1) on [a, b ] i f u is given by the variation of constants formula, i.e. u(t) - S ( t - s)~ +

S ( t - s)dg(s)

for each t C [a,b], where the integral is considered in the a ( X A , X O ) topology. See Section 9.1.

226

Linear Evolution Problems with Measures as data

3. It would be of great interest to use the results here established in order to get information on control problems of the form:

C (u s (t ) ) dt ; f E g =

minimize subjected to

{ u~ - A u f + f (t ) -

-

where A : D(A) C_ X -~ X is the infinitesimal generator of a compact C0semigroup of contractions, and the set 9" of admissible controls is the unit closed ball centered at zero in X. Here X is assumed to be a nonreflexive real Banach space. 4. Another problem is to find necessary and/or sufficient conditions in order for a locally closed subset 2 D in X to be invariant with respect to the evolution equation du = { A u } d t + dg, (a) where A : D(A) C_ X -+ X is the infinitesimal generator of a compact C0-semigroup and g C B V ( [ a , b ] ; X ) . We recall that D is invariant with respect to (E) iffor each c C [a, b) and each ~ E X with g(c+O)-g(c)+~ E D, the unique L~-solution u : [a, c] --+ X satisfies u(t + O) C D for each t C [c,b). We notice that, in the case in which g(t) - f ta f ( s ) d s with f C C([ a, b] ; X ) , D is invariant with respect to (E) if and only if li~$ionf ~1 d(S(h)~ + hf(t); D) -- 0

(9")

for each t C [a,b) and each ~ C D. Here d(x;D) denotes the distance between x C X and D. For details on this interesting subject see Cs and Vrabie [34].

2We recall that D is locally closed if for each ~ E D there exists r > 0 such that B(~, r) n D is closed. Both closed subsets and open subsets in X are locally closed, but there are examples of sets which, although neither closed, nor open, are locally closed. An example of such set is an open 2-dimensional disk in F~3.

C H A P T E R 10

Some Nonlinear Cauchy Problems

In order to illustrate the power of the abstract methods already developed in the preceding chapters, we shall present here some remarkable results concerning the local existence problem for several classes of nonlinear differential equations in general Banach spaces. We begin with an extension of Peano's local existence theorem to infinite dimensions. Further, we prove another generalization of the later and present some basic facts about saturated and global solutions. As a first application, we consider the Klein-Gordon semilinear equation which, surprisingly, can be rewritten and studied as an ordinary first-order differential equation driven by a continuous right-hand side. We conclude with a partial differential equation from mechanics which also can be handled as an ordinary differential equation with continuous right-hand side

10.1. P e a n o ' s Local E x i s t e n c e T h e o r e m Let X be a real Banach space, 9 a nonempty and open subset in IR x X, and f : 9 ~ X. Let us consider the Cauchy problem

u'-f(t,u) u(a) - ~,

(10.1.1)

and let us recall that, if X is finite-dimensional, and f is continuous, then, for each (a,~) E 9 (10.1.1) has at least one local solution. This result, due to Peano, fundamental in the theory of ordinary differential equations, does not extend exactly in this form to the infinite-dimensional case, as we shall see in the counterexample below. To fix the ideas, we recall first that by a solution of (10.1.1) we mean a C 1 function u ' [ a , b) -+ X such that (t, u(t)) C 9 for each t C [a, b) and which satisfies u'(t) - f(t, u(t)) for each t C [ a , b) and u(a) = ~.

C o u n t e r e x a m p l e 10.1.1. (Dieudonn6) Let X = co be the space of all real sequences (x~)neN. with lim~__+~ xn = 0, which, endowed with the supnorm defined by II(Xn)ncH, ]lc~ sup{]Xn] ; n C H*} for each (Xn)neN* C X , -

-

227

228

Some Nonlinear Cauchy Problems

is a real Banach space, which is infinite-dimensional. Let f : X --+ X be defined by

(f((Xn)nEN*)k)kcH*-

(2IX//~kl)kEN,

for each (Xn)neN* C co, let ~ = (1/n)neN*, and let us consider the Cauchy problem -

Let us observe that u : [0, 5) -+ X is a solution of the problem above if and only if (Uk)k~N. : [0, 5) --+ X is a solution of the system of infinitely many differential equations -

2v/1

1

uk(O) - 1/k

k-

1,2, . . . . Let us assume that the above Cauchy problem, or equivalently the above system, has at least one solution (Uk)kCN* : [0,5) --+ X, with 5 > 0. As this system contains infinitely many uncoupled differential equations with separate variables, whose solutions, if there exist, are necessarily of the form

uk(t) - (t + l / k ) 2 for each k C N* and each t C [0, 5), it follows that (uk)kcN* is defined by the above equalities. But, for all t > 0, we have limk-~c~ uk(t) = t 2, relation in contradiction with the fact that (uk(t))k~N, belongs to co, i.e. limk+oc uk(t) = 0. This contradiction can be eliminated only if the Cauchy problem in question has no local solution. Analyzing the proof of Peano's local existence theorem (see Corduneanu [39], Theorem 2.4, p. 33), we easily conclude that this nonexistence phenomenon is due to the lack of relative compactness of bounded subsets in infinite-dimensional Banach spaces. This counterexample shows that, in order to obtain an existence result applying to (10.1.1), besides the continuity condition on f, we need an extra-condition to compensate the lack of relative compactness of bounded subsets in a an infinite-dimensional Banach space. Roughly speaking, such an extra-condition would require f to c a r r y bounded subsets in 9 into relatively compact subsets in X. In fact, we will Use a slightly weaker condition. Namely, we introduce: D e f i n i t i o n 10.1.1. A function f : 9 ~ X is called b-compact if for each [a,b] C IR, ~ e X and r > 0 with [a,b] x B(Cs,r) C 9 f([a,b] • B(~,r)) is relatively compact in X. The function f is called compact if it carries bounded subsets in 9 into relatively compact subsets in X.

Peano 's Local Existence Theorem

229

Clearly each compact function is b-compact, and whenever 9 - I~ x X, the converse statement is also true. However, we notice that, if 9 does not coincide with I~ x X, there exists b-compact functions which are not compact. A simple example in the case of X - I~ is furnished by the 1 function f ' ( O , +co) x (0, +co) -+ ~, f ( t , u) - u, which is b-compact but not compact. In this section, we shall prove that, whenever f 9 9 --+ X is b-compact, for each (a,~) E 9 the Cauchy problem (10.1.1) has at least one local solution. This result, fundamental in the theory of ordinary differential equation in infinite dimensions, is an extension of the well-known Peano's local existence theorem. We analyze first the simplest case when 9 - ]I x X, with ]I a nonempty and open interval, and f is continuous on ]I x X with f(]I x X) relatively compact in X, and then we shall show how we can pass to the general case. So, let f - ] I x X - + X , let (a,~) E I [ x X , let ~ > 0 a n d l e t 5>0besuch that [a, a + 5] C I[. Let us consider the integral equation with delay

/a

for t Ia-- ,al t

u~ (t) -

~+

f (7-, u)~ (7- - )~)) dT

for t C (a, a + 5 ].

(10.1.2)

The following result will prove useful in the sequel. L e m m a 10.1.1. If f " I[ x X ~ X is continuous, then, for each ~ > O, each (a, ~) E 1I x X and each 5 > 0 w i t h [ a , a + 5] C I[, (10.1.2) has one and only one solution defined on [ a - ~, a + ~ ].

P r o o f . Obviously, u)~ is uniquely determined on [ a - )~, a] by (10.1.2). Let then t C [a, a + ,~ ]. Let us remark that, for each ~- C [a, t ], we have 7-- A e [aa ], and therefore u~ (~-- ,~) - ~. accordingly, -

+

and ua is uniquely determined on [a, a + ~ ]. Proceeding analogously, we can successively determine u,x on [a + ~, a + 2)~ ], [a + 2)~, a + 3~ ], and so on. Obviously, after m steps, with m)~ _> a + 5, we can define u~ on the whole interval [a, a + 5 ]. We complete the proof by observing that ua is continuous and uniquely determined. [:3 As we already have noticed, we shall prove first the following existence result which, although auxiliary, is interesting by itself. L e m m a 1 0 . 1 . 2 . If f : ~ x X is continuous and f(I[ x X ) is relatively compact, then, for each (a, ~) e I[ x X and ~ > 0 with [ a, a + 5 ] C I[, (10.1.1) has at least one solution defined on [a, a + 5].

Some Nonlinear Cauchy Problems

230

P r o o f . Let (a,~) C lI x X, and 5 > 0 such that [ a , a + 5] C I[, let rn C N*, and let us consider the integral equation with the delay 5m = 5 / m

urn(t)--

~+

t

f(T, Um(T--Sm))dT

(10.1.3)

fortC(a,a+5].

Let us remark that, by virtue of Lemma 10.1.1, for each rn C N*, (10.1.3) has a unique continuous solution Um: [ a - 5m, a + 5] ~ X . We shall prove next that {urn; rn C N*} satisfies the hypotheses of Theorem A.2.1. Namely, we shall show that, for each t C [a, a + 5 ], {urn (t) ; rn E N* } is relatively compact in X, and {urn; rn C N* } is equicontinuous on [ a, a + 5 ]. Let us recall that f(]I x X) is relatively compact in X and, therefore, from Lemma A.1.3, we conclude that

{ /a

{Um(t);rneN*}-

~+

f(S, U m ( S - S m ) ) d s ; r n e N *

}

is relatively compact in X for each t C (a, a + 5 ]. Next, let us observe that, also from (10.1.3), we have ]lure(t) - um(s)[[ _<

IIf(T, u m ( ~ - ~m))ll d~ <_ M I t - sl,

for each rn C N* and t,s E [a,a + 5]. Consequently, {Urn; rn C N*} is equicontinuous on [a, a + 5 ]. From Theorem A.2.1, it follows that (Um)m~N* has at least one subsequence, denoted for simplicity again by (Urn)mEN*, which is uniformly convergent on [a, a + 5] to a continuous function u. Obviously, we have lim Um(7" -- (~m) = U(T), m--+oo

uniformly for ~- C [a, a + (f ]. Since f is continuous on ]I x X, the relation above and Corollary A.2.2 show that we may pass to the limit in (10.1.3) for rn -+ oc. Moreover, u satisfies

u(t) - ~ +

/a

f (T, u(r)) dT, for each t C [ a, a + 5 ],

which shows that u : [a,a + 5] --+ X is a solution for (10.1.1), and this achieves the proof. W1 R e m a r k 10.1.1. In the hypotheses of Lemma 10.1.1, we can prove that, for each initial datum (a, ~) E ]I x X, (10.1.3) has at least one global solution. We may now proceed to the statement of the main result in this section. To this aim, let 9 be a nonempty and open subset in R x X and f : 9 --+ X a given function.

The problem u' = f (t, u)+ g(t, u) Theorem

231

1 0 . 1 . 1 . If f : 9 -+ X is b-compact, then for each (a,~) C 9

(10.1.1) has at least one local solution. P r o o f . Let (a, ~) C 9 . As 9 is open, there exist d > 0, a n d r > 0, such that [a - d, a + d] x B(~, r) C 9 where

Let us define p : X --+ X by y p(y)

for y e B({, r)

-

Ily -

(y -

+

for y c x

\

We may easily see t h a t p maps X to B(~, r) and is continuous on X. Next, let us define g : ( a - d, a + d) x X --+ X by

g(t, y) = f (t, p(y)), for each (t, y) C (a - d, a + d) x X. As f is b-compact, f ( [ a - d, a + d] x B(~, r)) is relatively compact. Hence g is continuous and g((a - d, a + d) x X ) is relatively compact. T h a n k s to L e m m a 10.1.1, we know that, for each d ~ E (0, d), the Cauchy problem

{

4

-

g(t,

u(a) -

has at least one solution u : [a, a + d] --+ X. I n a s m u c h as u is continuous at t = a and u(a) = ~, for r > 0 there exists 5 E (0, d'] such that, for each t C [a,a + 5], ] ] u ( t ) - ~11 -< r. But in this case g(t, u(t)) = f(t, u(t)), and therefore u : [ a , a + 5 ] ~ X is a solution o f t h e problem (10.1.1). T h e proof is complete. [-1 10.2.

The Problem

u' = f (t, u) + g(t, u)

Let X be a real B a n a c h space and 9 a n o n e m p t y and open subset in ]R x X. In this section we shall prove a local existence result referring to a class of Cauchy problem of the type u' - f(t, u) + g(t, u)

(10.2.1)

--

where f : 9 ~ X is a b-compact function, and g : 9 -~ X is continuous on 9 and locally Lipschitz with respect of its last argument. We recall t h a t g : 9 -~ X is locally Lipschitz with respect to its last argument if for each (a, ~) E ~B C 9 there exists b > a, r > 0 and L - La,~ > 0 such t h a t [a,b] x B(~,r) C 9 and

]]g(t,u)

- g(t, v) II <- Liiu -

vii

232

Some Nonlinear Cauchy Problems

for each (t, u), (t, v) e [a, b] x B(~, r). As expected, by a solution of (10.2.1) we mean a C~-function u" [a, b] ~ X satisfying (t, u(t)) E 2), u'(t) - f ( t , u(t)) + g(t, u(t)) for each t E [a, b ], and u(a) - ~. T h e o r e m 10.2.1. Let f : 2) ~ X be b-compact, and let g : 9 ~ X be continuous on 9 and locally Lipschitz with respect to its last argument. Then, for each (a,~) C 9 there exists b > a such that (10.2.1) has at least one solution defined o n [ a , b ]. We shall prove this theorem with the help of three lemmas. First, we prove a variant of the well-known Gronwall's lemma. Having in mind our later purposes, we shall formulate this lemma in a more general framework t h a n t h a t needed here. L e m m a 10.2.1. (Gronwall) Let x , k " [a,b) --~ R+ be measurable with s ~-~ k ( s ) x ( s ) locally integrable on [a, b), let m >_ 0 and assume that

~(t) _< m + fa ~ a.e. for t C [ a, b). Then

z(t) <_reef: k(~) e~ a.e. f o r t E [a,b). P r o o f . Let y(t) - m + f ta k ( s ) x ( s ) d s . Then y is a.e. differentiable on [a, b) and we have

y'(t) = k(t)x(t) <_ k(t)y(t) a.e. for t E [ a, b). Hence

d( -~ y(t)~- ft k(s) ds) <_ o a.e. for t E [a, b). Integrating from a to t, after some rearrangements, we get

y(t) <mef: k(~)d~ for each t E [a, b). Since x(t) <_ y(t) a.e. for t E [a, b), this achieves the proof. [~ T h r o u g h o u t the next two lemmas, ]I denotes an interval with nonempty interior but which is not necessarily open.

The problem u' = f ( t ,u)

+ g(t,u)

233

Lemma 10.2.2. Let g : I1 x X + X be continuous on II x X and Lipschitz on X . Then, for each (a, J ) E II x X , each b > 0 with [ a , b] c 11, and each h E L' (a, b ; X), the problem

has a unique solution defined on [ a , b]. In addition, for each (a, J), and b, fixed as above, the mapping h t+ S(h), i.e. the unique solution of the problem (10.2.2) corresponding to h, satisfies

5

IIS(hl) - S ( h 2 ) I I ~ ( [ a , b ] ; ~ )

where L

6L llHl

-

H211~([a,b];~),

(10.2.3)

> 0 is the Lipschitz constant of g, and

for i = 1,2 and t E [ a , b ] .

Proof. Let a E II, b > a with [ a ,b] C 1, J E X and h E L1(a,b ; X ) . Let us define Q : C ( [ a ,b]; X ) + C ( [ a ,b]; X ) by

for each u E C ( [ a ,b]; X ) , and each t E [ a ,b]. Let us observe that, for each u,v E C ( [ a , b]; X ) and t E [ a ,b], we have From this inequality, by a simple inductive argument, we deduce that Ln(t - a)n IIu - v l l ~ ( [ a , b ] ; ~ ) n! for each n E N*, each u, v E C ( [ a ,b]; X ) , and each t E [ a , b]. Consequently

Il (Qnu)(t) - (Qnv)(t)ll 5

for each n E N*, and each u,v E C ( [ a , b]; X ) . But, this shows that, for n E N*large enough, Qn is a strict contraction. By virtue of Banach fixed point theorem, Qn has a unique fixed point u E C ( [ a ,b]; X ) . But llQu - u11 = llQnQu - Qnull 5 qllQu - 4, where q E ( 0 , l ) . Hence Qu = u. But u is the only fixed point of Q, because each fixed point of Q is a fixed point for Qn, which is a strict contraction. As each fixed point of Q is a solution of the problem (10.2.2) and conversely, this proves the first part of lemma.

Some Nonlinear Cauchy Problems

234

Let now hi and h2 be two elements in Ll(a, b ; X ) . Let us observe that [l(S(hl))(t) - ( 8 ( h 2 ) ) ( t ) [ I <_

IIg(~,(8(hl))(s)) - g(s, (8(h2))(s))ll ds

+ <_ IIH

-

(hi(s) - h2(s)) ds

g211C(Ia,b];X) + L

-

(S(h2))(s)ll

ds

for each t E [a, b]. From this, and Gronwall Lemma 10.2.1 we deduce 11(8(hl))(t) - (8(h2))(t)LL < e~LLLHI - It2[[C([a,b];x). Passing to the supremum for t C [a, b ], we obtain (10.2.3), and this achieves the proof. D L e m m a 10.2.3. Let f :][ • X ~ X be a continuous function, with f (~ • X ) relatively compact, and let g : ~ x X --+ X be continuous on IT • X , and Lipschitz on X . Then, for each (a,~) C ]I • X, and each b > a with [a, b] C ]I, the problem (10.2.1) has at least one solution defined on [a, b]. P r o o f . Let A > O. In view of Lemma 10.2.2, it follows that the problem

{ u'~ - g(t, u~) + f (t, u~(. - ),)) u~(t) - ~

for t e [ a - A,a],

which is equivalent with the integral equation with the delay A in [a - A, a]

]'at

-

+

g(s, u~(s)) ds +

f ( s , u~(s

A)) ds

in [a, b],

(10.2.4) has a unique continuous solution u~. Let now A = l / n , and let us denote by ~t n the corresponding solution of (10.2.4). Let us denote by Fn: [a, b] -+ X the function defined by

Fn(t)-

f (s, Un(S

l/n))ds

for each n C N* and each t E [a, b ]. As f (]I x X) is relatively compact, from Lemma A.1.3, it follows that, for each t e [a, b l, the set {Fn(t); n e N*} is relatively compact in X. Moreover, {Fn; n C N*} is equicontinuous on [a, b]. Indeed, as f is bounded on I[ • X, there exists M > 0 such that

JJf (t, u)[[ ~ M

The problem u' = f (t, u ) + g(t, u)

235

for each (t, u) E I[ x X. We then have

liEn (t) - Fn (s) II ~- M I t - sl for each n C N* and each t, s C [a, b]. In view of Theorem A.2.1, there exists F C C([ a, b l; X) such that, on a subsequence at least, lim F~(t) = F(t), n--+~

uniformly for t C [a,b]. Let us observe that, by virtue of (10.2.4), we have

Un(t) -- ~ +

/a

g(s, Un(8)) ds +

/a

f (s, Un(8 - l / n ) ) d s ,

(10.2.5)

relation which, according to (10.2.3) and to the definition of the function Fn, leads to

IlUn -- UmllC([a,b];X) ~_ e~LIIFn -- FmllC([a,b];X) for each n, m C N*. Inasmuch as (Fn)neN* is a Cauchy sequence, it follows that (Un)ncN* has the same property, and therefore it is convergent in C([ a, b]; X) to some function u. Passing to the limit in (10.2.5), and using Corollary A.2.2, we deduce that u is a solution of the problem (10.2.1) defined on [a, b ], and this completes the proof. K] We can now proceed to the proof of Theorem 10.2.1. P r o o f . Let (a, ~) C 9 c < a < b, such that

Since 9 is open, there exist c, b and r > 0 with

[c,b] x B ( ~ , r ) C 9 In addition, as g is continuous on 9 and locally Lipschitz with respect to its last argument, diminishing r if necessary, we may assume that there exist M > 0 and L > 0 such that IIg(t, u)ll _< M

for each (t, u) C [c, b] • B(~, r), and

]lg(t, u) - g(t, v)[I <- LII u - vii for each (t, u), (t, v) C [c, b] x B(~, r). Let us define p : X --~ X by y p(y) -

for y C B(~, r)

r

( y l-l y

-

+

for y C X \ B({, r),

and let us observe that p maps X in B({, r) and is Lipschitz continuous on X with Lipschitz constant 2. Next, we define fr :[c, b] x X -+ X by

fr (t, y) = f (t, p(y)), for each (t, y) e [ c, b ] x X,

236

Some Nonlinear Catchy Problems

and gr : [ c, b] x X -+ X by gr (t, y) = g(t, p(y)) for each (t, y) e [ c, b] x X. As f is b-compact, f([ c, hi x B(~, r)) is relatively compact. Accordingly, fr is continuous and fr((C, b) x X ) is relatively compact. Analogously, from the choice of c, b and r, we deduce that gr is continuous and bounded on [ c, b ] x X, and Lipschitz continuous on X with Lipschitz constant 2L. From Lemma 10.2.3, we know that the Cauchy problem u' -- f r

(t, u)

-~- gr

(t, u)

--

has at least one solution u : [a, b] ~ X. As u is continuous at t = a and u(a) = ~, diminishing b > a if necessary, we have l i t ( t ) - ~I] -< r for each t E [a, b]. But this means that we have both fr(t, u(t)) = f ( t , u(t)), and gr(t,u(t)) = g(t,u(t)). Consequently u : [a,b] -+ X is a solution of the problem (10.2.1), and this achieves the proof. D 10.3.

Saturated Solutions

Let 9 be a nonempty and open subset in R x X, let f : 9 --+ X be a given function, and (a, ~) C 9 Let us consider the C a t c h y problem

u'-f(t,u) u(a) - ~. A solution u : [a,b) --+ X of (10.3.1) another solution v : [a, c) --+ X with each t E [a, b). A solution u : [a, b) continuable. If the projection of 9 on global if it is defined on [a, +c~).

(10.3.1)

is called continuable if there exists b < c and such that u(t) = v(t) for -+ X is called saturated if it is not I~ contains I~+, a solution u is called

R e m a r k 10.3.1. We emphasize that all the results within this section hold true, with essentially the same proofs, also for the C a t c h y problem -

f(t,

+ g(t,

u(a) -- ~, where f and g are as in Section 10.2, but, for the simplicity of writing, we consider here only this simpler case corresponding to g - 0 and analyzed in Section 10.1. L e m m a 10.3.1. Let f : 9 --+ X be a b-compact function on 9 Then a solution u : [a, b) -+ X of (10.3.1) is continuable if and only if there exists u* = limu(t)

(10.3.2)

(b, u*) e 9

(10.3.3)

t$b

and

237

Saturated Solutions

P r o o f . The necessity is obvious, while the sufficiency is a consequence of Theorem 10.1.1. [3 R e m a r k 10.3.1. From Lemma 10.3.1 it follows that each saturated solution of (10.3.1) is necessarily defined on an interval of the form [a, b), i.e. on an interval which is open at the right. A sufficient condition for the existence of the limit (10.3.2) is stated below. P r o p o s i t i o n 10.3.1. Let u : [a, b) --+ X be a solution of (10.3.1) and let us assume that b < +c~, and there exists M > 0 such that

IIf (~, u(~-)) II _< M, for each ~- C [ a , b). Then there exists

u* = lim u(t). t$b

P r o o f . Let us observe that, for each t, s C [a, b), we have

Ilu(t) - ~(s)ll _<

]If(r, u(~-))ll&- _

Mlt

-

s],

and thus u satisfies the hypothesis of the Cauchy test on the existence of the limit at b. [3 We may now proceed to the statement of a characterization of continuable solutions of (10.3.1 ). T h e o r e m 10.3.1. Let f : 9 --+ X be b-compact and let u : [a, b) --+ X be a solution of (10.3.1). A necessary and sufficient condition in order for u to be continuable is that the graph of u, i.e. graph u = { ( t , u ( t ) ) C I~ x X; t E [a,b)} is included in a compact subset in 9

P r o o f . Necessity. Assume that u is continuable. Then u may be extended by continuity to [ a, b ]. Denote this extension by v and let us observe that the mapping t ~ (t, v(t)) is continuous from [ a, b] to 9 Therefore, its range, which coincides with graph v is compact and included in 9 Since graph u C graph v, this completes the proof of the necessity. Sufficiency. Assume that graph u is included into a compact subset of 2). Then f is bounded on graph u, i.e. there exists M > 0 such that I[f(T, u(T))]] _< M, for each 7 C [a, b). The conclusion is an easy consequence of Lemma 10.3.1 and Proposition 10.3.1. D

238

Some Nonlinear Cauchy Problems

We continue with a fundamental existence result concerning saturated solutions for (10.3.1). T h e o r e m 10.3.2. If f " 9 --+ X is b-compact and u " [a,b) ~ X is a solution of (10.3.1), then either u is saturated, or u can be continued up to a saturated one. P r o o f . If u is saturated we have nothing to prove. Thus, let us assume that u is continuable and let us define g as the set of all solutions of (10.3.1) which extend u. Obviously, u E g and thus g is nonempty. Moreover, since u is continuable, g contains at least two elements. On g let us define the relation " ~_ " by v ~ w if w extends v. It is a simple exercise to show that (g, ~) is an inductively ordered set. So, from Zorn's Lemma, there exists at least one maximal element Um E g such that u ~_ urn. From the definition of " ~ ", and from the maximality of Um it follows that Um is a saturated solution of (10.3.1) which coincides with u on the common part of the domains, thereby completing the proof. D From Theorems 2.3.1 and 10.3.2 it follows: C o r o l l a r y 10.3.1. If f 99 -+ X is b-compact, then for each initial data (a,~) E 9 (10.3.1) has at least one saturated solution. We recall that a limit point of a function u : [a, b) --+ X as t tends to b is any element u* in X for which there exists a sequence (tk)kcN in [a, b) tending to b and such that limk_~o~ u(tk) = u*. We denote the set of all limit points of u as t tends to b by Limt$bu(t). Concerning the behavior of saturated solutions at the right end point of their interval of definition, we have the following fundamental result. T h e o r e m 10.3.3. Let f : 9 --+ X be b-compact and let u : [ a, b) ~ X be a saturated solution of (10.3.1). Then either (i) u is unbounded or (ii) u is bounded and, either u is global, or (iii) u is bounded and non-global, and in this case, either Limt$bu(t) is empty, or for each u* E Limt?bu(t), (b, u*) C 0 9 P r o o f . If u is unbounded on [a, b), or if it is bounded and global, or bounded and Limt~bu(t) is empty, we have nothing to prove. Thus, let us assume that u is bounded on [a, b) and not global, i.e. b < + e c and Limt$bu(t) is nonempty. To prove (iii) let us assume by contradiction that there exists at least one sequence (tk)kcr~ in [a,b) tending to b and such that (u(tk))k~N is convergent to some u* C X, but (b,u*) ~ 0 9 So, we necessarily have (b, u*) C 9 In view of Lemma 10.3.1, to get a

Saturated Solutions

239

contradiction, it suffices to show that there exists limtl-b u(t), which of course must coincide with u*. To this aim, let us observe that, since 9 is open and (b,u*) e 9 there exists c > b and r > 0 such that [a,c) x B ( u * , r ) C 9 Furthermore, inasmuch as b < +oc, and f is continuous on 9 diminishing r > 0 if necessary, it follows that there exists M > 0 such that

IIf (~', v)II < M, for each (~-, v) e [ a, b] x B (u*, r). As l i m k ~ we may choose k C N such that

b - tk < 2 M

(10.3.4)

tk -- b and limk~oc u(tk) -- u*,

r

(10.3.5)

II~(tk) - u* II < ~.

Fix k with the properties above. We shall show that for each t C [tk, b), we have u(t) C B(u*, r). Let t* - s u p { t E [tk, b); u(s) E B ( u * , r ) , for s E [tk,t]}. If t* -- b, the statement above is obviously true. Let us assume by contradiction that t* < b. This means that u(t) E B(u*, r) for each t C [tk, t*], I l u ( t * ) - u * l l - r and there exist points t > t*, as close to t* as we wish, satisfying I l u ( t ) - u* II > r. In other words, t* is the "first moment in (tk, b) after which u leaves the set B ( u * , r ) " . The condition L l u ( t * ) - u*ll - r signifies that, at t*, u must cross the boundary of B(u*, r). Next, from the remark above combined with (10.3.4), and (10.3.5), we get r - Ilu(t*) - u* II _< Ilu(t*) - u(tk)ll + Ilu(tk) - u* II _<

IfI(~, u(~-))lld~- + Ilu(tk) - u*ll _< (t* - tk)M + II~(tk) - ~*ll

<_ ( b - tk)M + II~(tk) - u*ll < ~ + ~ -- ~. This contradiction (r < r) is a consequence of our supposition that, for at least one t e [tk, b), u(t) r B(u*, r). Then, for each t e [tk, b), we must have u(t) e e(u*, r). In view of (10.3.4) combined with Proposition 10.3.1 and Lemma 10.3.1, it follows that u is continuable thereby contradicting the hypothesis. This contradiction can be eliminated only if u* C 0 9 The proof is complete. [3 Under an additional hypothesis on f we shall prove a sharper result. 10.3.4. Let f " 9 -+ X be b-compact and let us assume that it maps bounded subsets in 9 into bounded subsets in X . Let u " [ a, b) --+ X be a saturated solution of (10.3.1). Then either

Theorem

240

Some Nonlinear Cauchy Problems

(i') u is unbounded and, if b < +oo, limtTb Ilu(t)ll- +oo, or (ii) u is bounded and, either u is global, or (iii') u is bounded and non-global, and in this latter case there exists limtTb u ( t ) - u*, and (b, u*) E 02). P r o o f . In view of Theorem 10.3.3, the only fact we have to prove here is t h a t whenever (ii) does not hold, then one of the two conditions (i') or (iii') must hold. So, let us assume for the beginning that (i') and (ii) do not hold. Then, from Lemma 10.3.1 and Proposition 10.3.1, it follows that there exists lim u(t) = u*, tTb

while from (iii)in Theorem 10.3.3 we know that (b, u*) E oq9 Thus (iii') holds. Let us assume now that (ii) and (iii') do not hold and that b < +c~. To show that limtTb Ilu(t)ll = +oo, let us assume by contradiction that, under these circumstances, we have lim inf Ilu(t)II < +oc. tTb

In other words, there exist at least one sequence (tk)kEN in (a, b), convergent to b, and r > 0 such that ]lu(tk)]] _< r, for each k E N. Let C = {v E ft; Ilvll _< r + 1}. Since f maps bounded subsets in 2) into bounded subsets in X, b < +oo and C is bounded, there exists M > 0, such that IIf(r, v)lt <_ M, (10.3.6) for each (r, v) E [a, b) • C. Now let us choose a number d > 0 satisfying d M < 1,

(10.3.7)

and fix k E N such that b - d < tk < b. Since u is unbounded on [a, b), it is necessarily unbounded on [tk, b). Then there exists t* E (tk, b) such that

]lu(r) II < r + 1,

(10.3.8)

for each r E [tk, t*) and ]]u(t*)l ] = r + 1. In fact, t* is "the first moment in (tk, b) at which u leaves B(0, r + 1)." Let us observe that

+ 1 - II~,(t*)ll - II~(tk) +

fl

/ ( ~ , ~(~-))&-II

-< II~(tk)ll + f~i* IIf(;, ~(;))lld;.

Saturated Solutions

241

Taking into account (10.3.6), (10.3.7), (10.3.8), and the obvious inequality t * - t k < d, we get r + 1 <_ r + (t* - t k ) M < r + d M < r + 1.

This contradiction can be eliminated only if lim Ilu(t)ll - +co ttb

and this completes the proof.

89

Remark 10.3.2. I f 9 - IR+ x X, then, each compact function f " 9 --~ X maps bounded subsets in 9 into bounded subsets in X.

C o r o l l a r y 10.3.2. Let f 9 It(+ x X --+ X be compact. T h e n a saturated solution u "[a, b) --+ X of (10.3.1) is either global, or it is not global, and in this case there exists

lim Ilu(t)ll - +co. tSb

P r o o f . Let us observe first that, if b < +co, u is necessarily unbounded on [ a, b). Indeed, if we assume the contrary, u has at least one limit point u* as t tends to b. By Theorem 10.3.4, (b, u*) must lie on the boundary of I~ x X which is the empty set. Thus, the supposition that u is bounded on [a, b) is false. The conclusion of Corollary 10.3.2 is then an easy consequence of Remark 10.3.2 combined with Theorem 10.3.4. [2 If b < +oc and limttb Ilu(t)II

=

we

say that u blows up in finite time.

C o r o l l a r y 10.3.3. Let f : IR+ x X --+ X be compact.

A necessary and sufficient condition in order for a solution u : [a, b) --+ X of (10.3.1) to be continuable is that b < +oc and u be bounded on [a, b).

P r o o f . The necessity is trivial, while the sufficiency is a reformulation of Corollary 10.3.2. D We conclude this section with a sufficient conditions on f ensuring the existence of global solutions of (10.3.1). Theorem

10.3.5. Let f : R + x X --+ X be compact, and let us assume that there exist two continuous f u n c t i o n s 1 h, k " I~+ --+ IR+ such that Ill (T, v)II <- k(T)IIvll + h(T),

(10.3.9)

for each (~-,v) C IR+ x X . Then, for each ( a , ~ ) C I~+ x X , (10.3.1) has at least one global solution.

l In fact we can only ask that these functions belong to Lr

242

Some Nonlinear Catchy Problems

P r o o f . T h a n k s to Corollary 10.3.1, it suffices to show t h a t each saturated solution of (10.3.1) is global. To this aim, let u : [a,b) ~ X be a s a t u r a t e d solution of (10.3.1). From (10.3.9), we get

Ilu(t)ll ~ II~ll + s

h(r)d~- + ~a t

k(~)llu(~)ll&,

for each t E [a, b). We will show next t h a t b = +oc. Indeed, if we assume the contrary, inasmuch as [a,b] is compact and h,k are continuous on [a, b] C R+, there exists M > 0 such t h a t

h(t) <_ M

and k(t) <_ M,

for each t C [a, b]. The preceding inequalities and Gronwall's L e m m a 10.2.1 show t h a t Ilu(t)[I <-- [ll~]]-4- M(b - a)]e M(b-a), for each t C (a, b). Hence u is b o u n d e d on [a, b) and therefore it has at least one limit point u* as t tends to b. By T h e o r e m 10.3.3, (b, u*) must lie on the b o u n d a r y of R+ x X which is the e m p t y set. This contradiction can be eliminated only if b = + ~ , and this completes the proof. [-1

10.4. The Klein-Gordon Equation In this section we shall present an interesting application of T h e o r e m 10.1.1. Let X be a real Banach space and A : D(A) C X --+ X the generator of a C0-group {G(t) ; t c IR}, let 9 be a n o n e m p t y and open subset in IR x X, let f : 2) --+ X be a continuous function and let us consider the problem

u'-

Au + f ( t , u )

~(a) -- ~.

(10.4 1)

We begin with the following simple, but useful lemma. L e m m a 10.4.1. A function u 9[ a, b] -+ X is a C~ .for the problem (10.4.1) if and only if the function v: [a, b] --+ X , defined by

v(t) = a ( - t + a)~(t) for each t E [a, b], is a Cl-solution of the problem v' - g(t, v)

~(a) -- ~,

where g: E ~ X is defined by

g(t, v) = a ( - t + a)y(t, a ( t - a)v)

So~ ~ach (t, v) c Z, ~ h ~

~ = {(t, a ( t - a)v); (t, v) C ~}.

(10.4.2)

The Klein-Gordon Equation P r o o f . First, let us recall that u 9 [a,b] ~ (10.4.1) if (t, u(t)) e 9 and ~(t) - a ( t - a){ +

243 X is a C~

of

a ( t - ~)f(~, ~(~))d~

for each t C [a, b ]. Consequently we get

a ( - t + a)~(t) - ~ +

a ( - ~ + a)/(~, u(~)) d~

for t E [a, b]. Let v : [a, b] ~ X be defined by v(t) = a ( - t t C [a, b]. Obviously it satisfies v(t) - ~ +

a(-~

+ a)~(t) for

+ a)f(~, a ( ~ - a)~(~))d~

for each t C [a, b], and this achieves the proof.

D

T h e o r e m 10.4.1. Let A : D ( A ) C X -+ X be the generator of a Co-group {G(t) ; t C IR}, 9 a nonempty and open subset in X , and let f : 9 --+ X be a b-compact function. Then, for each (a, ~) C 9 there exists b > a such that (10.4.1) has at least one C~ solution defined on [a, b). P r o o f . As f is b-compact, we deduce that the function g, defined in Lemma 10.4.1, is b-compact too. An appeal to Theorems 10.1.1 and 10.3.2 completes the proof. V] E x a m p l e 10.4.1. Let ~t be a nonempty, bounded and open subset in IRn whose boundary F is of class C 2, let QT - (0, T) x Ft and ET -- (0, T) x F. Let A > 0 be the first eigenvalue o f - A on H - l ( f t ) , i.e. A - inf{l[Vull~(~) ; u ~ Hl(f~), I[ullL~(~) -- 1} and let m > -A. We consider the nonlinear hyperbolic equation ~(t, x) - 0 u(O,x) - u o ( x )

(t, ~) e r~T x C ~,

~(O, x) - v o ( x )

x e

(10.4.3)

where g : IR+ x ~t x I~ --+ IR is a continuous function. This equation is known as the semilinear Klein-Gordon equation. T h e o r e m 10.4.2. Let g : IR+ x ~ x IR --+ IR be a continuous function for which there exist c > O, d > O, and a C IR, such that

Ig(t, x, u)] _< clul ~ + d

(10.4.4)

244

Some Nonlinear Cauchy Problems

for (t, x, u) C IR+ x ~t x IR, where c~ > 0 if n - 2, and c~ < n / (n - 2) if n >_ 3. Then, for each uo e H~(~) and each vo e L2(~), there exists T > 0 such that the problem (10.4.3) has at least one saturated solution u satisfying (i) u e C([O,T);HI(~))

(ii)

ut e

C([ 0, T); L2(~)).

In addition, if T < +cxD, then lim

sST

If n -

(llu(8)llH~(a) -Jr-IlUt(8)llL2(a)) -- -~-CX:).

1, the conclusion remains valid without condition (10.4.4).

P r o o f . First, we observe that (10.4.3) can be rewritten as a first-order ordinary differential equation in an infinite-dimensional Hilbert space. Let H-

H~(~) x L2(~)

which, endowed with the inner product (., .), defined by ((u, v), (~, ~)) - ~ u ' ( x ) ( t ' ( x ) d x + m / ~ u ( x ) ( t ( x ) d x + f~ v(x)~(x)dx for each (u, v), (~, ~) C H, is a real Hilbert space. We define the operator A" D(A) C H ~ H by D(A) = (H2(~t)N H~(~)) x H~(~) A(u, v ) = (v, A u - mu) for each (u, v) e D(A). Furthermore, let us define f - I R + x D ( f ) -+ H by

O f = {(u,v) e H ; g(t, . , u ( . ) ) e n2(~t) for each t e [0, +c~)} f(t, (x) - (0, g(t, x, for each t C [0, + ~ ) , each (u, v) C Df and a.e. for x C ~. At this point, let us observe that the problem (10.4.3) can be rewritten under the equivalent form z ' - Az + f ( t , z ) z(a) - ~, where A and f are as above, z(t)(x) - (u(t,x),v(t,x)) a.e. for (t,x) in (0, T) x ~t and ~ - (u0, v0). In order to prove that A and f satisfy the general conditions in Theorem 10.4.1, let us remark first that, by virtue of Theorem 4.6.4, A generates a C0-group of isometries. As concerns f, from (10.4.4), Theorem 1.5.4 and Lemma A.6.1, it follows that Df - H and f is continuous on IR x H. Again from Theorem 1.5.4, we know that H01(~t) is compactly imbedded in: C(~) if n - 1, in Lq(~t) for each q _> 1 if n - 2, and q < 2 n / ( n - 2) if n _> 3. Thus f is compact, and we are in the hypotheses of Theorem 10.4.1, and Corollaries 10.3.1 and 10.3.2. The proof is complete. D

An Application to a Problem in Mechanics

245

R e m a r k 10.4.1. Using the same arguments and Theorem 10.3.4, one may prove a global existence result for the Sine-Gordon equation U t t - AU-4-sin u

~(t, x) - 0 ~(0, ~) - ~o(~) ~ ( o , ~ ) - ~0(x)

(t, x) E QT (t, ~) e x ~ 9 e ~, 9~ ~

This equation is important in the study of the transformation of surfaces with constant negative curvature. 10.5.

An Application

to a Problem

in M e c h a n i c s

The movement of a continuous medium, having the domain of reference a nonempty open and bounded subset ~ in R n, is described by the following pseudoparabolic partial differential equation

I ut-aAut+/3Au+f(t, u- 0 ~(0, ~ ) -

x)

{ttlx) cQoo x) E Eo~

g(~)

9~

~,

(10.5.1)

wh~r~ ~ > 0, 9 > 0, f e C(R+; L2(f~)), g ~ H 2 ( ~ ) n H I ( ~ ) , 0 ~ - R+ • and u is the field of velocities. 10.5.1. In the general hypotheses above, the problem (10.5.1) has a unique solution

Theorem

U E CI(R+; H2(~)

Hol(~)).

P r o o f . Let us observe that (10.5.1) can be rewritten as the following implicit differential equation in the space L2(~) [ ( I - a A ) u ] ' - / 3 A u + f(t) ~ ( o ) = 9,

(10.5.2)

where A . D ( A ) C_ L2(~) --, L2(f~) is defined by D(A) - Hl(fi) N H2(~) Au for each u E D(A).

AuLet us denote by v -

( I - aA)u and let us observe that

/ 3 A ( I - aA) -1 - --fl [I - ( I

aA)] ( I - aA) -1 - / 3 ( I - aA) -1

--~I. OL

So, the problem (10.5.2) can be reformulated as

I - aA) -1 v - coy + f(t) v ' - - (co v(O) - ( I - aA)g,

(10.5.3)

246

Some Nonlinear Cauchy Problems

where w - / 3 a -1 > 0. As w ( I - a A ) -1 - w I is linear bounded, it is globally Lipschitz on L2(f~). By virtue of Lemma 10.2.2, the problem (10.5.3) has a unique solution v C C1(1~+; L2(f~)). The fact that u - ( I - a A ) - l v satisfies U E cl(]~+;

H2(a) n H i ( a ) )

follows from the remark that ( I - a A ) -1 is linear continuous from L2(ft) to H2(f])N H~(f~), and this completes the proof. V] Let us consider now the semilinear version of the problem (10.5.1) I ut - a A u t + / ~ A u + f ( t , x , u ) u-O

(t,

c c

(10.5.4)

xE~,

x)

where a > 0,/3 > 0, f " IR+ x ~ x IR is continuous and g C H 2 (~)N H~ (f~). T h e o r e m 10.5.2. In the general hypotheses above, let us assume that there exists c > O, d > 0 such that

If(t, z, u) l < clul + d for all (t, z, u) C R+ x f~ x R. Then there exists T > 0 such that problem (10.5.4) has at least one solution

u C CI([o,T];H2(O, Tr)). If in addition, for each (t,x) C IR+ x f~, f ( t , z , . ) " IR -+ N is decreasing, then u is the only solution of the problem (10.5.4) on [0, T].

P r o o f . Repeating the arguments in the proof of Theorem 10.5.1, we deduce that (10.5.4) can be rewritten under the form v' - w(I - a A ) - l v - wv + f (t, (I - a A ) - l v ) v(O) - (I -

(10.5.5)

A)g,

where w - ~o1-1 > 0. Since ( I - a A ) -1 is continuous from L2(ft) to H2(f~) M H~(ft), and H2(gt)N H~(ft) is compactly imbedded in C(ft), it follows that the function F ' R + x L2(f~) -+ L2(f~), defined by F(t, v)(z) - f ( t , z , ((I - a A ) - l v ) ( x ) )

a.e. for x E ft, is compact. Since ( I - a A ) -1 - w I is Lipschitz continuous, we are in the hypotheses of Theorem 8.5.1, and so there exists at least one local solution u C CI([O,T];H2(f~) A H ~ ( D ) ) . Next, let u and v be two solutions of the problem (10.5.4). Subtracting side by side the corresponding two equalities, taking the inner product of

Problems

247

b o t h sides by u - v, recalling t h a t f(t, x, .) is decreasing, and integrating over ~, we obtain ld

2dt (

Pl~(t)

-

--

v(t)ll~(~) + ~ll~x(t) - v~(t)ll2~(~)) -~llux(t)-

vx(t)ll~ (~).

Therefore, 7d / [ll 2 dt

9

(t)

-

+ II

(t)

~

-

<_ o

for each t > 0, where -), = min{1, a} > 0. Consequently, the function

t ~ ~ lit(t) - v(t)ll~(a) + Ilux(t) - v~(t)ll~(a) is decreasing on [0, T ]. As this function vanishes at t - 0, it is identically 0 on [0, T ], and this achieves the proof. D

Problems P r o b l e m 10.1. Let X be a Banach space, ]I a n o n e m p t y and open interval, ~t n o n e m p t y and open subset in X, and f : ]I x ~ --+ X a function satisfying: (i) for a.e. for t C ]I, f(t,-): ~ --+ X is continuous ; (ii) for each x C ~t, f(., x ) : ][ --+ X is measurable ; (iii) for each ~ C ~, there exists r > 0, and a compact subset K in X such t h a t f(t, x) C K for each x E B(~, r ) ~ ~, and a.e. for t C ]I. Then, for each a C I[, and ~ E ~, there exists b > a with [a, b] C ]I, and such that the C a t c h y problem

~(a) has at least one absolutely continuous solution u ' [ a, b] ~ f~. P r o b l e m 10.2. Let X be a Banach space, lI a n o n e m p t y and open interval, and f~ a n o n e m p t y and open subset in X. Let f, g : lI x f~ --+ X be two functions satisfying (i) and (ii) in P r o b l e m 10.1. a s s u m e , in addition, t h a t f satisfies (iii) in the same problem, and g satisfies: (iv) for each ~ E f~ there exists r > O, and a locally integrable function [:1I ~ R+, such t h a t IIg(t, u) - g(t, v)II -- [(t)II ~ - vii for each u, v C B(~, r) M ~, and a.e. for t E I[,

248

Some Nonlinear Cauchy Problems

and g(., ~) :1I ~ X is locally integrable. Then, for each a E I[, and each E f~, there exists b > a with [a, b] C 1I, and such that the Cauchy problem

{ has at least one absolutely continuous solution u : [a, b] ~ ft. N o t e s . Theorem 10.1.1 in Section 10.1 is the most known extension to a r b i t r a r y infinite-dimensional Banach spaces of Peano's local existence theorem. For a survey of such extensions see Bogachev [27]. Our proof given here differs from the usual ones using either the Euler's polygonal lines method, or Schauder fixed point theorem in that is more direct and elementary. The proof of Theorem 10.2.1 in Section 10.2 is elementary, and as far as we know, new, avoiding the use of Krasnoselskii fixed point theorem in [80] which we recall below. T h e o r e m (Krasnoselskii) Let K be a n o n e m p t y closed convex subset in a B a n a c h space X and let F, G : K ---+X be two mappings satisfying:

(i) F(K) + G(K) C K; (ii) F is continuous and compact; (iii) G is a strict contraction, i.e. there exists k E (0, 1) such that ]IG(u) - G(v)l I < kllu - vii f o r all u, v C K . Then F + G has at least one fized point in K . For a proof using the latter method see Frigon and O'Reagan [57]. The results in Section 10.3 are simple extensions to the infinite-dimensional case of some qualitative results known for many decades within the theory of ordinary differential equations. As far as we know, in this form allowing non-uniqueness, Theorem 10.3.3 seems to be new. Regarding Section 10.4, the m e t h o d of reducing a second order hyperbolic equation to an ordinary differential equation in an infinite-dimensional Hilbert space, equation which can be handled by Peano's local existence theorem, is due to Vrabie (1998) and described for the first time here. The application included in Section 10.5 is adapted from Showalter and Ting [113]. The two problems included are extensions to the Carath6odory case of the abstract existence results in Sections 10.1 and 10.2

C H A P T E R 11

The Cauchy Problem for Semilinear Equations

Section 11.1 is mainly concerned with an existence and uniqueness result for the semilinear evolution equation u ~ = Au + f(t, u) in the case in which A generates a C0-semigroup of contractions and f is locally Lipschitz with respect to its last argument. In Section 11.2 we reconsider the same problem by assuming that A generates a C0-compact semigroup of contractions and f is merely continuous. In Section 11.3 we present some basic results on the continuation of the solutions, while in Section 11.4 we prove a Poincar6-Liapunov-type theorem concerning the asymptotic behavior of solutions. In Sections 11.5 and 11.6 we include several examples illustrating the abstract theory.

11.1. T h e P r o b l e m Let X be generator nonempty and (a, ~)

u ' = Au + f ( t , u) w i t h f L i p s c h i t z

a real Banach space, and A : D ( A ) C_ X --+ X the infinitesimal of a C0-semigroup of contractions {S(t); t > 0}. Let 9 be a and open subset in R x X, let f : 9 --+ X a continuous function E 9 We consider the Cauchy problem

u ' - Au + f ( t , u ) u(a) - ~.

(11 1.1)

Excepting for Section 11.4, here we confine ourselves to the simpler case of C0-semigroups of contractions, although all the results in this chapter hold true in the general case, i.e. of the C0-semigroups of type (M, w). In analogy with the linear non homogeneous case, we introduce: D e f i n i t i o n 11.1.1. A C~ of the problem (11.1.1) on [a,b] is a continuous function u : [a, b l--+ X satisfying (t, u(t)) C 9 and

u(t) - S(t - a)~ + for each t C [a, b].

S(t - s ) f (s, u(s)) ds

250

The Cauchy Problem ]'or Semilinear Equations

Since the notions of classical, as well as of strong, solution for (11.1.1) are simply defined in analogy with the case f(t, u) - f(t) (see Definitions 8.1.1 and 8.1.2), we do not give their explicit statements here. D e f i n i t i o n 11.1.2. The function f " 9 --+ X is called" (i) locally Lipschitz with respect to its last argument if for each (a, ~) in 9 there exist b > a, r > 0 and La,( - L > 0 such that

[a,b] x B ( ~ , r ) C 9 and IIf (t, u) - f (t, v)II <- Lllu - vii for each (t, u), (t, v) E [a, b] x B(~, r); (ii) locally Lipschitz with respect to both arguments if for each (a, ~) in 9 there exist b > a, r > 0 and La,~ - L > 0 such that

[a,b] x B(~,r) C 9 and Ilf(t, u) - f ( s , v)l I <_ L ( l t - sl +

I1 - vii)

for each (t, u), (s, v) c [a, b] x B(~, r); (iii) globally Lipschitz with respect to its last argument if there exists L > 0 such that

IIf (t, u) - f (t, v)II <- Lllu - vii for each (t, u), (t, v) C 9 (iv) globally Lipschitz with respect to both arguments if there exists L > 0 such that

Ilf (t, u) - f (s, v)l I <_ L(lt - s I + Ilu - vii ) for each (t, u), (s, v) C 9 As we shall see later on, the next abstract local existence result concerns abstract semilinear problems of both "parabolic" and "hyperbolic" nature. T h e o r e m 11.1.1. If A " D(A) C_ X -+ X generates a Co-semigroup of contractions, {S(t); t > 0}, and f " 9 ~ X is continuous and locally Lipschitz with respect to its last argument, then, for each (a,~) E 9 there exists b > a such that the problem (11.1.1) has a unique C~ defined on [a,b]. For the proof, we need the following lemma which is interesting by itself. L e m m a 11.1.1. If A " D(A) c X --+ X generates a Co-semigroup of contractions, {S(t) ; t > 0}, and f " [ a, b] x X --+ X is continuous, bounded and globally Lipschitz with respect to its last argument then, for each ~ C X , the problem (11.1.1) has a unique G~ defined on [a,b].

The Problem u' = Au + f (t, u) with f Lipschitz

251

P r o o f . Let [ E X and let us define Q : C([ a, b]; X) -~ C([ a, b]; X) by S ( t - s ) f ( s , u(s)) ds

(Qu)(t) - S ( t - a)~ +

for each u C U([ a, b]; X), and each t E [a, b]. Let us observe that, for each u , v E C ( [ a , b ] ; X ) and t C [a,b], we have II(Qu)(t)

-

(Qv)(t)ll

a)llu

<_ L ( t -

- vllC([a,b];X).

From this inequality, by a simple inductive argument, we deduce that Ln(t-a)

II(Q~u)(t) - (Qnv)(t)ll <-

n

llu --

n!

VllC([a'b];X)

for each n C N*, each u, v E C([ a, b]; X), and each t C [a, b]. Consequently Ln(b-a) n IIQnu - O~vllc([ a,b ];X) <--

I1~ - vllc(E a,b ];X)

n!

for each n E N*, and each u , v E C([a, b];X). But, this shows that, for n C N* large enough, Qn is a strict contraction. By virtue of Banach fixed point theorem, Qn has a unique fixed point u C C([a, b]; X). But ]]Qu -

ul]

= ] ] Q ~ Q u - Q n u l ] <_ q ] l Q u -

ul],

where q E (0, 1). Hence Qu = u. But u is the only fixed point of Q, because each fixed point of Q is a fixed point for Qn, which is a strict contraction. As each fixed point of Q is a solution of t h e p r o b l e m (11.1.1) and conversely, this achieves the proof. D We may now proceed to the proof of Theorem 11.1.1. P r o o f . Let (a,~) C 9 As 9 is open, there exist b > a and r > 0 with [a,b] x B ( ~ , r ) C_ 9 In addition, as f is continuous and locally Lipschitz with respect to its last argument, diminishing b > a a n d / o r r > 0 if necessary, we may assume that there exists M > 0 such that IIf (t, u)II <- M for each (t, u ) C [ a , b] x B(~, r ) a n d llf(t, ~) - I(t, ~)II _< Lll~ - ~ll

for each (t, u), (t, v) C [a, b] x B(~, r). Let p : X -~ X be defined by y

for y C B({, r)

p(y) -

-----~(ylIB - ~1 - ~) + ~ for y E X \ B(~, r).

The Cauchy Problem for Semilinear Equations

252

We may easily see that p maps X to B(~,r), and is Lipschitz continuous on X with Lipschitz constant 2. Next, let us define g : [ a, b] x X ~ X by

g(t, y) = f(t, p(y)) for each (t, y) C [a, b] x X. Since f is continuous and globally Lipschitz with respect to its last argument on [a, b] x B(~,r), we conclude that g enjoys the very same properties. From Lemma 11.1.1, we know that the Cauchy problem u' - Au + g ( t , u ) u(a) has a unique C~ u "[a, b] --+ X. As u(a) - ~ and u is continuous at t = a, diminishing b > a if necessary, we get that u(t) C B(~, r) for each t C [ a, b]. Accordingly g(t, u(t)) = f ( t , u(t)), and therefore u : [ a, b] -~ X is a C~ of the problem (11.1.1). The proof is complete. D In the case in which X has good geometrical properties and f is locally Lipschitz with respect to both arguments, then, for each ~ E D(A), the C~ of (11.1.1) is even a classical one. More precisely we have: T h e o r e m 11.1.2. Let A : D(A) C X --+ X be the generator of a Cosemigroup of contractions and f : 9 --+ X locally Lipschitz with respect to both arguments. If X has the Radon-Nicod~m property, then, for each (a,~) E 9 with ~ E D(A), there exists b > a such that the problem (11.1.1) has a unique classical solution defined on [ a, b ]. P r o o f . Let u : [a, b] --+ X be the unique C~ whose existence is ensured by Theorem 11.1.1 and let v : [a, b] --+ X be defined by

~(t) -

s ( t - ~) f (~, ~(~)) d~.

Let t C (a, b), let h > 0 and let us observe that

v(t + h) - v(t) -

ft+h

S(t + h - s) f (s, u(s) ) ds -

fa t S(t

- s) f (s, u(s) ) ds

da

-

la-h

S ( t - s)f(s + h, u(s + h))ds +

[a-h

s(t-

la

S ( t - s)f(s, u(s))ds

~)/(~ + h, ~(~ + h))d~

S ( t - s)[f(s + h, u(s + h ) ) - f ( s , u(s))] ds.

253

The Problem u' = Au + f (t, u) with f Continuous

Inasmuch as t ~-+ f ( t , u(t)) is bounded on [a, b ], say by M > 0, we deduce

Ilv(t + h) - v(t)l I <_ M h + L Since u(t) = S ( t -

/a

([h I + Ilu(s + h) - u(s)l])ds.

a)~ + v(t) and

IlS(t + h - a)~ - S ( t -

a)~l] <_ I I S ( h ) ~

-

~11_

IIA~llh,

from the preceding inequality, we get

II~(t + h) - ~(t)ll _< [IIA~II + M + L(b

-

a)]h + L

/a

I1~(~ + h) - ~(s)ll d~.

From Gronwall Lemma 10.2.1, it follows

Ilu(t + h) - u(t)ll <_ eL(b-a)[llA~ll + M + L ( b - a)]h for each t, t + h C [a, b ]. Clearly this inequality shows that u is Lipschitz on [ a, b] and consequently t ~ f (t, u(t)) is Lipschitz on [ a, b ]. Since X has the Radon-Nicod:~m property, it follows that t ~ f(t, u(t)) is in wl'l(a, b;X) (in fact in W 1'+~ (a, b ; X ) ) . Now the conclusion follows from Theorem 8.1.4 and this completes the proof. K1 Since reflexive Banach spaces, as well as separable duals have the RadonNicodj~m property, we easily deduce" T h e o r e m 11.1.3. Let A : D(A) C_ X -+ X be the generator of a Cosemigroup of contractions and f : 9 --+ X locally Lipschitz with respect to both arguments. If X is either reflexive, or a separable dual, then, for each (a,~) C 9 with ~ C D(A), there exists b > a such that the problem (11.1.1) has a unique classical solution defined o n [ a , b ]. 11.2. T h e P r o b l e m u ' = Au + f ( t , u) w i t h f C o n t i n u o u s In this section we shall present a useful consequence of Theorem 6.3.2. Let X be a real Banach space, and A : D(A) C_ X ~ X the infinitesimal generator of a C0-semigroup of contractions {S(t) ; t _> 0}. Let 9 be a nonempty and open subset in R x X, and let f : 9 --+ X a continuous function. Let (a, ~) C 9 , and let us consider the Cauchy problem

u ' - Au + f ( t , u ) ~(a) --

(11.2.1)

As we shall see in the sequel, the following abstract local existence result concerns abstract semilinear problems "of parabolic type".

254

The Catchy Problem for Semilinear Equations

T h e o r e m 11.2.1. (Pazy) If A " D ( A ) c_ X --+ X generates a compact Co-semigroup of contractions, {S(t) ; t _> 0}, and f " 9 --+ X is continuous then, for each (a,~) C 9 there exists b > a such that the problem (11.2.1) has at least one C~ defined on [a, b]. For the proof, we need the following lemma which is interesting by itself. L e m m a 11.2.1. If A " D ( A ) C_ X --+ X generates a Compact Co-semigroup of contractions, {S(t) ; t _> 0}, and f " [a, b] • X -+ X is continuous and bounded then, for each ~ C X , the problem (11.2.1) has at least one C Osolution defined on[a, b ]. P r o o f . Let ~ E X, let )~ > 0 and let us consider the integral equation with the delay tE[a-A,a] u~(t)-

S(t_a)~+

t s(t

-

-

tC[a,b].

(11.2.2) Let us observe that (11.2.2) has a unique continuous solution which is defined successively on [a, a + ~ ], [a + ~, a + 2~] and so on. For n C N*, let us denote by Un the unique solution of the problem (11.2.2) corresponding to ~ - 1/n. As f is bounded, it follows that ~" - {f(., U n ( ' - l / n ) ) ; n E N*} is bounded, and therefore uniformly integrable. In view of Theorem 6.3.2, the set {Un ; n C H*} is relatively compact in C([a, b]; X). Then we may assume without loss of generality that there exists lim u~(t) - u(t), n-+c~

uniformly for t E [a, b ]. We also have lim Un(S - 1/n) - u(s), n---~(x)

uniformly for s C [a, b ]. From Corollary A.2.2, we deduce that lim f (s, u~(s - l / n ) ) - f (s, u(s)), n---~oo

uniformly for s C [a, b ]. Therefore, passing to the limit both sides in Un(t) - S ( t -

a)~ + ~a t S(t - s ) f (s, Un(S - l / n ) ) d s ,

we obtain

fat -

s(t

-

+

s(t

-

for each t C [a, b], which shows that u is a C~ (11.2.1) on [ a, b ]. The proof is complete.

of the problem D

Saturated Solutions

255

We may now proceed to the proof of Theorem 11.2.1. P r o o f . Let (a,~) C 9 As 9 is open, there exist b > a and r > 0 w i t h [ a , b] x B(~, r) C_ 9 In addition, as f is continuous, diminishing r if necessary, we may assume that there exists M > 0 such that ]]f(t,u)l ] _< M for each (t, u) E [ a , b] x B(~, r). Let p" X -~ X be defined by y p(y)

-

for y C B(~, r)

~l----~(y[[y _ - ~) + ~ for y C X \ B(~, r).

We may easily see that p maps X to B(~, r), and is continuous on X. Next, let us define g ' [ a, b] x X -+ X by

g(t, y) - f (t, p(y) ) for each (t, y) C [a, b] x X. Inasmuch as f is continuous and bounded on [a, b] x B(~, r), we conclude that g is continuous and bounded on [a, b] x X. From Lemma 11.2.1, we know that the Catchy problem

u' - An + g(t,u) x(a)

-

has at least one C~ u "[ a, b] ~ X. As u(a) - ~ and u is continuous at t - a, diminishing b > a if necessary, we get that u(t) C B(~, r) for each t C [a, b]. Accordingly g(t, u(t)) - f ( t , u(t)), and therefore u "[ a, b] --+ X is a C~ of the problem (11.2.1). The proof is complete, rl 11.3.

Saturated Solutions

Let 9 be a nonempty and open subset in R x X, let f : 9 --+ X be a given function, and (a,~) E 9 and A : D ( A ) C_ X ~ X. Let us consider the semilinear Catchy problem

u ' - Au + f ( t , u ) u(a) --~.

(11.3.1)

of (11.3.1) on [a, b) we mean a function u "[a, b) --+ X By a C~ whose restriction to any interval[a, c] C [a, b) is a C~ of (11.3.1) on [a,c] in the sense of Definition 11.1.1. A C~ u " [a,b) --+ X of (11.3.1) is continuable if there exists another C~ v ' [ a , c ) --, X with b < c and such that u(t) - v(t) for each t E [a, b). A C~ u ' [ a , b) --+ X is called saturated if it is not continuable. If the projection of 9 on R contains R+, a solution u is called global if it is defined on [a, +co). We begin with a very simple but useful lemma.

256

The Catchy Problem for Semilinear Equations

L e m m a 11.3.1. Let A 9 D ( A ) C_ X --+ X be the generator of a Cosemigroup of contractions and let f 9 9 -+ X be continuous. A s s u m e that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Then a C~ u " [a,b) --+ X of (11.3.1) is continuable if and only if there exists u* - lim u(t)

(11.3.2)

(b, u*) E 9

(11.3.3)

tSb

and

P r o o f . The necessity is obvious, while the sufficiency is a consequence of Theorems 11.2.1 and 11.1.1. [5 R e m a r k 11.3.1. By virtue of Lemma 11.3.1, it follows that each saturated solution of (11.3.1) is necessarily defined on an interval of the form [a, b), i.e. on an interval which is open at the right. A sufficient condition for the existence of the limit (11.3.2) is stated below. P r o p o s i t i o n 11.3.1. Let A " D ( A ) C_ X -+ X be the generator of a Cosemigroup of contractions and let f " 9 --~ X be continuous. Furthermore, let u " [a, b) --+ X be a C~ of (11.3.1) and assume that b < +c~, and f (., u ( - ) ) C Ll(a, b ; X ) . Then there exists u* -- lim

t~b

P r o o f . Since g -

u(t).

f(., u(. )) is integrable, it follows that

v(t) - S ( t - a)~ +

S ( t - s)g(s) ds

is continuous on [a, b] and thus there exists limt~b v(t) -- v(b). Inasmuch [--] as v(t) - u(t) for each t E [a, b), this achieves the proof. We proceed next to the statement of a useful characterization of continuable C~ of (11.3.1). T h e o r e m 11.3.1. Let A " D ( A ) C X --+ X be the generator of a Cosemigroup of contractions and let f " 9 -+ X be continuous. A s s u m e that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. A necessary and sufficient condition in order that a Cosolution u " [a, b) --+ X of (11.3.1) be continuable is that the graph of u, i.e. graph u - {(t, u(t)) e I~ x X; t e [a, b)} be included in a compact subset in 9

Saturated Solutions

257

The proof is a simple copy of that of Theorem 10.3.1, and so we do not give details. We continue with a fundamental existence result concerning saturated C~ for (11.3.1). T h e o r e m 11.3.2. Let A " D ( A ) C_ X ~ X be the generator of a Cosemigroup of contractions and let f 99 -+ X be continuous. A s s u m e that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Let u " [a, b) -+ X be a G~ of (11.3.1). Then either u is saturated, or u can be continued up to a saturated one. The proof follows the same lines as those in the proof of Theorem 10.3.2 and therefore we omit it. From Theorems 2.3.1 and 11.3.2 it follows" C o r o l l a r y 11.3.1. Let A " D ( A ) C_ X --+ X be the generator of a Cosemigroup of contractions and let f " 9 --+ X be continuous. A s s u m e that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Then for each initial data (a,~) C 9 (11.3.1) has at least one saturated C~ We recall that a limit point of a function u 9 [a, b) --+ X as t tends to b is any element u* in X for which there exists a sequence (tk)kcN in [a, b) tending to b and such that l i m k ~ u(tk) -- u*. We denote the set of all limit points of u as t tends to b by Limtsbu(t). The behavior of saturated C~ at the right end point of their interval of definition is described by the following fundamental result. T h e o r e m 11.3.3. Let A " D ( A ) C_ X --+ X be the generator of a Cosemigroup of contractions and let f " 9 --~ X be continuous. A s s u m e that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Let u " [a,b) --+ X be a saturated C~ of (11.3.1). Then either (i) u is unbounded or (ii) u is bounded and, either u is global, or (iii) u is bounded and non-global, and in this case, either Limt$bu(t) is empty, or for each u* C Limt$bu(t), (b, u*) C Off). P r o o f . Up to one point, the proof parallels that one of Theorem 10.3.3. So, if u is unbounded on [a, b), or if it is bounded and global, or bounded and Limt$bu(t) is empty, we have nothing to prove. Thus, let us assume that u is bounded on [a, b) and non-global and Limt$bu(t) is nonempty. To prove (iii) let us assume the contrary, i.e. that there exists at least one sequence (tk)kcN in [a,b) tending to b and such that (u(tk))kEN is convergent to some u* C X~ but (b, u*) ~ 0 9 . So, a f o r t i o r i (b~ u*) C 9 Thanks to Lemma 11.3.1~ to get a contradiction, it suffices to show that

The Cauchy Problem for Semilinear Equations

258

there exists limtl-b u(t), which clearly must coincide with u*. To this goal, let us observe that there exists c > b and r > 0 such that [a, c) x B(u*, r) C 9 Furthermore, inasmuch as b < + e c , and f is continuous, diminishing r > 0 if necessary, we may assume t h a t there exists M > 0 such t h a t [If(T, v)ll < M,

(11.3.4)

for each (r,v) e [a, b] x B(u*,r). As limk+cc tk -- b, the semigroup is strongly continuous at t - 0 and l i m k + ~ u(tk) -- u*, we have

limS(h)u(tk) - u(tk) h40 uniformly for k C N. So we may choose k E N such t h a t

b-tk < 2M r IIS(t- tk)u(tk) -- u(tk)ll + Ilu(tk) - u*ll < ~

for t C [tk b) , 9

(11.3.5)

Fix k with the properties above. We shall show that for each t E [tk, b), we have u(t) E B(u*, r). Let t* - sup{t E [ tk, b); u(s) C B(u*, r), for s C [ tk, t l}. If t* -- b, we have nothing to prove. So, let us assume by contradiction t h a t t* < b. In other words, u(t) C B(u*,r) for t C [tk, t*], I l u ( t * ) - u*ll - r and there exist points t > t*, as close to t* as we like, such t h a t u(t) does not belong to B(u*, r). More precisely, t* is the "first m o m e n t in (tk, b) after which u leaves the set B(u*,r)". The fact that I l u ( t * ) - u*ll - r has a simple geometrical meaning, i.e. it signifies that, at t*, u must cross the b o u n d a r y of B(u*,r). From the considerations above combined with (11.3.4), and (11.3.5), we conclude r _< IIS(t* -

Ilu(t*) - u* II _< Ilu(t*) - u ( t k ) l l + Ilu(tk) - u* II

tk)u(tk)

-- u ( t k ) l l +

<_ ( b - tk)M +

[IS(t* -

Si*

IIS(t* -- ~ ) f ( ~ ,

tk)u(tk) --

u(~))lld~

+ I l u ( t k ) - u*ll

u ( t k ) l l + Ilu(tk) -- u*ll < r.

This contradiction (r < r) is a consequence of our supposition that, for at least one t E [tk, b), u(t) r B(u*, r). Then, for each t C [tk, b), we must have u(t) C B(u*, r). T h a n k s to (11.3.4) combined with Proposition 11.3.1 and L e m m a 11.3.1, it follows t h a t u is continuable, assertion which does not agree with the hypothesis. This contradiction can be eliminated only if (b, u*) C O9 and this achieves the proof. [q Under an additional hypothesis on f we shall prove a sharper result.

259

Saturated Solutions

T h e o r e m 11.3.4. Let A : D ( A ) semigroup of contractions and let that f maps bounded subsets in 9 the semigroup is compact, or f is argument. Let u" [ a, b) ~ X be a either

c_ X ~ X be the generator of a Cof : 9 ~ X be continuous. Assume into bounded subsets in X and, either locally Lipschitz with respect to its last saturated C~ of (11.3.1). Then

(i') u is unbounded and, if b < +cx~, limtTb Iiu(t)II- +c~, or (ii) u is bounded and, either u is global, or (iii') u is bounded and non-global, and in this later case there exists limtTb u ( t ) = u*, and (b, u*) E 0 9 P r o o f . In view of Theorem 11.3.3, we have merely to prove that whenever (ii) does not hold, then one of the two conditions (i') or (iii') must hold. So, let us assume first that (i') and (ii) do not hold. Then, from L e m m a 11.3.1 and Proposition 11.3.1, it follows that there exists lim u(t) = u*, tTb

while from (iii) in Theorem 11.3.3 we know that (b, u*) E 0 9 and so (iii') holds. Let us assume now that (ii) and (iii') do not hold and that b < +c~. To show that limtTb Iiu(t)II- +oc, let us assume the contrary, i.e. that, under these circumstances, we have lim inf Ilu(t) II < +oc. tTb

Equivalently, there exist at least one sequence (tk)kEN in (a, b), convergent to b, and r > 0 such that Ilu(tk)fl < r, for each k E N. Let C = {v E X; Ilvl] _ < r + l } . Since f maps bounded subsets i n 9 bounded subsets in X, b < +oc and C is bounded, there exists M > 0, such that I[/(~-, v)[ I _< M, (11.3.6) for each (T, v) E ([ a, b) • C) M 9

Now let us choose d > 0 satisfying d M < 1,

(11.3.7)

and fix k E N such that b - d < tk < b. Since u is unbounded on [a, b), it is necessarily unbounded on [tk, b). Then there exists t* E (tk, b) such that Ilu(r) [[ < r + 1,

for each r E Irk, t*) and II

(t*)ll = r + 1.

(11.3.8)

260

The Cauchy Problem for Semilinear Equations

Let us remark that r + 1 -Ilu(t*)ll-

<_ IIs(t* -

tk)u(tk) +

lIS(t* -

tk)u(tk)ll +

S(t*

-- ~)f(~,

fi"

_< Ilu(tk)ll +

LIS(t* - ~)l~(x)ll/(~,

Ilf(~,

u(~-))d~ll

u(~))lld~

u(~'))lldT.

Taking into account (11.3.6), (11.3.7), (11.3.8) and the obvious inequality t* - tk < d, we get r + 1 <_ r + (t* - t k ) M < r + d M < r + 1.

This contradiction can be eliminated only if lim Ilu(t)ll - +oc t?b

and this completes the proof.

[B

The proofs of the next results follow exactly the same lines as those of their counterparts in Section 10.3, and therefore we do not give details. We only notice t h a t there, due to the fact that f : R x X -~ X is compact, it maps bounded subsets in R x X into bounded subsets in X, and therefore it was no need to assume this explicitly, as we shall do here. C o r o l l a r y 11.3.2. Let A : D ( A ) C_ X ~ X be the generator of a Cosemigroup of contractions and let f : R x X -+ X be continuous. A s s u m e that f maps bounded subsets in 9 into bounded subsets in X and, either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Then a saturated C~ u " [a, b) --, X of (11.3.1) is either global, or it is not global, and in this case there exists lim ]]u(t)] I - + o c . ttb

C o r o l l a r y 11.3.3. Let A " D ( A ) C X ~ X be the generator of a Cosemigroup of contractions and let f " R x X -+ X be continuous. A s s u m e that f maps bounded subsets in 9 into bounded subsets in X and, either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Then a necessary and sufficient condition in order that a C O. solution u " [a, b) --+ X of (11.3.1) be continuable is that b < + e c and u be bounded on [ a, b). We conclude this section with a sufficient conditions on f ensuring the existence of global C~ of (11.3.1).

Asymptotic Behavior

261

T h e o r e m 11.3.5. Let A : D ( A ) C_ X --+ X be the generator of a Cosemigroup of contractions and let f :ll~+ • X --+ X be continuous. Assume that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Let us assume further that there exist two continuous functions h, k:IR+ --+ R+ such that ]If(% v)l [ _< k(~)llvll + h(~-),

(11.3.9)

for each (7-, v) C IR+ • X. Then, for each (a,~) e IR+ • X , (11.3.1) has at least one global C~

11.4. A s y m p t o t i c B e h a v i o r Let ~ be an open neighborhood of 0 E t~, f 9 It~+ • t~ --+ X a continuous function with f ( t , 0) - 0 for each t C R+, and let A " D ( A ) C_ X --+ X be the generator of a Co semigroup {S(t) ; t _> 0}. We consider the perturbed problem u' - Au + f (t, u). (11.4.1) In this section we will show that the asymptotic behavior of the null solution of the linear system u' - A u is inherited by the null solution of the system (11.4.1), provided that the perturbation f is dominated by the semigroup generated by A. First we recall: D e f i n i t i o n 11.4.1. The null solution x - 0 of (11.4.1) is asymptotically stable if for each a >_ 0 and each e > 0 there exists 6(a, e) > 0 and #(a) > 0 such that: (i) for each ~ E ~ with ][~]l <-- 6(a, c), each saturated solution u(., a, ~) of (11.4.1) with u(a,a,~) - ~ is global and Ilu(t,a,~)ll _< e for all t_>0; (ii) for each ~ E ft with ]]~]] _< p(a), each saturated solution u(., a, ~) of (11.4.1) with u(a, a, ~) - ~ is global and l i m t _ ~ ]it(t, a, ~)]] - 0. We notice that, in our general setting, we have no uniqueness and this explains why, the conditions in Definition 11.4.1 refer to all global solutions. We begin with the following fundamental result.

T h e o r e m 11.4.1. (Poinca%-Liapunov) Let A " D ( A ) C_ X --+ X be the generator of a Co-semigroup {S(t) ; t _> 0} and let f " IR+ • ~ --+ X be a continuous function. We assume that there exist M >_ 1, w > 0 and L > 0 such that (11.4.2) IlS(t)ll (x) for each t E It~+,

IIf(t,x)ll

Lllxll

(11.4.3)

262

The Catchy Problem for Semilinear Equations

:for any (t, x) C IR+ x ~t and L M - w < 0.

(11.4.4)

If either the semigroup is compact, or f is Lipschitz with respect to its last argument, then the null solution of (11.4.1) is asymptotically stable. P r o o f . Let ~ C f~, a > 0 and let u ( . , a , ~ ) " [a, Tm) --+ f~ be a given saturated solution of (11.4.1) satisfying the initial condition u(a,a,~) - ~. We notice that, excepting the case in which f is Lipschitz with respect to its last argument, this solution might not be unique. We will prove first that, if I1~11is sufficiently small, then u(., a, ~) is global, i.e. it is defined on [a, + ~ ) . To this aim, let us recall that for each t E [a, Tm), we have

u(t,a,~) - S ( t - a ) ~

+

S(t- s)f(s,u(s,a,~))ds.

From this relation, we deduce

lit(t, a, ~)II < IIS(t - a)II~(x)I1~11 +

IIS(t - s)II~(x)IIf (s, (s, a, ~))[I ds

from where, by virtue of (11.4.2) and (11.4.3), it follows that

lit(t, a, ~)II -< Me-W(t-a)I1{11 +

I'

LMe -w(t-s) lit(s, a, ~)]l ds

for any t c [a, Tin). Multiplying both sides of the inequality above by e wt, we get

e~tl[u(t,a,~)ll < Me-alibi ] +

fa t LMe~llu(s,a,~)]l

ds

for each t C [ a , Tin). Let v : [a, Tin) --+ IR+ be the function defined by

v(t) - e~t]lu(t , a, ~)11 for each t C [a, Tin). The preceding inequality takes the equivalent form

v(t) < Me~all{ll +

LMv(s) ds

for each t C [a, Tm). From Gronwall Lemma 10.2.1 it follows that

v(t) < Mewa]]~]]eLM(t-a) from where, recalling the definition of the function v, we deduce ][u(t, a, ~)]] for any t C [a, Tm).

<

M]]~[]e(LM-w)(t-a)

(11.4.5)

Asymptotic Behavior

263

Let now p > 0 be such that B(0, p) C f~ and let It(a) > 0 be defined by #(a)-

p 2M" Then, by virtue of (11.4.4) and (11.4.5), for each ~ C f~ with I1~11 -< p(a), we have lit(t, a, ~)II < p-2 for any t C [a, Tin). Assuming that Tm < +c~, from this inequality and Proposition 10.3.1, it follows that there exists lim u(t, a, ~) - u*

t~Tm

and u* e B(0, ~) C ~, relation which, by virtue of (iii) in Theorem 11.3.3, shows that u(., a, ~) is not saturated. This contradiction can be eliminated only if, for each ~ C ~t satisfying I1~11 -< #(a), we have Tm - +c~. Now, taking (~(a,c) - m i n { p ( a ) , ~ } , we conclude that Ilu(t,a,~)ll <_ c for each t >_ a. Finally, let us observe that, from the above proof and (11.4.5), it follows that, for each ~ C f~ with I1~11-< p(a), we have lim u ( t , a , ~ ) - O, t?+~ and this completes the proof.

D

A useful consequence is: T h e o r e m 11.4.2. Let A " D ( A ) C_ X --~ X be the generator of the Cosemigroup of type ( M ~ - w ) , {S(t) ; t _> 0}, with M >_ 1 and w > 0 and let f " I~+ x f~ --+ X be continuous. We assume that there exists ~ " I~+ --~ I~+ such that

]If(t, )11 _< (11 11) for each (t, u) C R+ x ~t, and

lim c~(r) = 0. r$0 r If either {S(t) t _> 0} is compact, or f is Lipschitz with respect to its last argument, then the null solution of (11.4.1) is asymptotically stable.

P r o o f . Let us fix L > 0 satisfying (11.4.4) and let us choose ~ > 0 such that ~(r) ~ Lr

for r C [0~ (~). Let us consider the restriction of f to I~+ x {x E ~ ; Ilxll < (~}, and let us observe that we are in the hypotheses of Theorem 11.4.1. The proof is complete. [:3

264

The Cauchy Problem for Semilinear Equations

11.5. T h e K l e i n - G o r d o n E q u a t i o n R e v i s i t e d

In this section we reconsider the Klein-Gordon equation in Example 10.4.1 in a different frame, i.e. in the case in which the domain ~ is not necessarily bounded and/or smooth, i.e. with boundary F of class C 2. So, let ~ be a nonempty and open subset in R n, let us denote QT = (0, T) x ~t, and F~T -- (0, T) x F. For the sake of simplicity, throughout this section, we consider only a perturbation g which does not depend on t and x, i.e. g : IR --+ I~. Let us consider the Klein-Gordon semilinear equation

~(t, x) - 0 ~(0,~) -~o(x) ~(0,x)~o(x)

(t, x) e r~T 9e 9e

(11.5.1)

where g- I~ --+ I~ is a continuous function. T h e o r e m 11.5.1. Let g" I~ -+ I~ be a continuous function for which there exist c > 0 and c~ C 1~+ such that

g(0) - 0 Ig(~) - g(v)l <_ c(1 + I~ I~ + Ivl ~)1~ - ~l

(11.5.2)

for each u , v C I~, where ~ >_ O, if n - 2, and a < 2 / ( n - 2), /f n >_ 3. Then, for each uo e H~(~), and each vo e L2(~), there exists T > O, such that the problem (11.5.1) has a unique saturated solution u satisfying

(i) u e C([ O, T); Hlo (~)) ; (ii) ut e C([ 0, T); L2(~)) ; (iii) u e C 2 ( [ O , T ] ; H - I ( ~ ) ) . In addition, if T < + ~ , then

sST ([lu(s)[[H~(a)-Jr-IlUt(S)l[L2(a))-- -~(:xD.

lira

P r o o f . We note that (11.5.1) can be rewritten as a differential equation of the form (11.1.1) in a suitably chosen infinite-dimensional space. So, let H - H01 (~) • L 2 (~t) which, endowed with the inner product (.~ .} defined by ((~tl,Vl),(U2,V2)

} --

/ Ull(X)Ul2(x)dx -Jr-/ Vl(X)V2(x)dx

for each (Ul, Vl), (u2, v2) E H, is a real Hilbert space. We define the operator A" D ( A ) C H --+ H by D(A) = (H2(~t) N Hol(~)) x (H~(~) N L2(~)) A(u, v ) = (v, Au) for each (u, v) e D(A).

A Parabolic Semilinear Equation

265

Next, let us define f : D ( f ) ~ H by D(f)-

{(u,v) e H ; g(u(.)) e L2(f~) for each t e [0,+cc)}

f ((u, v))(x) - (0, g(u(x))) for each (u, v) e D ( f ) . One may easily see that the problem (11.5.1) can be rewritten under the form

z'-Az+f(z) z(a) = ~ , where z ( t ) ( x ) = ( u ( t , x ) , v ( t , x ) ) a.e. for (t,x) e (0, T) x (f~), ~ = (u0, v0), and A and f are as above. In order to prove that A and f satisfy the general hypotheses in Theorem 11.1.1, let us remark first that, by virtue of Theorem 4.6.2, A generates a C0-group of isometries, and therefore it generates a C0-semigroup of contractions as well. On the other hand, let us observe that, from (11.5.2), Theorem 1.5.4 and Lemma A.6.1, it follows that D ( f ) = H and f is continuous on H. Let us observe that in fact f is locally Lipschitz on bounded subsets in H. This easily follows again from the condition (11.5.2) and Lemma A.6.1. We are thus in the hypotheses of Section 4.6, of Theorem 11.1.1 and Corollaries 11.3.1 and 11.3.2 and this achieves the proof. [3

11.6. A P a r a b o l i c S e m i l i n e a r E q u a t i o n Our aim here is to illustrate the power of Theorem 11.2.1 by means of a simple, but non-trivial application. More precisely, as we shall show next, Theorem 11.2.1 is very appropriate in the study of some parabolic semilinear partial differential equations. Let f~ be a nonempty, bounded, and open subset in IRn, whose boundary F is of class C 2, let QT = (0, T) x f~, and ET = (0, T) x F. Let us consider the semilinear parabolic partial differential equation -

+ g ( t , x,

u(t,x)x) -

0

(t,

c QT

(t,x) C ET

(11.6.1)

x c

where g : ~ + x gt x R --+ IR is a continuous function. Using Pazy's Theorem 11.2.1, we shall prove that the problem (11.6.1) has at least one local solution. More precisely, we have:

11.6.1. If g : ] R + x f~ x IR --+ IR is a continuous function for which there exist c > 0 and d > 0 such that

Theorem

]g(t,x, u)l < clul + d

(11.6.2)

The Cauchy Problem for Semilinear Equations

266

for each (t,x,u) C I~+ • f~ • I~, then, for each ~ C L2(f~), there exists at least one solution u "[0, +:x~) -+ L2(f~) of (11.6.1) satisfying for T > O" (i) u e C([O,T];L2(f~)); (ii) t ~+ v/tu'(t, .) C L2(O,T;L2(gt)) ; (iii) t~-+ IIVu(t, ")]lL2(a) e LI(0, T ) N AC(5, T) for each 5 e (0, T) ; (iv) u e L2(3, T;H2(f~)) N WI'I(~,T;H~(a)). If ~ e H2(a) N H i ( a ) , then (v) u e WI'2(O,T;L2(~)) NL2(O,T;H2(~)) N WI'I(O,T;H~(~)). If, in addition, d = 0 and c is small enough, then we have also (vi) lim I[u(t)l[L2(fl)--O. t-+cxD

P r o o f . Let A ' D ( A )

C_ L2(f~) -+ L2(f~) be the operator defined by D(A) - H I ( ~ ) A H2(f~) An-An foruED(A).

Then, by virtue of Theorems 4.1.1 and 7.2.5, A is the infinitesimal generator of a compact Co semigroup of contractions on L2(ft). Let us define the mapping f ' [ 0 , +co) • D ( f ) C_ L2(f~) -+ L2(f~) by D(f)f (t,

{u e L2(f~); g(t, . , u ( . ) ) e L2(~t) for each t e I~+} - g(t,

for each u C D ( f ) each t C [0, + ~ ) and a.e. x C f~. Let us observe that, since 9 is continuous and satisfies the sublinear growth condition (11.6.2), it follows that D ( f ) = L2(f~). Again, from (11.6.2) and Lemma A.6.1, we deduce that f is continuous on R+ • L2(f~). Finally, let us observe that the problem (11.6.1) can be rewritten in the space L2(f~) under the form

u ' - Au + f ( t , u ) u(0) - ~,

(11.6.3)

with A and f defined as above. In view of Corollary 11.3.1, combined with Theorem 11.3.5, we know that, for each ~ C L2(f~), there exists at least one C~ of (11.6.3) defined on [0, +c~), solution which obviously satisfies (i). From Theorem 4.1.2, we have that A is self-adjoint, and thus we are in the hypotheses of Theorem 8.2.1, from where we deduce (ii), (iii), (iv) and (v). Since (vi) is a direct consequence of Theorem 11.4.1, the proof is complete. D R e m a r k 11.6.1. Using a truncation procedure for the function g and estimates of the gradient of the function u(t, .) obtained directly by (11.6.1), and making use of Theorem 1.5.4, one can show that Theorem 11.6.1 can be extended to the more general case in which, instead of (11.6.2), g satisfies

Ig(t, x,

_< r

+ d

Problems

267

where p > 1 depends on n. Problems P r o b l e m 11.1. Let A 9 D(A) C_ X ~ X be the generator of a compact C0-semigroup of contractions, let ]I be a n o n e m p t y and open interval, ~ a n o n e m p t y and open subset in X, and f 9 ]I x ~t ~ X a function satisfying" (i) for a.e. for t E ]I, f(t, .)" ~ --+ X is continuous ; (ii) for each x E f~, f(., x)" I[ --+ X is measurable ; (iii) for each ~ E f~, there exists r > O, and a compact subset K in X such t h a t f ( t , x ) E K for each x E B ( ~ , r ) M f~, and a.e. for t E ]I. Then, for each a E]I and each ~ E f~, there exists b > a with [a, b] C ]I, and such t h a t the Cauchy problem

u ' - Au + f ( t , u ) u(a) has at least one C~

u ' [ a, b] ~ f~.

P r o b l e m 11.2. Let A : D(A) C X -+ X be the infinitesimal generator of a C0-semigroup of contractions, let I[ be a n o n e m p t y and open interval, ~t a n o n e m p t y and open subset in X, and f : ]I x ~ --+ X a function satisfying: (i) for a.e. for t E]I, f(t, .): ~t ~ X is continuous ; (ii) for each x E ~, f(-, x ) : ]I ~ X is measurable ; (iv) for each ~ E ~ there exists r > 0, and a locally integrable function t~: ]I -+ I~+, such t h a t IIg(t, u ) - g(t, v)l I < t ~ ( t ) i i u - vii for each u, v E B(~, r) M gt, and a.e. for t E ]I, and g(., ~) :I[ --+ X is locally integrable. Then, for each a E ]I and ~ E ~, there exists b > a with [a, b] C ]I, and such t h a t the Cauchy p r o b l e m

u'-

Au + f ( t , u ) -

-

has a unique C~ u 9 [a, b] -+ ~. If, in addition, ~ - X, and the function t~ in (iv) is independent of r > 0, then this solution can be continued up to a global one, i.e. defined on [a, b), where b = sup 1I. P r o b l e m 11.3. Let a E ]~3, and f 9 IR+ x ]~3 ___+]~3 & function satisfying (i), (ii) in P r o b l e m 11.1, and (iv) in P r o b l e m 11.2. If the function t~ in (iv) is independent of r > 0, then, for each u0 E Lp(~3), the semilinear Cauchy problem

ut+a'Vu-f(t,u)

(t,x) EI~+ xlR 3

u(O,x) --Uo(X )

X e ]~3

has a unique solution defined on I~+.

268

The Catchy Problem for Semilinear Equations

P r o b l e m 11.4. Let A be an n x n matrix with real elements, and let f : R+ x R n --+ It(n be a function satisfying (i), (ii)in Problem 11.1 and (iv) in Problem 11.2. If the function g in (iv) is independent of r > 0, then, for each u0 C LP(IR3), the semilinear C a t c h y problem

ut+(Ax, V u ) - f ( t , u )

(t,x) E I R + x I R 3 9 e

has a unique solution defined on R+. P r o b l e m 11.5. Let a > 0, c > 0 and let c > 0 be "very small". Let 99: I~ -+ R be a continuous function satisfying I~(x)] < c for each x C R and c if Ix I _< a

~(x)-

0 iflxl > a + c

and let us consider the C a t c h y problem for the neutron transport in a stab equation

, I ut(t, xy)+yux(t, x y, ) - ~ (2x ) ; 1 u(t,x,z) dz v) -

v).

Prove that, for each ~ E Cub(]~ x [ - 1 , 1 ]), the problem above has a unique solution u : JR+ --+ C~b(R x [ - 1 , 1 ]). Notes. The main result in Section 11.1 is due to Segal [111] and the one in Section 11.2 was proved by Pazy [99]. A nonlinear version of the latter was obtained by Vrabie [124]. For other results of this type see Pavel [97] and Vrabie [127]. Both Sections 11.3 and 11.4 contain slight extensions of some classical results on the continuation of the solutions and on their asymptotic behavior. See Sections 2.4 and 5.3 in Vrabie [129], and Cazenave and Haraux [36], Theorem 10.2.2~ p. 157. The example analyzed in Section 11.5 is a reformulation of Theorem 6.2.2 in Cazenave and Haraux [36], while that in Section 11.6 is inspired from Pazy [101]. The first two Problems 11.1 and 11.2 extend Theorems 11.2.1 and 11.1.1 respectively to the case of Carath~odory perturbations. Problems 11.3 and 11.4 are classic, while Problem 11.5 is inspired from Richtmyer [110].

C H A P T E R 12

Semilinear Equations Involving Measures

Here, we consider the Cauchy problem for the semilinear differential equation with distributed and/or spatial measures

d u - {Au}dt + dg~ u(a) - ~. In Section 12.1 we prove a local existence and uniqueness result in the case when u ~ gu is Lipschitz from L ~ ( a , b ; X ) to BV([a,b] ;X) and A: D(A) C_ X -+ X generates a C0-semigroup of contractions, continuous from (0, +co) to L ( X ) in the uniform operator topology. In Section 12.2 we study the problem of local existence assuming that A generates a compact C0-semigroup of contractions and u ~ g~ is continuous in some weak sense. In Section 12.3 we present the main results concerning saturated solutions and in Section 12.4 we reconsider the problem in Section 12.2 within a more general setting, i.e. allowing ~ C XA and g C BV([a,b];XA). Finally, in the last two Sections 12.5 and 12.6 we include some examples.

12.1. T h e P r o b l e m

du = { A u } d t + dgu w i t h u ~ gu L i p s c h i t z

Let X be a real Banach space, let A : D ( A ) C_ X ~ X be the generator of a C0-semigroup {S(t) ; t >_ 0}, continuous in the uniform operator topology from (0, + o c ) t o L ( X ) , and let G : LCC(a,b;X) --+ B V ( [ a , b ] ; X ) be a given function. Let { E X, and let us consider the Cauchy problem

du - { A u } d t + dgu u(a) - ~,

(12.1.1)

where gu - G(u). As in the preceding chapters, here, we deal only with C0-semigroups of contractions, but we emphasize that all the results hold true for general C0-semigroups. 269

Semilinear Equations Involving Measures

270

D e f i n i t i o n 12.1.1. The mapping G : L~176 b;X) --+ BV([a, b]; X) is called hereditary if for each u , v E L~176 satisfying u = v a.e. on [a, c ], we have also Var (G(u) - G(v)), [a, c ]) = 0. D e f i n i t i o n 12.1.2. The function G : L~176 b;X) --+ BV([a, b];X) is called locally Lipschitz if there exists a function t~ : (0, +oo) --+ (0, + c o ) with lim t~(x) = 0 and such t h a t x;0 Var (G(u) - G(v), [a, c]) _< t~(c -

aDllu vllLoo(a,c;X > -

for each u, v E L~176 b ; X ) and each c E [ a , b]. R e m a r k 12.1.1. It is easy to observe t h a t each locally Lipschitz function is hereditary. D e f i n i t i o n 12.1.3. Let G : L~176 b ; X ) -+ BV([a, b]; X ) be hereditary. (12.1.1)on [a,c] if it A function u : [a,c]--+ X is called an L~~ is an L~176 on [a, c] of the Cauchy problem below

du - {Au}dt + dg~, -

where g E L~176 b;X) satisfies g = u a.e. for t E [a, c]. Remark

12.1.2. We notice that, since G is hereditary, for each g and ~ in

L~176 b ; X ) with g = 5 = u a.e. for t E [a, c], it follows t h a t G ( u ) - G(v) is constant on [a, c]. Therefore, dg~, = dg~ in the sense of vector measures on [a, c]. So, without any danger of confusion, we can denote both these measures by dgu. A very simple but useful local existence and uniqueness result is: 12.1.1. Let A : D(A) C_ X --~ X be the generator of a Cosemigroup of contractions, {S(t) ; t >_ 0}, continuous from (0, +oc) to L ( X ) in the uniform operator topology. Let G: L~ b;X) ~ BV([a, b l; X ) be a locally Lipschitz mapping. Then, for each ~ E X, there exists c E (a, b] such that the problem (12.1.1) has a unique L~-solution defined on [a, c].

Theorem

P r o o f . Let c E (a, b] be such t h a t

g ( c - a ) < 1. We define the operator Q : L ~176 c; X) --+ L o~(a, c ; X ) by

Q(u)(t) - S(t - a) +

S(t - s)dg~(s)

for each u E L~ c ; X ) and each t E [a, c], where dgu is defined as in R e m a r k 12.1.2. At this point let us observe t h a t u is an L ~ - s o l u t i o n of

The Problem du = {Au}dt + dgu with u H gu Lipschitz

271

(12.1.1) on [a, c] if and only if u is a fixed point of Q. So, to complete the proof we have merely to show that Q has a unique fixed point. We shall do this with help of Banach fixed point theorem. To this aim let us observe that

ilQ(u)(t) - Q(v)(t)[[ <_

s(t-

s)d(g~ - g~)(s)

V~r (g~ - g~, [a, t]) _< Vat (g~ - g~, [a, ~]) ___ e ( c - a ) l l ~ ~llL~(a,~;X) for each u, v E L ~ ( a , c ; X ) . But this inequality clearly shows that _<

IIQ(u)

-

Q(v)IIL~(a,~;x) <_ g ( c -

a)llu

-

vllL~(a,~;X)

for each u, v E L~176 c ;X). Since g.(c-a) < 1, Q is a strict contraction and therefore, by virtue of Banach fixed point theorem, it has a unique fixed point as claimed. The proof is complete. K] Let us consider next a useful application of Theorem 12.1.1. Let 2) be a nonempty and open subset in R x X, let f : 9 --~ X a continuous function, g E B V ( [ a, b, ]; X) and (a, ~) E R x X. We consider the Cauchy problem

du - {Au + f ( t , u ) } d t + dg ~(a) - ~.

(12 1.2)

D e f i n i t i o n 12.1.4. By an L ~ - s o l u t i o n of (12.1.2) on [a,c] we mean a function u : [a, c] ---, X which satisfies: (i) for each t E [ a , c), there exists u(t + 0); (ii) for each t C [a, c), (t, u(t + 0)) E 9 (iii) t H f ( t , u(t + 0)) E Ll(a, c ;X) and u is an L~-solution on [a, c] of

du -- { A u } d t + dh(t) u(a) in the sense of Definition 9.1.1, where, for each t E [ a, c ],

h(t) -

f ( s , u(s + 0))ds + g(t).

R e m a r k 12.1.3. If u : [a, c]--+ X is an L~176 of (12.1.1) then u is piecewise continuous on [a, c] and consequently f ( t , u(t + 0)) = f ( t , u(t)) excepting for an at most countable subset in [a, c ]. This shows that h in Definition 12.1.4 is given in fact by

h(t) -

f (s, u(s)) ds + g(t)

for each t E [a, c]. So, we may question why, in Definition 12.1.4, instead of (ii), we do not ask (t,u(t)) C 9 for each t C [a, c], as in the case of

Semilinear Equations Involving Measures

272

C~ The answer rests in that, for L~176 it may happen that, at some point of discontinuity t E (a, c], (t, u ( t - 0)) ~ 9 and thus, in order to enter again 9 (t, u(t)), which is uniquely determined by the "left jump" condition u(t) - u(t - O) = g(t) - g(t - 0), must be, not only outside 9 but even outside its closure. 12.1.2. If A : D(A) C_ X ~ X generates a Co-semigroup of contractions, {S(t) ; t > 0}, which is continuous in the uniform operator

Theorem

topology from (O,+oc) to L ( X ) , f : 9 ~ X is continuous and locally Lipschitz with respect to its last argument, and g E BV([a, b ] ; X ) then, for each (a, ~) E R x X satisfying (a, g(a + O) - g(a) + ~) E 9 there exists c > a such that the problem (12.1.2) has a unique Lee-solution defined on [ a, c ]. P r o o f . Let (a, ~) E R x X with (a, g(a + O) - g(a) + ~) E 9 and let us denote by ~ = g(a + O) - g(a) + ~. Inasmuch as 9 is open, there exist b > a and r > 0 such that [a, b] x B(r/, r) C 9 In addition, as f is continuous and locally Lipschitz with respect to its last argument, diminishing b > a a n d / o r r > 0 if necessary, we may assume that there exists L > 0 such that Ilf(t, u) - f(t, v)ll < Lllu - vii for each (t, u), (t, v) E [a, b] • B(~, r). Let p: X ~ X be defined by y

for y E B(r/, r)

p(y) -

Ily - vii (y - v) +

for y c X \ B(V,

We may easily see that p maps X to B(r/, r), and is Lipschitz continuous on X with Lipschitz constant 2. Next, let us define f~ : [a, b] x X --~ X by

fr(t, y) = f (t, p(y)) for each (t, y) E [a, b] x X. Since f is continuous and globally Lipschitz with respect to its last argument on [a, b] x B(~, r), we conclude that f~ enjoys the same properties. Let us define G : L ~ ( a , b ; X ) --+ B V ( [ a, b]; X) by

/a We note that G is well-defined because, for each u E L ~ ( a , b ; X ) , the mapping s ~ fr(s, u(s)) is strongly measurable and bounded. While the boundedness is obvious, the measurability follows from the observation that it can be approximated a.e. by a sequence of countably-valued functions

273

The Problem du - {Au}dt + dgu with u H gu Continuous

simply because u enjoys this property and f~ is continuous. See (i) and (iii) in Definition 1.1.1. Now, let us remark that the problem (12.1.2) can be rewritten under the form (12.1.1), where g, - G(u), with G defined as above. Let us observe that

/a c

Var ( G ( u ) - G ( v ) , [ a , c ] )

IIf~(s, u ( s ) ) - f ~ ( s , v ( s ) ) l l d s

<

<_ 2L(c - a)II u

-

vllL~(a,c

;x)

for each c E (a, b] and each u, v E L c~ (a, c ;X). This shows that G is locally Lipschitz (with g(x) = 2 L x ) . By virtue of Theorem 12.1.1, we know that the Cauchy problem du - { A u } d t + dg~(t)

has a unique L~ u : [a, c] ~ X provided c E (a, b] is small enough. Recalling that, by virtue of (9.2.1) in Theorem 9.2.1, + 0 ) - - g ( a + 0) -

+

-

diminishing c > a if necessary, we get that u(t) E B(rl, r) for each t E [a, c ]. Hence (t, u(t + 0) E [a, c] x B(r/, r) C 9 for each t E [a, c). Accordingly f~(t, u(t + 0)) = f ( t , u(t + 0)), and therefore u : [a,c] ~ X is in fact an L~-solution of the problem (12.1.2). The proof is complete. [3

12.2. T h e P r o b l e m du - { A u } d t + dg~ w i t h u ~ g~ C o n t i n u o u s Let X be a Banach space and let A " D ( A ) c_ X -~ X be the infinitesimal generator of a compact C0-semigroup of contractions {S(t); t _> 0}, let G 9 L ~ ( a , b ; X ) ~ B V ( [ a , b ] ; X ) be a given function, ~ E X and let us consider the Cauchy problem (12.2 1)

du - { A u } d t + dgu -

where gu - G(u). As in the preceding section, the meaning of the L ~ solution is that given by Definition 9.1.1. D e f i n i t i o n 12.2.1. The mapping G 9 n ~ ( a , b ; X ) ~ B V ( [ a, b ] ; X ) is bounded and weakly pointwise continuous if its range is of equibounded variation and for each sequence (Un)nENsatisfying (Un)nEN is bounded in L ~ ( a , b ; X ) lim U n - U in L l ( a , b ; X ) n----+ o o

and

Semilinear Equations Involving Measures

274 we have

lira G(un)(t) - G(u)(t) n-----~cx:)

in the or(X, X~

for each t E [a, b ].

The main result in this section is" 12.2.1. Let A : D(A) C X ~ X be the generator of a compact Co-semigroup of contractions and let G : L~(a, b;X) ~ BV([a, b]; X)

Theorem

be bounded and weakly pointwise continuous. Then, for each ~ E X, the problem (12.2.1) has at least one L~-solution defined on [a, b]. P r o o f . Let M > 0 be such that Var (gu, [a, b]) < M for each u E LC~(a, b;X). Let ~c E X, let r = I1~11+ M and let K be the closed ball with center 0 and radius r in LC~(a, b;X). We define the operator Q : K ~ L~(a, b ; X) by

Q(u)(t) - S ( t - a)~ +

S ( t - s)dgu(s)

for each u E K and each t E [a,b]. Obviously u is an L~-solution of (12.2.1) if and only if u is a fixed point of Q. So, to complete the proof, it suffices to show that Q has at least one fixed point in K. In order to prove this, we will show that Q satisfies the hypotheses of Schauder fixed point theorem A.1.5. First, let us observe that

IIQ( )(t)ll

IIS(t- a) l[

+

fa s(t -

11411+ Var (g, [a, t l) _< 1141 +Var(g,[a,b]) < r. So Q maps K into itself. Since G is weakly pointwise continuous and bounded, by virtue of Theorem 9.4.2, it follows that Q(K) is relatively compact in Ll(a, b;X) (in fact in any LP(a, b ; X ) , provided p E [ 1, +c~)). Summing up, K, as a subset of Ll(a, b ; X ) , is closed, convex and bounded and Q ( K ) is relatively compact. To achieve the proof it remains merely to check that Q is continuous from K to K in the norm topology of L l(a, b;X). But, this follows from the weak pointwise continuity and boundedness of G combined with the relative compactness of Q(K), along with Theorem 9.5.1. Indeed, let u E K and let (Uk)kEN be a sequence in K which converges in L l(a, b;X) to u. Since G is bounded and weakly pointwise continuous and Q ( K ) is relatively compact, by Theorem 9.5.1, it follows that lim Q(uk) - v

in Ll(a b ' X ) and

lim Q ( u k ) ( t ) - v(t)

in the ~ ( X , X ~ - t o p o l o g y on X,

k---+cx3

The Problem du - {Au}dt + dgu with u ~ gu Continuous

275

where v : [a, b] -+ X is the unique generalized L~-solution of the problem

dv - {Av}dt + dgu v(a) = ~. Inasmuch as gu belongs to B V ( [ a, b] ;X), the problem above has a unique L~-solution which must coincide with v. So v = Q(u) and Q is continuous from K to K in the norm topology of Ll(a, b ; X ) . Consequently Q satisfies the hypotheses of Schauder fixed point Theorem A.1.5 and this completes the proof. E:] A useful application of Theorem 12.2.1 is given below. Throughout in this chapter, BV(I~; X ) denotes the space of all functions g" I~ -+ X whose restrictions to any interval [a, b] belong to B V ( [ a, b]; X). Let 9 be a nonempty and open subset in I~ x X, let f 99 --+ X be a continuous function and g C B V ( i ~ ; X ) . We consider the C a t c h y problem

du - { A t + f ( t , u ) } d t + dg u(a) = ~.

(12.2.2)

The next local existence theorem, extending Pazy's Theorem 11.2.1, is useful in the study of semilinear parabolic problems with distributed measures as we shall see in Section 5. We notice that Pazy's Theorem 11.2.1 refers to the special case g - 0. T h e o r e m 12.2.2. If A " D(A) C_ X --+ X is the infinitesimal generator of a compact Co-semigroup of contractions {S(t) ; t > 0}, f " 9 --+ X is continuous, and g C BV(IR; X ) , then, for each (a, ~) C I~ x X satisfying (a, g(a + O) - g(a) + ~) C 9

there exists b > a such that the problem (12.2.2) has at least one L ~162 solution defined on[a, b]. P r o o f . Let (a, ~) r IR x X with (a, g(a + O) - g(a) + ~) C 9 and let us denote by r / - g(a+O)-g(a)+~. Inasmuch as 9 is open and f is continuous, there exist c > a, r > 0 and M > 0, such that [a,c]. x B(~?,r) C 9 and ]If(t, u)]] < M

(12.2.3)

for each (t, u) C [ a , c] x B(~?, r). Let us define p" X --+ X by y

for y e B(~, r)

p(y) x \

276

Semilinear Equations Involving Measures

Clearly, p maps X to B(rl, r) and is continuous. function fr :IR x X --+ X by

I f(a,p(u)) fr(t,u) f(t,p(u)) f(c,p(u))

i f t c ( - o c , a] i f t C (a,c) iftc[c,+ec)

Now, let us define the anduCX and u C X anduCX.

From (12.2.3), we conclude that f~ is bounded. Since both functions f and p are continuous, it follows that fr is continuous. Finally, let us define the mapping G : L ~ ( a , c ; X ) --+ BV([a,c] ;X) by a(**)(t)

- g

(t) -

+ g(t)

for each u C L ~(a, c ; X ) and t C [a, c]. It is easy to see that G is bounded, continuous with respect to the variation seminorm on BV([a, c] ;X), and thus weakly pointwise continuous in the a(X, X~ By virtue of Theorem 12.2.1, it follows that the Cauchy problem

du - {Au + fr(t,u)}dt + dg -

u : [a, c] --+ X. Since, u(a) = ~ and, by (9.2.1)

has at least one L~176 in Theorem 9.2.1,

u(a + O) = g(a + O) - g(a) + u(a) = ~, there exists b e (a, c] such that, for each t e (a, b], u(t) e B(rl, r). But in this case p(u(t)) = u(t) for each t e (a,b], and consequently f~(s, u(s)) must coincide with f(s, u(s)) for each s e (a, b]. Since u : [a, b] --+ X is piecewise continuous and (t,u(t)) e (a,c] x B(rl, r) C 9 it follows that (t, u(t + 0)) e [a,c] x B(rl, r) C 9 and thus, by Remark 12.1.3, u is an L~-solution of the problem (12.2.2). This completes the proof. D

12.3.

Saturated

L~-Solutions

In this section, we shall say a few words about saturated L~176 the semilinear Cauchy problem

du - {Au + f(t,u)}dt + dg u(a) = ~.

for

(12.3.1)

Here, as in the preceding sections, 9 is a nonempty and open subset in R x X, f : 9 --+ X is a continuous function and g C BV(IR;X).

277

Saturated L ~ -Solutions

D e f i n i t i o n 12.3.1. An L ~ solution u : ] I --+ X of (12.3.1), ]I = [a,b], or ]I = [a, b), is called continuable if there exists an LCC solution v : [a, c] --+ X of (12.3.1), with C > b and such that u(t) = v(t) for all t E]I. A solution which is not continuable is called saturated, or noncontinuable. R e m a r k 12.3.1. In contrast with the case considered Chapter 11, where each saturated C~ were necessarily defined on an interval of the form [a, b), here it may exist saturated L~-solutions of (12.3.1) which are defined on a closed interval [a, b]. Clearly this happens only if the right jump of g at b is such that (b, g ( b + O ) - g ( b ) + u ( b ) ) ~ 9 See Theorems 12.1.2 and 12.2.2. Therefore, in all that follows, we confine ourselves only to the study of saturated L~-solutions defined on intervals of the form [a, b). L e m m a 12.3.1. Let A : D ( A ) C_ X --+ X be the generator of a Cosemigroup of contractions and let f : 9 -+ X be continuous. A s s u m e that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Let g E B V ( R ; X ) . Then an L ~ - s o l u t i o n u : [a, b) --+ X of (12.3.1) is continuable if and only if there exists u* = limu(t)

(12.3.2)

(b, g(b + O) - g(b - O) + u*) e 9

(12.3.3)

tSb

and

P r o o f . Necessity. Assume that u is continuable. (9.2.1) and (9.2.3), we have

Then, in view of

g(b + O) - g(b - O) + u* = u(b + O) - u(b - O) + u(b - O) = u(b + O)

and, by (i) in Definition 12.1.4, (b, u(b + 0)) C 9 Thus (12.3.3) holds. Sujficiency. Let us assume that (12.3.2) and (12.3.3) hold. Take = g(b) - g(b - O) + u*

and let us observe that g(b + O) - g(b) + ~ = g(b + O) - g(b - O) + u*. Then, by virtue of (12.3.3), we are in the hypotheses of either Theorem 12.1.2, or Theorem 12.2.2~ and accordingly the problem dv - { A v + f ( t , v ) } d t + d g v(b) - g(b) - g(b - O) + u*

has at least one L~-solution v : [b,c) -~ X. By Proposition 9.1.1, we conclude that the concatenate function w : [ a, c) -+ X, defined by w(t)-

u(t) v(t)

forte[a,b) for t c [ b , c )

is an L~-solution of (12.3.1) on [a,c) which coincides with u on [a,b). Thus u is continuable and this completes the proof. [Z]

278

Semilinear Equations Involving Measures

P r o p o s i t i o n 12.3.1. Let A 9 D ( A ) C X --+ X be the generator of a Co-semigroup of contractions which is continuous in the uniform operator topology from (0,+cr to L ( X ) , let f 9 9 --+ X be continuous and let g 6 B V ( I ~ ; X ) . Let u " [a,b) ~ X be an L~ of (12.3.1). If b < +e~ and t ~-~ f ( t , u ( t + 0 ) ) belongs to L I ( a , b ; X ) , then there exists u(b - 0) - lim u(t). ttb P r o o f . Since u is given by u(t) - S(t - a)u(a) +

/a'

S(t - s) f (s, u(s + 0))ds +

/a'

S(t - s) dg(s)

and the first two terms on the right-hand side have limit as t tends to b as the semigroup is continuous and t ~ f ( t , u ( t + 0)) belongs to L l ( a , b ; X ) , it remains only to show that there exists lim t?b

S(t - s)dg(s).

But this is a direct consequence of the second part of Theorem 9.2.1. The proof is complete. [3 R e m a r k 12.3.2. The condition t ~ f ( t , u(t + 0)) belongs to Ll(a, b; X ) in Proposition 12.3.1 above can be replaced with, either t ~ f ( t , u ( t ) ) , or t ~-~ f ( t , u ( t - 0)) belongs to L l ( a , b ; X ) . This follows from the simple observation that these three mappings coincide a.e. on (a, b) simply because u is piecewise continuous on [a, b). T h e o r e m 12.3.1. Let A 9 D ( A ) C_ X --+ X be the generator of a Cosemigroup of contractions, let f 99 --~ X be continuous and g 6 BV(IR; X ) . Assume that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Let u 9 [a,b) --+ X be an L ~ - s o l u t i o n of (12.3.1). Then either u is saturated, or u can be continued up to a saturated one. The proof follows the same lines as those of Theorem 11.3.2 and therefore we omit it. From Theorems 12.1.2, 12.2.2 and 12.3.1 we deduce" C o r o l l a r y 12.3.1. Let A " D ( A ) C_ X -+ X be the generator of a Cosemigroup of contractions, let f 99 --+ X be continuous and g 6 BV(I~; X ) . Assume that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Then, for each (a, ~) 6 I~ x X with (a, g(a + O) - g(a - 0) + ~) 6 9 , the problem (12.3.1) has at least one saturated L~-solution.

Saturated L ~176 T h e o r e m 12.3.2. Let A : D(A) C_ X semigroup of contractions, let f : 3) --+ X Assume that f maps bounded subsets in either the semigroup is compact, or f is last argument. Let u : [a, b) --+ X be a Then either

279

--+ X be the generator of a Cobe continuous and g C BV(I~; X ) . 9 into bounded subsets in X and, locally Lipschitz with respect to its saturated Lee-solution of (12.3.1).

(i) t ~-+ u(t + O) is unbounded on (a, b) and, if b < +c~, there exists lim Ilu(t + O)ll - + ~

or

tSb

(ii) t ~ u(t + O) is bounded and, either u is global, or (iii) t ~ u(t + O) is bounded and u is non-global, case in which there exists lim u(t) = u(b - 0) and t$b

(b, g(b + O) - g(b - O) +

- 0)) r

R e m a r k 12.3.3. Unlike the case of semilinear evolution equations already considered in Chapter 11, here, under the circumstances described in (iii), it may happen that, not only (b, g ( b + O ) - g ( b - O ) + u ( b - O ) ) ~ 9 but even (b, g(b + O) - g(b - O) + u(b - 0)) ~t 9 This apparently strange situation is due to the (possible) existence of an "excessive" right jump of g at b. For instance, let us consider the following simple example. Take X = R, 3) = I~ • ( - 1 , 1), A - 0, g : I~ --+ I~ defined by 0 t~ 3

g(t) -

ift<0 iftc[0,1] ift>l

and f : 9 ~ R, f - 0. Then one may easily verify that the unique saturated L~-solution of the Cauchy problem

du - { A t + f ( t , u ) } d t + dg -o

is u : [0, 1) --+ I~, given by u(t) = g ( t ) for each t E [0, 1). But in this case, (1,g(l+0)-g(1-0)+u(1-0))=(l~3)~9. We may now pass to the proof of Theorem 12.3.2. P r o o f . Up to one point, the proof parallels that of Theorem 11.3.4. So, let us assume first that t ~-~ u ( t - 0) is bounded and non-global. Since, except for an at most countably subset, t ~-~ f (t, u(t)) and t ~-~ f (t, u ( t - O ) ) coincide and the latter is bounded, from Proposition 12.3.1, it follows that there exists limt~b u(t) = u ( b - 0). Thanks to Lemma 12.3.1, we conclude that ( a , g ( b + O ) - g ( b - O ) + u ( b - O ) ~ 9 and thus we have (iii). To complete

280

Semilinear Equations Involving Measures

the proof, we have only to check (i). So, let us assume that u is unbounded. To show that limt?b II~(t- 0 ) I I - + ~ , let us assume the contrary, i.e. that, under these circumstances, we have lim inf Ilu(t - 0)II < +oc. tSb Equivalently, there exist at least one sequence (tk)kcN in (a, b), convergent to b, and r > 0 such that Ilu(tk -0)11 <_ r, for each k C N. Let C - {v c X; Ilvll _< r + 1}. Since f maps bounded subsets in 9 into bounded subsets in X, b < +ec and C is bounded, there exists M > 0, such that Ill (7, v)II -< M, for each (~-,v) e ([ a, b) x C ) A 9

(12.3.4)

Now let us choose d > 0 satisfying

1 d M < ~, 1 Var (g, [ b - d, b) ) < -~

(12.3.5)

and fix k C N such that b - d < tk < b. Since t ~ u(t + O) is unbounded on [a, b), it is necessarily unbounded on [tk, b). Let t*--sup{te[tk,

b); Ilu(s +O) ll < - r + l,Vs e [tk,t)}.

Recalling that u is piecewise continuous on [a, b) (see Theorem 9.2.1), it readily follows that t F-+ u(t + 0) is continuous from the right on [a, b). This clearly implies that t* C (tk, b). By the definition of t* it follows that there exists a sequence (t;)keN in (t*, b) with t; $ t* and such that r + 1 < Ilu(t; + 0)ll for each k C N. Again from the right continuity of the mapping t ~ u(t + 0), we conclude that r+l_<

Ilu(t*+0)l I.

A simple calculation argument shows that r + 1 _< II~(t* + 0)ll _< Ilu(tk + O)ll + Var (g, [tk, t*)) + (t* -- tk)M. But (12.3.4), (12.3.5) and the obvious inequalities b - d < tk < t* < b, yield r + l <_ Ilu(t*+0)ll_ I l u ( t k + O ) l l + V a r ( g , [ b - d , b ) ) + d M

1 1 < r+~+~ -r+l.

This contradiction can be eliminated only if lim Ilu(t + 0)[I - +co t$b and this completes the proof.

D

Saturated L ~ -Solutions

281

C o r o l l a r y 12.3.2. Let us assume that A " D ( A ) C_ X --+ X generates a Cosemigroup of contractions, f " I ~ x X --+ X is continuous and g C BV(I~; X ) . Assume further that f maps bounded subsets in 9 into bounded subsets in X and, either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Then a saturated Lee-solution u " [a, b) --+ X of (12.3.1) is either global, or it is not global, and in this case there exists lim ]lu(t + O)ll - +co. t$b

P r o o f . If u is global we have nothing to prove. So, let us assume that u is non-global. This means that b < +co and thus g E B Y ( [ a , b ] ; Z ) . According to Theorem 12.3.3, either t ~ u(t + 0) is unbounded and the conclusion holds, or t ~ u(t + 0) is bounded. In the later case, since g e B V([a, b f; X) and inasmuch as f maps bounded subsets in R x X into bounded subsets in X, we conclude that t ~ f ( t , u(t + 0)) is bounded. Thanks to Proposition 12.3.1, there exists lim u(t) - u(b - 0). teb By virtue of Theorem 12.3.3, we must have (b, g(b + O) - g(b - O) + u(b - 0)) ~ I~ x X

which is absurd. This contradiction can be eliminated only if t ~-~ u(t + O) is unbounded and this completes the proof. U] The next corollary is a reformulation of the preceding one. C o r o l l a r y 12.3.3. Let us assume that A " D ( A ) C X --+ X generates a Cosemigroup of contractions, f " I ~ x X -+ X is continuous and g C BV(I~; X ) . Assume further that f maps bounded subsets in 9 into bounded subsets in X and, either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Then a necessary and sufficient condition in order that an L ~ - s o l u t i o n u " [a, b) --+ X of (12.3.1) be continuable is that b < +co and t ~ u(t + O) be bounded on [a, b). We conclude this section with a sufficient conditions on f ensuring the existence of global Lee-solutions of (12.3.1). T h e o r e m 12.3.3. Let A " D ( A ) C X --+ X be the generator of a Cosemigroup of contractions, f 9 I~ x X --+ X a continuous function and g E BV(]~; X ) . Let us assume that either the semigroup is compact, or f is locally Lipschitz with respect to its last argument. Further, let us assume that there exist two continuous functions h, k" I~+ -+ ~+ such that IIf (~-, v)I] -< k(~-)]IvII + h(~-),

(12.3.6)

282

Semilinear Equations Involving Measures

for each (% v) e I~+ x X. Then, for each (a,~) C I~+ x X , (12.3.1) has at least one global Lee-solution. P r o o f . Let u : [a, b) --+ X be a saturated solution of (12.3.1) whose existence is ensured by Corollary 12.3.1. To complete the proof it suffices to show that b = +co. Thus, let us assume by contradiction that b < +c~. From the definition of the L~-solution (see (iii) in Definition 12.1.4) and the growth condition on f we deduce Ilu(t

fat IIs(t -

O)ll < IlS(t

~)ll~(x)llf(~, ~(s

a)~ll

o))11 ds

it+0 s ( t

- ~)dg(s)

dot

<_ I1~11+ Var (g, [ a, b]) +

k(s) Ilu(~ + 0)II ds

for each t C [ a, b ). By virtue of Lemma 10.2.1, it follows that t ~ u(t +0) is bounded on [a, b). From Corollary 12.3.3, we conclude that u is continuable thereby contradicting the hypothesis. This contradiction can be eliminated only if b = +co and this achieves the proof. [:3 R e m a r k 12.3.4. All the results in this section can be reformulated to handle the more general case du - { A t + f ( t , u ) } d t + dg~ u(a) -- ~, where A and f are as above and, for each interval [a, b ], either u ~-+ gu is locally Lipschitz from Lee (a, b ; X) to BV([ a, b ] ; X) whenever f is locally Lipschitz, or is bounded and weakly pointwise continuous from L ~ (a, b; X) to B V ( [ a, b]; X) whenever f is continuous. See Definition 12.2.1. However, we notice that in both cases, we have to assume that u ~ g~ is hereditary (see Definition 12.1.1) and there exists a function h E BV(I~) such that, for each a, b E I~ with a < b and each u C L ~ ( a , b ; X ) , we have

12.4. T h e C a s e of S p a t i a l M e a s u r e s Let f : I~ • X --+ X be a continuous function, g E BV(IR; X A ) and let us consider the Catchy problem du - { A t + f (t, u) }dt + d g u(a) - ~A.

(12.4.1)

The Case of Spatial Measures

283

Here X A is the space of all admissible measures for A, i.e. X A -- (X| where X ~ is the sun dual of X and ~A C XA. See Sections 9.1 and 9.5.

*,

T h e o r e m 12.4.1. Let A" D(A) C_X --+ X be the infinitesimal generator of a compact Co-semigroup of contractions {S(t) ; t >_ 0} and f " R x X --+ X a continuous function for which there exists c~,/~ C L~oc (R) satisfying ]If(t, u)]] _< ~(t)llull + ~(t)

(12.4.2)

a.e. for t C ~ and for every u C X. Then, for each a, b E I~ with a < b, each ~ E XA, and g C BV([a,b]; XA) which is the pointwise limit in the a(XA, X ~ topology of a sequence of functions (gn)neN in B V ( [ a , b ] ; X ) with equibounded variation, there exists at least one Lee-generalized solution of (12.4.1) defined on [a,b]. P r o o f . First, let us observe that we may assume without any loss of generality that g C B V(IR; XA) and it can be pointwise approximated in the a(XA,XO)-topology by a sequence of functions (gn)neN in B V ( ~ ; X ) with equibounded variation. Indeed, if g C BV([a,b] ; X ) satisfies the hypotheses of Theorem 12.4.1, then the function ~, defined by

~(t) -

I g(a) g(t) g(b)

ift b,

belongs to BV(]~; XA), it can be pointwise approximated in the a(XA, XG) topology by a sequence of functions (gn)u'eN in BV(I~; X) with equibounded variation and coincides with g on [a, b]. Let (~n)n~N be a sequence in X which converges in the weak-| topology to ~A and let (Un)nE N the sequence of saturated solutions of (12.4.1) corresponding to (~n)neN and (gn)neN. We notice that the existence of these solutions is ensured by Corollary 12.3.1. We claim that un is global, i.e. it is defined on [a, +c~). Indeed, let us assume by contradiction that, for some n C H, Un is defined on [a, b) with b < +c~. By (12.4.2) and Lemma 10.2.1, it readily follows that Un is bounded on [a, b). By Corollary 12.3.3, we conclude that u~ is continuable, thereby contradicting the fact that Un is saturated. This contradiction can be eliminated only if Un is global. So~ for each b E I~, b > a and n C H, Un is defined at least on [a, b]. By the uniform boundedness principle, it follows that (~n)neN is bounded in X, while, by hypotheses, (gn)neN is of equibounded variation on [a, b]. Again by (12.4.2) and Lemma 10.2.1, we conclude that (Un)nCHis uniformly bounded on [a, b l, and therefore the family of functions

{ /a t ~-~

f(s, Un(S))ds + gn(t); k e N

}

Semilinear Equations Involving Measures

284

is of equibounded variation on [ a, b]. By Theorem 9.4.2, we conclude that, on a subsequence at least, (Un)ncN is convergent in LP(a,b; X ) , for each p C [1, +c~), to a certain function u : [a, b] -+ X. Since f is continuous and we may assume without loss of generality that (Un)ncN converges a.e. on [a, b] to u, a simple computational argument shows that u is an L ~ generalized solution of (12.4.1) and this completes the proof. D

12.5. Two Examples We conclude this chapter with two illuminating examples. Let ~ be a bounded domain in I~n whose boundary F is of class C 2. In this section we consider the semi-dynamic elliptic problem -An-0 du + {u, + fl(t, u)}dt - d~ u(O)

-

uo

in QT on ET on F,

(12.5.1)

as well as the parabolic problem with dynamic boundary conditions

du + { - A n + a(t, u) }dt - d~ du + {u, +/~(t, u)}dt - de u(0) -- u0~

in QT on ET in ~

u(0)

on F

(12.5.2)

where QT = (0, T) • ~, ET = (0, T) • F, a,/3 : [0, T] • R --+ R are continuous, u0, u0r e L2(F). u0~ e L2(~). u. is the conormal derivative of u at points of F, ~, r e BV([O, T ] ; L2(F)) and ~ C BV([O, T ] ; L2(~)). We will show next how (12.5.1) and (12.5.2) can be analyzed by means of Theorem 12.2.2. We start with the problem (12.5.1). Namely we have: T h e o r e m 12.5.1. Assume that/~ : [0, T] • I~ --+ R is continuous, has sublinear growth and ~ e BV([O,T] ;L2(F)). Then, for each u o e L2(F), there exists To E (0, T) such that the problem (12.5.1) has at least one solution u "[0, To]-+ L2(~) satisfying" (i) for each t C[0, To) and each s E (0, To ], we have

{

~ r ( t + 0) - ~ r ( t ) ~lr(~)

- ~tr(~

-

- 0) -

~ ( t + 0) - ~ ( t ) ~(~)

- ~ ( ~ - 0)

'

(ii) for each interval ~ of absolute continuity of ~ for which ~' lies in L2(~ ; L2(F)). we have" u r E C(~ ; H1/2(F)) n W~.2(I[; L2(F)), u e C(I[; H I ( ~ ) ) and u satisfies (12.5.2) a.e. on ~. P r o o f . Take H -

L2(F) and let us define A ' D ( A ) C_ H ~ H by

D(A)-

{y e

H~/2(F) ; u. ~ L2(F)}.

285

Two Examples

where u C Hi(D) is the unique solution of the elliptic problem -An - 0 u-y

in onF

and Ay = - u , for each u C D(A). By virtue of Theorem 7.5.1, it follows that A generates a compact analytic C0-semigroup of contractions on H. Let f 9 [0, T] • L 2(F) --+ L 2(F) and g 9 [0, T] -+ L 2(F) be defined by f ( t , u ) ( a ) - - ~ ( t , u ( a ) ) for (t,u) C [0, T] x L2(F) and a.e. for a C F and respectively by g(t) = ~o(t) for each t C [0, T]. Then (12.5.1) rewrites as an abstract Cauchy problem of the form (12.2.2) in the Hilbert space H, with ~ = u0, and Theorem 12.2.2 applies. An appeal to Theorem 9.2.1 shows that (i) holds. Since (ii) follows from well-known regularity results concerning elliptic equations along with the Green's formula, the proof is complete. D T h e o r e m 12.5.2. If g,~ : [0, T] x I~ --+ ~ is continuous, has sublinear growth, ~7 e B V ( [ O , T ] ; L2(~)) and r e B V ( [ O , T ] ; L2(F)), then, for each U~o e L2(~t) and Uro e L2(F), there exists To e (O,T) such that the problem (12.5.2) has at least one solution u" [0, T0]-+ L2(~) satisfying" (i) for each t E[0, To) and each s C (0, To ], we have ~ ( t + o) - ~(t) - ~(t + o) - ~(t) r ( t + 0) - ~ r ( t ) ~(~)

- ~ ( ~ - 0) -

r(~)

-

r

~(~)

- ~ r ( ~ - 0) -

+ 0) - r - ~ ( ~ - 0)

r

- r

- 0);

(ii) for each interval ~ of absolute continuity of both ~ and r for which V' e L2(I[; L2(~)) and r C L2(]I;L2(F)), we h a v e " u C C(]I; H I ( ~ ) ) A W1'2(I[ ; L2(~t)) u r c C(~;H1/2(F)) n W1,2(~; L2(F)) and u, u r satisfy (12.5.2) a.e. on I[.

P r o o f . Take H -

L2(~) • L2(F) endowed with the inner product

<(~,v), (~, ~>H - <~, ~>L~(~I+ L~(rl, for each (u. v). (~. ~) C H and let us define A ' D ( A ) C_H --+ H by D(A) - {(u.v) C H; Au C L2(D). u~ C L2(F) and u. - v on F}

and A(u, v) - ( A n , - u ~ ) for each (u, v) E D(A). From Theorem 7.4.1, it follows that A is a densely defined, self-adjoint operator which generates a compact, analytic C0-semigroup of contractions on H. Let us define both f" [O.T]xL2(~)xL2(F) --+ L2(~t)xL2(F)and g" [0. T]-+ L2(~t)xL2(F) by

Semilinear Equations Involving Measures

286

f ( t , u , v ) ( x ) - -(a(t,u(x)),fl(t,v(a))) for (t,u,v) e [O,T]xL2(ft)xL2(F), a.e. for x E f~ and a.e. for a E F, and respectively by g(t) = (rl(t), r for t C [0, T]. Then (12.5.1) can be rewritten as an abstract C a t c h y problem of the form (12.3.1) in the Hilbert space H, with ~ - (u0a, u0r) and we are in the hypotheses of Theorem 12.2.2. The conclusion follows from Theorem 9.2.1 combined with well-known regularity results concerning parabolic equations. [3

12.6. One M o r e E x a m p l e In order to show how Theorem 12.4.1 applies, in this section we analyze a simple example of semilinear parabolic problem with distributed measures with respect to both time and space. Let f~ be a nonempty, open and bounded subset in R n whose boundary F is of class C 1, let b : R x f~ x R --+ R, let g C BV(IR; (C0(f~))*), ~A E (C0(ft))* and let us consider the problem

I du-{Au+b(t,x,u)}dt+dg(t) u - 0 U(0, X) -- ~A

inQ~ on Ecc in f~,

(12.6.1)

where Q ~ = [0, +co) x f~ and E ~ = [0, +co) x F. As usual, we rewrite (12.6.1) as an abstract C a t c h y problem of the form

du - { A t + f(t,u)}dt + dg u(0) - ~A,

(12.6.2)

satisfying the hypotheses of Theorem 12.4.1. To this aim, let us assume that b : IR x f~ x IR --+ IR is continuous and there exist two continuous functions k, h : I R --+ R+ such that

Ib(t, x, u) l _< k(t)]u I + h(t)

(12.6.3)

for each t C R, each u C R and a.e. for x E f~. Now, take X = Ll(ft) and A : D(A) C_ X -+ X, defined by D(A)-

Au-

{u e w~'l(~t); A u e Ll(f~)}

Au

for u E D(A),

and let us recall that A generates an analytic and compact C0-semigroup of contractions. See Theorem 7.2.7. Furthermore, let us recall that in this case X ~ - Co (~) and XA -- (Co (~))*, i.e. the space of all Radon measures concentrated in f~. See Example 9.5.1. Next, let us define f : R x X ~ X by

f (t, u)(x) = b(t, x, u(x)) for each u C L~(f~), each t C IR and a.e. for x C 12. Thanks to (ii) in L e m m a A.6.1, f is well-defined and continuous. At this point it is easy to

One More Examples

287

see that (12.6.1) can be rewritten in the abstract form (12.6.2), where A, f, g and ~A are as above. From Theorem 12.4.1 we deduce" T h e o r e m 12.6.1. Let us assume that b" R x f~ x R -+ R satisfies (12.6.3) and g C B V ( R ; (Co(ft))*) is the pointwise limit in the a((Co(ft))*, Co(f~))topology of a sequence (gn)n~N of equibounded variation in BV(IR; Ll(ft)). Then, for each ~A E (Co(a))* which is the limit in the a((Co(ft))*, Co(a))topology of a sequence (~n)n~N bounded in Ll(ft), the problem (12.6.2) has at least one generalized L~ u" [ O, +ec) --+ LI(f~). We describe below a typical instance in which both g and ~A satisfy all the hypotheses of Theorem 12.6.1. More precisely, let xo C f~ and let ~A = ~(x--xo), i.e. the Dirac delta concentrated at xo. Then ~A is the limit in the a((Co(ft))*, Co (ft) )-topology of a sequence (~n)ucr~ which is bounded in L~(f~). Indeed, let co > 0 be sufficiently small such that B(xo, eo) C f~, let c C (0, co), and let us define the function ( ~ ( . - xo) : ft ~ R by 1

(f~(x - xo) -

#(B~)

if

IIx

- ~011 <

if 9 C

0

IIx- x011 > e.

Here #(B~) is the Lebesgue measure of the closed ball in ]~n centered at 0 and having radius E. For any e C (0, co), we have 115~(.- Xo)llL~(n)- 1 and therefore { ( ~ ( . - Y o ) ; c E (0, co)} is bounded in Ll(f~). On the other hand, we have lim~E(x e40

-- xo)

-- 5(x

--

xo)

in the a((Co(f~))*, Co (f~) )-topology. Indeed, this means that

limfa e$0

-

xo)

(x) d x -

(xo)

m

for each ~ E Co(f~), which clearly holds true. So, ~A -- ( f ( x - x0) satisfies the hypotheses of Theorem 12.6.1. Next, let to C IR and Yo C ft and let g" R --+ (Co (f~))* be defined by

I - 89 g(t) 89

0 - Yo)

ift to.

We shall show that g satisfies the hypohteses of Theorem 12.6.1. It is easy to see that g C B V ( R ; (C(f~))*). Let co > 0 be such that B(yo,~o) C ft, let

Semilinear Equations Involving Measures

288

e E (0, co), and let us define g~" R ~ C(f~) by

/ -89 g~(t) -

ift
l(~e(X -- Yo)

if t - to if t > to,

where 5 ~ ( x - Yo) is defined as above. By the preceding proof, it readily follows that lim g~(t) - g(t), e$o m

for each t C R, in the cr((Co(ft))*, Co(ft))-topology. Therefore, g fulfils the hypotheses of Theorem 12.6.1, and thus, if b 9R x f~ x R ~ R satisfies (12.6.3), the semilinear parabolic problem

I du - {Au + b ( t , x , u ) } d t + 5 ( t - to) x 5(x - yo) u-O x)

-

in Q ~ o n Y]oc

in

-

f~,

has at least one global generalized L~-solution. By a generalized L ~ solution of the problem above, we mean in fact a generalized L~ of the corresponding problem (12.6.2).

Problems P r o b l e m 12.1. Let A 9 D(A) C_ X -+ X be the generator of a C0semigroup of contractions and let G 9L co(a, b ; X ) --+ BVA([a, b]; X) be a hereditary locally Lipschitz mapping. Then, for each ~ C X, there exists c C (a, b] such that the problem

du - { A u } d t + dg~ u(a) has a unique Lee-solution defined on [a, c]. We denote by BVA(R; X ) the space of all functions g C B V ( R ; X ) the property that, for each a < b, g C BVA([ a, b] ;X).

with

P r o b l e m 12.2. Let to C R be fixed and let h 9R --+ R be continuous. Using the result in Problem 12.1, show that, for each ~ C LI(R), the problem

ut-ux+h(x)(~(t-to) u(0, x) - ~(x)

for(t,x) eRxR, for x C IR

has an L~-solution defined on R+, although the C0-group generated by the operator Au - u' is not continuous from (0, +co) to L(LI(R)) in the uniform operator topology.

Problems

289

P r o b l e m 12.3. Prove t h a t if A : D(A) C_ X -+ X is the infinitesimal generator of a C0-semigroup of contractions {S(t) ; t _> 0}, f : R x X --+ X is continuous with f ( R x X) compact, and g C BVA(R; X ) then, for each (a, ~) C R x X and c > a, the problem

du - {Au + f (t, u) }dt + dg u(a) has at least one L~

defined on [a, c].

P r o b l e m 12.4. Let 9 be a n o n e m p t y and open subset in IR x X. Prove the following variant of T h e o r e m 12.2.2. If A : D(A) C_ X -+ X is the infinitesimal generator of a C0-semigroup of contractions {S(t) ; t >_ 0}, f : 9 ~ X is continuous and b-compact, and g E B V A ( R ; X ) , then, for each (a, ~) E R x X satisfying

(a, g(a + O ) - g(a) + ~) E 9 there exists b > a such t h a t the problem

du - {Au + f ( t , u ) } d t + dg u(a) has at least one L~

defined on [a, b].

P r o b l e m 12.5. Let ft be a nonempty, b o u n d e d and open subset in R n whose b o u n d a r y F is of class C 2, let QT = (0, T) x f~ and ET = (0, T) x F. We consider the nonlinear hyperbolic equation

dut - {Au + b(t, x, u)}dt + (~(t- to) ~(t, x) - o ~(O,x) ~,(O,x) - v o ( x )

(t, x) E QT (t, x) ~ r~r 9e x e a

where b 9 R+ x f~ x R --+ R is a continuous function and to C (0, +oc). Prove that, if b : R+ x Ft x R -+ R is a continuous function for which there exist c > 0 and d > 0 such t h a t

Ib(t, x, u)I <_ ciu I + d for (t,x, u) E R+ x ft x R, then, for each u0 C H~(f~) and each v0 C L2(ft), the problem above has at least one global L ~ - s o l u t i o n . Problem

12.6. Let us consider the following parabolic problem

/

d~ - {ZX~}dt + ~ ( t -

u--O ~(o, ~) - ~(x - xo)

to) • 6(x - x~)

in Q ~ on Z ~ in ft,

290

Semilinear Equations Involving Measures

where to C (0, +co), x0 C ~ and u ~-~ Xu is continuous from L ~ ( a , b ; L l ( ~ ) ) respectively to ~tg, the latter being a compact subset in ~t. Prove that this problem has at least one generalized L~-solution. We notice t h a t the problem above may be interpreted as a closed loop system whose feedback law u ~+ ~ ( t - to) • 5 ( x - Xu) takes values in the space of Radon measures over IR x f~. N o t e s . The existence results concerning semilinear differential equation with distributed measures in Sections 12.1 and 12.2, i.e. Theorems 12.1.1 and 12.2.1 were established by Grosu [65]. Theorems 12.1.2 and 12.2.2 are from Vrabie [131], but the proofs are new, and due to Grosu [65]. The qualitative results referring to continuable and saturated L ~ - s o l u t i o n s in Section 12.3 are in the same spirit as those in Sections 10.3 and 11.3 and extend the later to the general case of semilinear equations involving distributed measures. Theorem 12.4.1 and the examples in Section 12.5 are from Vrabie [131], while the example in Section 12.6 and the problems at the end of this chapter are new.

APPENDIX A

Compactness Results

The aim of this chapter is to present several necessary and sufficient conditions in order that a given subset in a certain function space be relatively compact. One of the most fundamental results in Real Analysis says that each bounded sequence in I~n has at least one convergent subsequence (Ces~ro's lemma). In addition, one knows that a Banach space is finite-dimensional if and only if the class of its relatively compact subsets coincides with the class of its bounded subsets. In other words, Ces~ro's lemma is a characteristic of the finite-dimensional frame. On the other hand, the usual function spaces employed in Nonlinear Analysis (C([ a, b ]; X), LP(a, b ; X ) , with X Banach space) are infinite-dimensional (even if X is finite-dimensional). From this reason at least, it is very important to get appropriate variants of Cess lemma for each one of this function spaces. As we shall see, these variants differ from one type of space to another, and are closely related to the nature of the space in question.

A.1. Compact Operators In this section we shall present some auxiliary concepts and results we need in t h a t follows. D e f i n i t i o n A . 1 . 1 . A subset C of a topological space (X, ~) is called:

relatively compact, if each generalized sequence in C has at least one generalized convergent subsequence; (ii) compact, if is relatively compact and closed; (iii) sequentially relatively compact, if each sequence in C has at least one convergent subsequence; (iv) sequentially compact, if is sequentially relatively compact, and closed. (i)

If (X, d) is a metric space, C C X is called precompact, or totally bounded, if for each e > 0 there exists a finite family of closed balls of radius c whose union includes C. 291

292

Compactness Results

R e m a r k A.1.1. Inasmuch as each countability i.e., each point has an of neighborhoods, in such a space, only if it is sequentially (relatively)

metric space satisfies the first axiom of at most countable fundamental system a subset is (relatively) compact if and compact.

R e m a r k A.1.2. One may easily show that a subset C C X is precompact if and only if, for each c > 0 there exists a finite family of closed balls centered in points of C and having radii c, whose union includes C. We recall, without proof, the following result due to Hausdorff. T h e o r e m A . I . 1 . If (X, d) is a complete metric space, then a subset of it is sequentially relatively compact if and only if it is precompact. We also recall: T h e o r e m A.1.2. (Tichonov) Let {(Xa, 9~a); c~ E F} be a given family of topological spaces. Then the product space X = IIa~pX~, endowed with the usual product topology, is compact if and only if, for each c~ C F, (Xa, 3"a) is compact. See Dunford and Schwartz [49], Theorem 5, p. 32. T h e o r e m A.1.3. (Mazur) The closed convex hull of a compact subset in a Banach space X is compact. See Dunford and Schwartz [49], Theorem 6, p. 416. T h e o r e m A.1.4. (Mazur) The weak closure of every convex subset in a Banach space coincides with its strong closure. See Hille and Phillips [70], Theorem 2.9.3, p. 36. Let (X0, II" I]0) and (X, II II) be two Banach spaces with X0 C X. We say that the inclusion X0 C X is compact, or that (X0, I]" II0) is compactly imbedded in (X, I1" II), if each bounded subset in X0 is relatively compact in X. We say that the inclusion X0 C X is continuous, or that (X0, II" I[0) is continuously imbedded in (X, II II), if there exists k > 0 such that

I111 <__kll llo for each z E X. L e m m a A . I . 1 . (Lions) Let (X0, II" II0), (X, I1" II), and (X1, I1" Ill) be three real Banach spaces with Xo C X C X1. If the inclusion Xo C X is compact and X C X1 is continuous then, for each c > 0, there exists rl(c) > 0 such that

I111 < 11 11o + ( )11 111 for each z E Xo.

(A.I.1)

Compact Operators

293

P r o o f . Let us assume by contradiction that this is not the case. T h e n there exists s > 0 such that, for each r; > 0 there exists x~ C X0 with the property Taking r / = n with n C N*, and denoting x~ by Xn, the last inequality can be rewritten under the form

IIXnll

giiXnlIo-Jr nllxniI1

for each n C N*. Obviously Xn ~ 0 and so, we can define 1

Yn

IIxnllo

Xn

for n E N*. From the preceding inequality we obtain

IlyniI

nllynlll

for each n E N*. As X0 is compactly imbedded in X and (Yn)nEN* satisfies Iiyni]o 1 for each n C N*, we may assume without loss of generality that there exists y C X such that -

-

lim Yn - Y

n--+c~

in the norm I]" II. The last inequality yields limn_+~ ]IYnI]I -- O, which shows that y - 0. On the other hand, ]]Yni] > S for each n C IN* and, thus ]]Yll >- s, relation which contradicts the preceding one. This contradiction can be eliminated only if (A.I.1) holds, and this completes the proof. [] We shall see in the next sections that L e m m a A.I.1 is useful in proving some sufficient conditions of compactness in several function spaces. Therein, the reader will find some examples of Banach spaces Xo, X and X1 satisfying the hypotheses of L e m m a A.I.1. Let X, Y be two real Banach spaces and let D a nonempty subset in Y. D e f i n i t i o n A . 1 . 2 . A (possible nonlinear) operator Q : D ~ X is called compact if it is continuous and maps bounded subsets in D into relatively compact subsets in X. L e m m a A . 1 . 2 . Let F be a nonempty subset in IR - I~ U { - c o } t2 {+cxD}, let p be an accumulation point of F, and {Q),; A c F} a family of operators from D C Y in X . If, for each A C F, Q~D is relatively compact, and lim Q~x = Qx, uniformly for x E D, then Q D is relatively compact.

(A.1.2)

294

Compactness Results P r o o f . Let c > 0. From (A.1.2) it follows that there exists ,~ C F such

that IIQ~x - Qxll <_ c

(A.1.3)

for each x C D. As Q~D is precompact, for the very same c > 0, there exists a finite family { x l , x 2 , . . . ,xk(~)} in D such that, for each x ~ D, there exists i = 1 , 2 , . . . , k ( ~ ) with lIQ~x - Q ~ x i l l < c.

From (A.1.3), and the preceding inequality, we deduce

QD is relatively compact in X, and this completes the proof.

D

Three useful consequences of L e m m a A.1.2 are listed below. C o r o l l a r y A . 1 . 1 . Let F be a nonempty subset in IR - I~ U { - o c } U {+oc}, let # an accumulation point of F, and {Q~; )~ c F} a family of compact operators from D C Y in X . If lim Q~x = Qx,

,k--+#

uniformly for x in each subset bounded in D, then Q : D --+ X is a compact operator. C o r o l l a r y A . 1 . 2 . Let K a bounded subset in X , F a nonempty subset in I~ - I~ U { - c o } U { + ~ } , # an accumulation point of F, and {Q~; )~ c F} a family of compact operators from K to X . If lim Q~x - x,

uniformly for x in K , then K is relatively compact. We recall that, if T : Y -+ X is a linear continuous operator, its operator norm IITIl~(y,x) is defined by sup

let # an accumulation point of F and {Q~ ; )~ c F} a family of linear compact operators from Y to X . If lim IIQ~ - QIIg(Y,X) - 0

then Q : Y --+ X is a compact operator.

Compactness in C([ a, b l; X)

295

L e m m a A.1.3. Let K be a compact subset in X and let 9= a family of continuous functions from [a, b ] to K. Then

{ L bf(t) dt; f e h:} is relatively compact in X . P r o o f . Using Riemann sums, if follows that

f(t) dt;

f c 5

c (b - a) onv K,

which, by virtue of Mazur's Theorem A.1.3, completes the proof.

D

L e m m a A.1.4. Let D be nonempty in X , Q : D -+ X and G : D -+ X. If G is compact, and there exists an increasing function c~ : IR+ -+ IR+ with lim c~(r) = 0, r$0

and such that IlQv - Qwll <_ c

(llav - a w l l )

for each v, w C D, then Q is compact. P r o o f . Each finite e-net {Gv~, G v 2 , . . . , Gvk} of G(E) defines a finite c~(e)-net {Qvl, Q v 2 , . . . , Qvk} of Q(E). Since lim~;0 c~(r) = 0, this achieves the proof. D T h e o r e m A.1.5. (Schauder) Let K be nonempty, convex and bounded in a Banach space X and let Q : K ~ K be compact. Then Q has at least one fixed point in K, i.e. there exists at least one u C K such that Q(u) = u. A.2. C o m p a c t n e s s

in C([ a, b]; X)

Let X be a real Banach space, and let [a, b] be an interval. In that follows, we denote by C([ a, b ]; X) the space of all continuous functions from [a, b] to X. Endowed with the sup-norm ]]-]]~, defined by Ilflloo = sup{llf(t)ll; t e [a,b]}, for each f C C([ a, b f; X), this is a real Banach space. D e f i n i t i o n A.2.1. A family 9" in C([a,b];X) is equicontinuous at t in [a, b], if for each e > 0 there exists 5(e, t) > 0 such that, for each s C [a, b] with I t - s I < 5(e, t), we have

]if (t) - f (s)II < c, uniformly with respect to f C 9".

Compactness Results

296

The family 9" is equicontinuous on [a, b ], if it is equicontinuous at each point t E [a, b], in the sense mentioned above. The family 9" is uniformly equicontinuous on [a, b], if it is equicontinuous on [a, b], and 5(c, t) can be chosen independent of t E [a, b]. R e m a r k A.2.1. We leave to the reader the proof of the fact that a family 9~ in C([ a, b]; X) is equicontinuous on [a, b] if and only if it is uniformly equicontinuous on [a, b]. A.2.1. (Arzel~-Ascoli) A subset 9 in C([a, b]; X) is relatively compact if and only if. (i) 9" is equicontinuous on [a, b]; (ii) there exists a dense subset D in [a, b] such that, for each t E D, 9~(t)- {f(t); f E 9~} is relatively compact in X.

Theorem

P r o o f . We begin by showing the necessity of (i) and (ii). To this end, let 9~ be a relatively compact subset in C([a, b]; X) and let c > 0. As 9" is relatively compact, it is precompact, and therefore there exists a finite family {fl, f 2 , . . . , fn(c)} included in 9~, such that, for each f E 9~, there exists i E { 1 , 2 . . . , n ( c ) } with [ i f - fillo~ < e.

(A.2.1)

Let t E [a, b]. Obviously, as {fl, f 2 , . . . , fn(c)} is finite, it is equicontinuous. Hence there exists 5(e) > 0 such that, for each s E [a, b] with It- sl < 5(e), we have Ilfi(t)-/~(s)ll < E, uniformly with respect to i E {1, 2 , . . . , n(e)}. Then I I f ( t ) - f(s)l I _< I I f ( t ) - f~(t)l I + Ilf~(t)- f~(s)l I + Ilf~(s)- f(s)l I < 2 1 I f - f~llo~ + I[/~(t)- f~(s)ll < 3e, for each s E [a, b] with I t - s I < ~(e), where fi is associated with f such that (A.2.1) holds. As the last inequality shows that J" is equicontinuous at t, arbitrarily chosen in [a, b ], it follows that 9" satisfies (i). We shall prove next that 9= satisfies (ii) as well, with D - [a, b ]. To this aim, let t E [a,b] and let (fn(t))nEN be a sequence in 9"(t). As :~ is relatively compact, the sequence (fn)nEN has at least one subsequence (denoted for simplicity again by (fn)nEN), uniformly convergent on [a, b] to some function f E C([a, b] ;X). Obviously, this subsequence is pointwise convergent on [a, b], also to f. Hence (fn(t))nEN is convergent to f(t) and therefore 9"(t) is relatively compact in X, which completes the proof of the necessity of both (i)and (ii).

Compactness in C([ a, b ]; X)

297

We may now proceed to the proof of the sufficiency. Let 9" be a family in C([ a, b]; X) satisfying (i) and (ii). By virtue of Hausdorff's Theorem A.I.1, it suffices to check that 9" is precompact. Let c > 0. As 9" is equicontinuous on [a, b], in view of Remark A.2.1, it is uniformly equicontinuous. Hence, there exists 5(e) > 0 such that, for each t, s C [a, b] with I t - s I < ~(e), we have Ilf(t) - f (s)II < c, for all f C 9". Let now to < tl < " " < tk in [a,b] with ti C D for i = 1 , 2 , . . . , k , and satisfying t o - a < 5(e), b - t k < 5(e), t i + l - t i < 5(e) (A.2.2) for i = 1, 2 , . . . , k. Let us define the set 9( = { ( f ( t l ) , f ( t 2 ) , . . . , f(tk)); f C :~}. Obviously, 9( is relatively compact being included in the product space 9"(tl) x ~(t2) x . . .

x 2F(tk),

which, by virtue of (ii) and Tichonov's Theorem A.1.2, is compact inasmuch as all the factors are compact. Let us observe that the product topology on X k can be equivalently defined by means of the norm II. lip, given by II(Xl, X2, "'" , X k ) l l p ---- max{llXl II, Ilx2il, ... , IlXkll}.

From Hausdorff's Theorem A.I.1, :g is precompact in X k. So, for e > 0 as above, there exists a finite family {fl, f 2 , . . . , fm(~)} in 9", so that, for each f E 9" there exists j C {1, 2 , . . . , re(e)} such that

[]f(ti) - fJ(ti)l[ < e

(A.2.3)

for each i = 1, 2 , . . . , k. We have

[If(t)- fJ(t)[[ <_ [If(t)- f(ti)[[ + Ilf(t~) - fJ(ti)[[ + [[fJ(ti) - fJ(t)[[ for each t E [a, b], where i C {1, 2 , . . . , k} is chosen such that t E [ti, ti+l ], and j C {1, 2 , . . . , re(e)} is associated to f by means of (A.2.3). From this inequality, (A.2.2) and (A.2.3), it follows that, for each e > 0, there exists a finite family {fl, f2 ..., fro(e)} in 9" such that, for each f C 9" there exists j C { 1 , 2 , . . . , r e ( e ) } with

IIf - fJllcc < de. Hence 9" is precompact in C([ a, hi; X) and this achieves the proof.

D

R e m a r k A.2.2. As we have seen from the proof of the necessity, if 5F is relatively compact in C([a, b]; X), (ii) holds with D = [a, b]. In fact, in this case, even a stronger condition holds as the next consequence shows.

298

Compactness Results

C o r o l l a r y A.2.1. If ~ c C([a, b];X) is relatively compact, then the set 9=([ a, b]) - {/(t); f E ~, t E [a, b]}

(A.2.4)

is relatively compact in X . P r o o f . Let (fn(tn))nEN be a sequence in 9:([a, b]). As both [a,b] and 9~ are compact in R, and respectively in C([ a, b ]; X), we may assume without loss of generality that, on a subsequence at least, we have lim tn - t, n---+oo

and lim f n -

n--~oo

f,

uniformly on [a, b ]. Let us observe that

Ilfn(tn)- f(t)ll _< Ilfn(tn)- f(tn)ll + IIf(tn)- f(t)ll

--< II/~ --/11~ + IIf(tn)- f(t)ll for each n E N. Clearly, the first term on the right-hand side of the last inequality goes to 0 for n tending to oo. Moreover, as f is continuous, the second term tends to 0 too for n tending to oo, and thus we have lim f ( t ~ ) -

f(t).

n--+oo

So ~([a, b]) is relatively compact in X and this completes the proof.

D

C o r o l l a r y A.2.2. Let U be nonempty and closed in X, g" [a, b ] x U ~ X a continuous function, 11:- {u E C([a,b]), u(t) E U for t E [ a , b ] } and let G " 1.[ ---, C([a, b]; X ) the superposition operator associated to the function g, i.e.

a(~) (t) - g(t, ~(t)) for each u E 1.[ and t E [a, b ]. Then G is continuous from 1.[ in C([ a, b ]; X), both the domain and range being endowed with the norm topology I1" I1~. P r o o f . Let (U~)nEN be a sequence in ~ convergent to u E II in the norm II" II~. Obviously, {Un ; n E N} is relatively compact in C([ a, b ]; X). Then, according to Corollary A.2.1, the set K-"

{Un(t) ; n E N,t E [a,b]}

is compact in X and consequently [a, b] x K is compact in R x X. Then the restriction of the function g to [ a, b] x K is uniformly continuous. This

Compactness in C([ a, b ]; X)

299

means that, for each c > 0, there exists ~(c) > 0 such that, for each (t, x), (s, y) e [a, b] • K with I t - s I + IIx - ylI <- 5(c) we have Ilk(t, x) - 9(t, y)II <

Let ~ > 0 and let 5(c) > 0 as above. As (Un)nc~ converges uniformly to u on [a, b], there exists n(~) C N such that

IlUn(t) - u(t)l I ~_ 5(c) for each n C N satisfying n >__n(c) and each t E [ a , b]. Therefore,

I[g(t,

(t) ) - g(t, u(t) ) ]l <_ e

for each n >_ n(c) and each t E [a, b]. Taking the supremum both sides of the above inequality for t E [a, b ], we obtain lIG( n) - C( )11

<

for each n >_ n(e). Thus limn--~ G(un) - G(u) in the norm [l" IIoc, and this completes the proof. [:] In order to state the next consequence of Theorem A.2.1, some notations and definitions are needed. If X is a real Banach space, we denote by X~ the set X endowed with the weak topology, and by C([a, b]; Xw) the set of all continuous functions from [ a, b] to Xw. D e f i n i t i o n A.2.2. A family 9" in C([a, b f; Xw) is weakly equicontinuous at a point t E [a,b], if for each x* E X* and each c > 0, there exists 5(c,x*,t) > 0 such that, for each s E [a,b, ] with I t - s I < ~(c,x*,t) we have

I x * ( f ( t ) - f(s))l < c, uniformly with respect to f C ~', i.e {x* o f ; f E 9=} is equicontinuous at t. The family 9" is called weakly equicontinuous on [a,b], if it is weakly equicontinuous at each t E [a, b ], in the sense mentioned in above. The family ~" is called uniformly weakly equicontinuous on [a, b f, if it is weakly equicontinuous on [a, b ], and 6(E, x*, t) can be chosen independent of t E [a,b]. C o r o l l a r y A.2.3. The family 9: in C([ a, b]; Xw) is included and relatively compact in C([ a, b]; X ) if and only if it is weakly equicontinuous on [a, b f,

and ~'([a,b]) - {f(t); f e 9",t e [a,b]}

is relatively compact in X.

300

Compactness Results

P r o o f . The necessity is a direct consequence of Corollary A.2.1. For the sufficiency, let us observe that, by virtue of Theorem A.2.1, it would be enough to show that 9" is equicontinuous on [a, b]. To this aim, let us assume by contradiction that this is not the case. Then there exist t C [a, b], E > O, (fn)ncN in 9~ and (tn)ncI~ in [a, b], convergent to t, such that I l f n ( t n ) - fn(t)ll _> c

(A.2.5)

for each n C N. As 9"([ a, b ]) is relatively compact, we may assume without loss of generality (by extracting a subsequence if necessary), that there exists x C X such that lim (fn (tn) - fn (t)) = X

n--+oo

strongly in X. But 9" is weakly equicontinuous, and consequently lim (fn(t~) - fn(t)) -- 0

n--+(x)

in the weak topology of X. This implies that x = 0, which contradicts (A.2.5). This contradiction can be eliminated only if 9" C C([a, b]; X) and is equicontinuous on [a, b] in the sense of Definition A.2.1, thereby completing the proof. D T h e o r e m A.2.2. Let Xo, X , and X1 be three real Banach spaces with the inclusion Xo C X compact and X C X1 continuous. If ~ is a family of functions from [a, b] to Xo, which is uniformly bounded in the norm I1" Iio, and equicontinuous in the norm I1" II1, then ~ is relatively compact in C([ a, b ]; X). P r o o f . As 9" is uniformly bounded in the norm I1"II0, there exists M > 0 such that IIf(t) ll0 _< M for each f C 9" and each t C [a, b ]. From the preceding inequality and the compactness of the inclusion X0 C X, it follows that, for each t E [a, b], 9"(t) = {f(t); f C 9"} is relatively compact in X. Hence 9" satisfies (ii) in Theorem A.2.1. Let c > 0. According to Lemma A.1.1, there exists ~ = ~(c) such that

llxll0 _< llxil +

( )llxlll

for each x C X0. Inasmuch as 9" is equicontinuous from [a, b] to X1, for each t E [a, b] there exists 5(t, c) > 0 such that C

Ill(t) - f( )ll

___

Compactness in C([ a, b ]; X~)

301

for each s E [a, b] satisfying It - s I _< 5(t, c). We then have Ilf(t) - f(s)ll _< ellf(t) - f(s)ll0 + w(c)llf(t) - f(s)lll < 2Me + r/(e)

c

_< (2M + 1)e

-

for each s E [a, b], It - s I _< 5(t, e). Hence 9" is equicontinuous from [a, b] to X, and therefore it satisfies (i) in Theorem A.2.1. Consequently 9" is relatively compact in C([ a, b ]; X), and the proof is complete. [-1 We conclude this section with two examples of triples X0, X, X1 satisfying the general condition in Theorem A.2.2. E x a m p l e A.2.1. Let X0 -- C 1 ([ 0, 1 ]), X - C([ 0, 1 ]), Xl - L 1(0, 1). In view of Theorem A.2.1, it follows that each bounded subset with respect to the norm II 9IIc1([0,1]), defined by Ilfllcl([0,1])

-

sup

xE[0,1]

I/(t)l+

sup [f'(t)I xE[0,1]

for each f E C1([0, 1 ]), is relatively compact in C([0, 1 ]) endowed with the norm I1" II~. In addition, each convergent sequence in C([0, 1]) is convergent in L 1(0, 1). Therefore X0 is compactly included in X which, in its turn, is continuously included in X1. E x a m p l e A.2.2. Let X0 - Hl(ft), X - L2(ft) and let X 1 - - H - l ( f t ) , where ft is a nonempty, bounded and open subset in R n whose boundary F is of class C 1. Then, by virtue of Theorem 1.5.4, it follows that X0 is compactly included in X, and this is continuously (in fact even compactly) included in X1. A.3. C o m p a c t n e s s

in C([ a, b]; Xw)

In this section we shall present a variant of Theorem A.2.1 referring to the space C([a, b]; X~) of all continuous functions from [a, b] to X. On this space we define a topology as follows. Let Fin (X*) be the class of all nonempty and finite subsets in X*, let F* E Fin (X*) and let us define II" liE* : C ( [ a , b ] ; X ~ ) ~ R+ by Ilfll/r. =

sup sup Ix*(f(t))l tE[a,b]z*EF*

for each f E C([a, b l; Xw). One may easily see that {II'IIF*; F* Fin (X*)} is a family of seminorms on C([ a, b]; X~) which defines a topology of a locally convex, separated space, called the uniform weak convergence topology. We emphasize that this topology (except for the case in which X is finitedimensional) is not metrisable. Therefore, the main result in this section

302

Compactness Results

referring to the sequential relative compactness of a subset in C([ a, b ]; Xw) is rather surprising because it cannot be obtained directly by means of the equivalence between relative, and sequentially relative compactness. T h e o r e m A.3.1. (Arzels Let X be a sequentially weakly complete 1 Banach space. A family ~" in the space C([a,b];Xw), endowed with the uniform weak convergence topology, is sequentially relatively compact if and only if" (i) 9" is weakly equicontinuous on [a, b]; (ii) there exists a dense subset D in [a, b] such that, for each t E D, the section 5:(t) - {f(t); f C 9"} is sequentially weakly relatively compact in X . P r o o f . Necessity. Let 9" C C([ a, b, ]; Xw) be sequentially relatively compact. In order to show that it satisfies (i), we proceed by contradiction. So, let us assume that there exist t E [a, b], c > 0, (tn)nCN in [a, b] convergent to some t, (fn)ncN in 9", and x* C X* such that

] x * ( f n ( t n ) - fn(t)) L >_ C

(A.3.1)

for each n E N. As 9~ is sequentially relatively compact, we may assume without loss of generality that there exists f in C([ a, b]; Xw), such that lim fn

-

f

n--~ co

in C([ a, b]; Xw). In particular, we have lim ]x*(fn(S) - f ( s ) ) ] -

u- +c<)

0,

(A.3.2)

uniformly for s C [a, b]. Let us remark that

c <_ I x * ( f n ( t n ) - fn(t))l <_ [x*(fn(tn) - f(t~))l + Ix*(f(tn) - f(t))l + ]x*(f(t) - fn(t))l for each n E N. In view of (A.3.2) and the weak continuity of the function f, it follows that, for n C N large enough, we have

and

C

C

Ix*(fn(tn) - f(tn))l < 5'

] x * ( f ( t ) - fn(t)) I < 3' g

] x * ( f ( t n ) - f(t))] < 5" But these inequalities, and (A.3.1), lead to c < c. This contradiction can be eliminated only if the family 9" satisfies (i). To complete the proof of 1Each reflexive Banach space is sequentially weakly complete, but there are nonreflexive spaces enjoying this property. For instance, L I ( ~ ) , w i t h ~t measurable in l~'~, although non-reflexive, is sequentially weakly complete.

Compactness in C([ a, b l; Xw)

303

the necessity it suffices to show that (ii) holds with D = [a, b]. To this aim, take t C [a, b] and (fn(t))ncN in 9"(t). By hypothesis, there exist fcr in C([a, b] ;Xw) and a subsequence (fnk)kcN of (fn)n~N with the property that l i m k ~ fnk = fcr in C([a, b];Xw). Therefore, for each x* e Z*, we have l i m k _ ~ x* (f~k (t)) -- x* (fcr (t)). Thus (ii) holds and this completes the proof of the necessity. Sufficiency. Inasmuch as [a, b] is compact, one may easily see that a family 9" is weakly equicontinuous on [a, b] if and only if it is uniformly weakly equicontinuous on [a, b]. See Definition A.2.2. Let 9" be a family which satisfies (i) and (ii). First, let us observe that, since [a, b] is separable, we may always assume that D is countable (by taking a subset if necessary). So, let D = {tk; k C N} be the dense set in [a,b] given by (ii). Let (fn)neN be a sequence in 9". Using (ii) and a standard diagonal procedure, we deduce that there exists fcr : D --+ X such that lim f n ( t k ) n~

f~(tk)

(A.3.3)

weakly in X for each tk C D. We shall show next that (fn)n~N is a Cauchy sequence in C([a,b];Xw). To this aim, let : > 0 and F* E Fin (X*). At this point, let us observe that we may assume without loss of generality that F* is a singleton, i.e. F = {x*}. By virtue of (i), there exists 5(:) > 0 such that, for each f C 9" and each t, s E [a, b] with I t - s I < 5(:), we have

] x * ( f ( t ) - f(s))[ < :. Let a < tl < t2 < . . . < tk(:) (_ b be such that ti C D, t : - a < 5, b--tk(:) < 5 and [ t i + l - t i < 5(:) for i = 1 , 2 , . . . , k ( : ) - 1 . Let t C [a,b] and let us choose i C { 1 , 2 , . . . , k ( e ) } so that I t - til < 5(e). Since (x*(fn(ti)))ncN* are Cauchy sequences, i = 1, 2 , . . . , k(:), there exists n(:) C N* such that, for each n, m C N with n, m _> n(:) and each i e {1, 2 , . . . , k(:)}, we have ] x * ( f n ( t i ) - f m ( t i ) ) l < e. So, for each t e [a,b], w i t h / a n d n , m as above, we have Ix* (fn (t) - fm (t))[ <_ Ix* (fn (t) -- fn (ti))l + ]X* (f~ (ti) -- fm (ti))l

+ l x * ( f m ( t i ) - fm(t))] < 3:. Thus (fn)n~N* is a Cauchy sequence in C([a,b] ;Xw). Inasmuch as X is sequentially weakly complete, we deduce that there exists f E C([ a, b]; Xw) such that lim Ilfn - fllx* = 0,

n--+cxD

and this completes the proof. We conclude this section with a weak variant of Theorem A.2.2.

[::]

Compactness Results

304

T h e o r e m A.3.2. Let Xo, X and X1 be three real Banach spaces with the inclusion Xo C X compact and X C X1 continuous. If 5: is a family of functions from [a, b] to Xo which is uniformly bounded with respect to I1" IIo, and weakly equicontinuous from [a, b] to X1, then ~: is relatively compact inC([a,b];X). P r o o f . As 9~ is uniformly bounded from [a, b] to X0, X0 is compactly imbedded in X, and the latter, in its turn, is continuously imbedded in X1, it follows that the set 9"([a,b]) = {f(t); f E 9", t C [a,b]} is relatively compact in X1. By virtue of Theorem A.3.1, it follows that 9" is relatively compact in C([a, b];X~), and from Theorem A.I.1, we deduce that ~" is equicontinuous with respect to the norm I1" I1~. By Lions' Lemma A.1.1 it follows that, for each c > 0 there exists 77(c) > 0 such that I]f (t) - f (s)fl ~ cllf (t) - f (s)IIx0 + ,(c)]lf (t) - f (s)IlXl for each f C 9", and all t, s C [ a, b ]. Since 9" is relatively compact in X0 and thus bounded, and 9~ is equicontinuous from [a, b] to X1, this inequality shows that 9" is equicontinuos from [ a, b] to X. An appeal to Theorem A.2.1 achieves the proof. D A.4. C o m p a c t n e s s in L p(a, b ; X ) In this section we shall prove several necessary and/or sufficient conditions in order for a given subset in LP(a, b; X) to be relatively compact. We begin by introducing some notions we need in that follows. Let X be a real Banach space and p C [1, +c~ ). D e f i n i t i o n A.4.1. A family 5: C LP(a, b; X ) is p-equiintegrable if lim h$O

~

b-h

IIf (t + h) - f (t)llPdt - O,

uniformly for f E 9". R e m a r k A.4.1. Each finite set in LP(a,b;X) is p-equiintegrable. This follows from the fact that C([a, b]; X) is dense in LP(a, b ; X ) , and each finite subset in C([ a, b]; X) has this property. R e m a r k A.4.2. A subset 9" in LP(a, b ; X ) is p-equiintegrable if and only if lim IIf (t - h) - f (t)IlPdt - 0 h40 +h uniformly for f E 9". Indeed, the condition above follows from the pequiintegrability condition by a simple change of variable, and conversely. We can now proceed to the first important result in this section.

Compactness in LP(a, b ;X)

305

A.4.1. (Kolmogorov-Riesz-Weil) A subset Y in LP(a, b ; X) is relatively compact if and only if: (i) 9" is p-equiintegrable ; (ii) for each interval [a,/3] included in [a, b ], the set

Theorem

{ fa ~ f (t) dt; f e 2[} is relatively compact in X. P r o o f . We begin by showing the necessity of the conditions (i) and (ii). Let ~ be a relatively compact subset in LP(a, b ; X ) , and let c > 0. As 9= is precompact, it has a finite c-net ~'~ - {fl, f 2 , . . . , fm(~)}" According to Remark A.4.1, ~ is p-equiintegrable. Therefore, there exists 5(c) > 0 such that, for each h C (0, 3(E)] and i C {1, 2 , . . . , n(E)}, we have

b-h

) lip Ilfi(t + h ) - fi(t)llPdt

(fkj,

< e.

(A.4.1)

As 9~ is an e-net for ~, for each f C 9", there exists i E {1, 2 , . . . , n(c)}, such that

b

) 1/p

(~a "f (t) - fi(t)"Pdt

< c.

(A.4.2)

From (d.4.1) and (A.4.2), it follows that

1/p IIf (t + h) - f (t)[IPdt

1/p IIf (t + h) - fi(t + h)[[Pdt) 1/p

+ (f\/o Ilfi(t + h ) - fi(t)llPdt

+

IIf~(t) - f(t)llPdt)

<2c+E for each h C (0, 5(c)]. Obviously, this inequality shows that 9" satisfies (i). In order to prove (ii), let [a,/3] be any interval included in [a, b], and let (fn)ncN be a sequence in 5=. As 9" is relatively compact, we may assume without loss of generality that this sequence is convergent in LP(a, b;X) to some element f. Then lira n--~oo

L

fn (t) dt -

which shows that 9" satisfies (ii).

f (t) dr,

Compactness Results

306

We can now proceed to the proof of sufficiency. Let 5" be a family satisfying (i) and (ii). Let f C 9", A > 0 with a < b - / ~ , and let f ~ - [ a , b - A] -+ X be defined by 1 it "t+~ f (s) ds. f ~ (t) - -~ We shall show next that the family 9"~ - {f~; f C 9"} is relatively compact in C([ a, b - ~ ]; X), and that lim IIf~ - fllLp(a,b-:~ ;X) -- 0 )%0

(A.4.3)

uniformly for f C 9". From (ii) it follows that, for each t C [a, b - ~] the t-section of the family 9"~, 9"~(t) - {f~(t); f C 5"}, is relatively compact in X. In addition, for each t , s C [ a , b - ~], with t < s and s - t _< ~, we have

f~(t) - f~(s)

-

I f t+~ f -s Jt

1L*+~

(r) d r - ~

f ( r ) dr.

The change of variable r - 0 + s - t in the last integral, leads to

i f t+~ f (O)dO- ~i f t+~ f (O + s - t)dO. f ~ ( t ) - f~(s) - -~ at Jt Therefore

1

f b+t-s

--~ ~ a a

IIf (o) - f (o + s - t)IIdO

l (b_ a + t _ s)ilq ( fb+t_s

-< X

)lip

Ill (o) - f (o + t - s)IlPdO

,

\aa

where 1/p + 1/q - 1. From this inequality, and (i), it follows that ~ satisfies the hypotheses of Arz~la-Ascoli's Theorem A.2.1, and therefore it is relatively compact in C([ a, b - ~ ]; X). In orde~ to check (A.4.3), let us observe that

IIf

f~l IpLp(a,b-~ ;X) _- f bJa- ~

-

f

- J,,.

)~P

rf ( t ) Jt

f (t) ds

II/(t) - f~ (t)IlPdt f ( s ) ds

Jt

IIpdt

f (s) ds

P

dt.

Compactness in LP(a, b ; X)

307

From this relation, by changing the variable s - t + h in the last integral, using Hhlder's Inequality and changing the order of integration, we obtain

1 ~b-A

1/ab (/0

<- Ap

AP/q

f (t) ds -

fo

P f (t + h) dh

dt

)

Ill(t) - f (t + h) llPdh dt

= ~lfo~fab-~ Ill (t) - f (t + h)IlPdt dh. But, this inequality, along with (i), proves (A.4.3). Finally, we shall show that, for each c C (a, b), the set 9"l[a,c], of restrictions to [a, c] of elements in 9", is relatively compact in LP(a, c;X). From the preceding arguments, and (A.4.3), we deduce that the family of operators {Q~; A e (0, b - c ) } , Q~" 9" ~ C([ a, c ]; X), defined by

Q~f - f ~ for each )~ c (0, b - c) and f C 9~, satisfies the hypotheses of Lemma A.1.2. Indeed, for each ~ C (0, b-c), (Q~9~)lEa,~] is relatively compact in LP(a, c;X) being relatively compact in C([ a, c ]; X), and lim IIQ~f

;%0

-

fllLp(a,c;X)

-- 0

uniformly for f c 9". In view of Lemma A.1.2, we conclude that 9"l[a,~] is precompact in LP(a, c; X). Finally, using Remark A.2.1, and reasoning as above, we deduce that, for each c E (a,b), 9"l[~,b] is relatively compact in LP(c, b;X). Therefore 9" is relatively compact in LP(a, b;X), and this achieves the proof. [3 T h e o r e m A.4.2. (Aubin) Let 1 _< p < +oc, and X0, X and X1 be three

real Banach spaces with Xo compactly included in X and X continuously included in X1. If 5: is bounded in LP(a,b;Xo) and p-equiintegrable in LP(a, b; X~), then it is relatively compact in LP(a, b;X). Proof. Let [a,/3] C [a, b]. follows that the set

Since 9" is bounded in LP(a, b;Xo), it

{ ~~ f (t) dt; f c 5=}

Compactness Results

308

is bounded in X0, and therefore relatively compact in X. Hence 9" satisfies (ii) in Theorem A.4.1. Let e > 0. According to Lemma A.1.1, there exists ~(e) > 0 such that I1 11 _< llxll0 + ( )llxll for each x C X0. Then, for each f E 9" and each h > 0, we have

b-h fa lif(t + h ) -

)lip < e (lab_h I[f(t + h ) f(t)llPdt

lip ) f(t)iiPodt

+,(e) ( lob-h 11f(t + h) - f (t)liPldt) 1/p As 9~ is bounded in

LP(a, b;X0), there (fa

b-h

exists M > 0, such that

Ilf(t + h) - f(t)"Pdt

) lip

b-h

<_2Me + rl(e) ( f\.l.

) liP Ilf (t + h) - f (t)II~dt

But 9" is p-equiintegrable in LP(a,b;X1), and consequently there exists 6(e) > 0, such that, for each f C 9" and each h E (0, 6(e)), we have

(~ab-h

II/(t

) l/pC + h) - f (t)li

dt

-<

9

From this inequality and the preceding one, we deduce b-h

( faa

) 1/p

IIf (t + h ) - f (t)llPdt

<_ (2M + 1)e

for each f E 9~ and each h C (0, 6(e)). Therefore 9~ is p-equiintegrable in LP(a, b; X), and, according to Theorem A.4.1, it is relatively compact in this space. The proof is complete, ff]

A.5. Compactness in Lp(a, b;X). Continued In this section we shall present a variant of Theorem A.4.1 which proves much easier to check in many practical situations.

Definition A.5.1. A family 9" in LP(a, b;X) is called uniformly integrable if for each e > 0 there exists 6(c) > 0 such that, for each measurable subset E in [a, b] whose Lebesgue measure #(E) < 6(e), we have

E IIf (tl lldt < e for all f E 9".

Compactness in LP(a, b ; X). Continued

309

R e m a r k A.5.1. As [a, b] can be decomposed into a finite union of sets of arbitrary small Lebesgue measure, it follows that each uniformly integrable subset is bounded. The converse statement is not true, as we can see from the example below. E x a m p l e A.5.1. Let 9" = {fn; n C N*}, where fn: [ - 1 , 1] --+ IR is defined by fn(t)

--

0, ~, n

for t e [ - 1 , - I / n ] U [i/n, 1] for t e [ - 1 / n , 1/n]

nC

N*.

Obviously 9" is bounded in L 1(-1, 1; R) (all elements having the norm 1), but it is not uniformly integrable "near 0". Namely, we have

1/u ]fn(t)l dt - 1

1/n

for each n E N*, relation which shows that the family 9" cannot satisfy the condition in Definition A.5.1. R e m a r k A.5.2. If p > 1 and 9" is bounded in LP(a,b;X), then it is uniformly integrable. Indeed, by virtue of HSlder's Inequality, it follows

dt < (#(E)) 1/q

( / E ) l idtp

for each f C 9" and each measurable subset E in [a, b], where q is the conjugate of p, i.e. lip + 1/q - 1. From this inequality we easily deduce that 9" satisfies the condition in Definition A.5.1. R e m a r k A.5.3. Let :T C Ll(a, b; X ) and let us assume that there exists g C L l(a, b), such that Iif (t)II _< g(t) for each f c 9" and a.e. t E (a, b). Then 9~ is uniformly integrable. Indeed, in this case, fE ]]f(t)]] dt <_ fE g(t)dt for each f C 9" and each measurable subset E in [a, b ]. As g is integrable, from this inequality, it follows that 9" is uniformly integrable. The main result in this section is: T h e o r e m A.5.1. (Gutman) A uniformly integrable family 9= in LP(a, b ; X)

is relatively compact if and only if: (j) 9~ is p-equiintegrable; (jj) for each e > O, there exists a compact subset CE in X , such that, for each f ~ 9= there exists a measurable subset E~,I in [a, b] with

#(E~,f ) <_ e, and such that f (t) E C~ for each t E [ a, b ] \ E~,f .

Compactness Results

310

P r o o f . Necessity. Let 9" C L p(a, b;X) be relatively compact. From Theorem A.4.1 it follows that 9" satisfies (j). To prove (jj), let e C (0, 1), and let E(c) C C([ a, b ]; X) be a finite e-net of 9" with respect to the norm II'llLP(a,b;X)" Let n C N*, and let

En(c) - E(c(n+I)/p/2n/p). Then, for each n C N* and each f C 9", there exists fn C En(e) such that Ilf -

fnllLp(a,b;X) ~---g.

Let us define

EEn,: -- {t E [a, b l; Ill (t)

-

fn(t)II

> Cn}"

We deduce that #(E~,:) <_ e/2 n. Let now

and let us observe that, for each t C [a, b ] \ Ee,f, and each n C N*, we have Ilf (t) -

:n (t)II

_~ Cn.

Finally, let

Ce,f - {f(t) ; t C [a,b] \ Ee,:} and C s -

U ce,:.

We shall prove that C~ is totally bounded. To this aim, let 5 > 0, and let us fix n E N* satisfying e ~ < ~/2. We can always do this, because e E (0, 1). As E~(e) is a finite family of continuous functions, the set

En(e)([a,b]) = {h(t) ; h C En(e), t E [a,b]} is relatively compact. Hence it has a finite 5/2-net {xl,x2,...,xk(5/2)}. Then, for each fn E En(s) and t C [a, b], there exists i E {1, 2 , . . . , k((~/2)}, such that we have IIf~(t)-

xill < ~.

From this inequality, recalling that p(E~,f) <_ el2 n, we deduce IIf(t) - x~ll _< I l l ( t ) -

fn(t)ll + IIfn(t)

- xgll < ~ + ~ - ~,

which shows that C~ is precompact and this completes the proof of the necessity. Sufficiency. Let 5 be a uniformly integrable family satisfying (j) and (jj) and let [c~,/~] be any interval included in [a, b]. By Theorem A.4.1, to complete the proof, it suffices to show that 9"~ - ( ( ~ f(t)dt; Lo~

f C 9:~) is

Compactness in LP(a, b ; X). Continued relatively compact in X. operator QE" 9"~ ~ X by

311

To this aim, let c > 0 and let us define the

Q~ ( L ~ f (t) dt) - I~,~]\E~,s f (t) dt, where E~,f is the subset whose existence is ensured by (jj). As f(t) C C~ for each t E [ a , / ~ ] \ E~,I and C~ is compact, from Lemma A.1.3, it follows that Q~(9~a~) is relatively compact. Let us observe that

f (t) dt

-

IIf (t)II dt,

f (t) dt < e,f

for each f E 9". Since 9" is uniformly integrable, this inequality shows that lira Q~x - x E$0 uniformly for x E 3"~. According to Corollary A.2.1, it follows that 3"~ is relatively compact. Hence ~ satisfies the conditions in Theorem A.4.1, and this completes the proof of Theorem A.5.1. V-] We conclude this section with a variant of Theorem A.4.2. T h e o r e m A.5.2. Let 1 _< p < +oc, and let X0, X and X1 be real Banach spaces with Xo compactly imbedded in X , and X continuously imbedded in X1. Let 9: be subset in LP(a, b; Xo) such that {IifiiP; f c 5=} is uniformly integrable and which satisfies" (1) ~" is p-equiintegrable in LP(a, b ; X1) ; (11) for each r > 0 there exists M~ > 0 such that, for each f E there exists a measurable subset E~,f in [a,b] with p(E~,f ) <_ r and such that [If (t)II0 _< M~ each t E [a, b ] \ Ec,f. Then ~ is relatively compact in L p(a, b ; X). Proof. As X0 is compactly imbedded in X, from (11), it follows that 9= satisfies (jj) in Theorem A.5.1. Let e > 0. Since {]]fllP ; f C 5"} is uniformly integrable, there exists 5 - 3(e) > 0 such that Ell f(t) llpdt <_ s p for each f E 9", and each measurable subset E in [a, b] with #(E) _~ 25. From (11), we deduce that, for 5 > 0, there exists M5 > 0 such that, for each f C 9", there exists a measurable subset Eb, I in [a, b l, with #(E~,f) <_ ~,

Compactness Results

312

and IIf(t)]lo < M5 for each t C [a, b] \ Es,I. Let us observe that, for each h > 0, we can choose

E~,f(.+h ) -- E~, S + h. Let us denote by E - Es,S U E&f(.+h ) and G - [a, b - h ] \ E. We then have 1/p IIf (t + h) - f (t)[IPdt) (f~ Ilf(t + h) - f(t)JJPdt <_

b-h

+

)liP (/E

(/G Ilf(t + h ) -

f(t)llPdt

)lip < 2e + (/G IIf(t + h ) -

f(t)llPdt

)l/p .

(A.5.1) Next, let A > 0 with 2AM5 <__e, and let ~(A) > 0 be given by Lemma A.1.1. We recall that r/(A) satisfies (A.5.2) for each x C X0. As ~" is p-equiintegrable in LP(a, b; X1), there exists ~y(r > 0, such that

b-hIIf(t + h) -

f(t) ll et

I 1/p<

(a.5.a)

for each f C 9" and each h > 0 with h _< 7(e). From (A.5.1), (A.5.2) and (A.5.3), it follows

([

b-hIIf(t + h ) -

1/p )l/p (/G IIf (t + h) f (t)]lPdt) f (t)llPdt <_ 2e + (/G [[f(t + h) - f(t)IlPdt )l/p b-h ) 1/p

_< 2e + 2AM5 + ~(A)

<_ 2e + e + ,(A) ( f\.l.

]]f (t + h ) - f (t)ilPldt

_< 4e,

for each f E 9% and each h > 0, with h <_ 3'(r Hence 9~ is p-equiintegrable in LP(a, b;X), and we are in the hypotheses of Theorem A.5.1. The proof is complete. [::]

A.6. The Superposition Operator In this section we shall prove several sufficient conditions in order that the operator obtained by composing a fixed continuous real function by elements in a given space of functions be well-defined and continuous. In the sequel, f~ is a bounded domain in R n, n >_ 1, g" R+ x ~ x R -+ R, and p C [1, §

313

The Superposition Operator

D e f i n i t i o n A.6.1. The function fp 9 IR+ • Dp C I~+ • LP(~) -+ LI(~) defined by fp(t, u)(x) - g(t,x, u(x)) for each t C I~+, u C Dp, and a.e. x C ~, where Dp - {u e LP(~); g ( t , . , u(. )) e LI(~) for each t e I~+}

is called the superposition operator on LP(~) associated to the function g. We say that the superposition operator on LP(~) associated to g is welldefined on L p (~), if Dp - Lp (~), and fp maps I~+ • Dp in L p (~t). We say that fp is well-defined on C ( ~ ) , if C ( ~ ) c Dp and fp maps ~+ • C(~) in We recall without proof the following result due to Vitali. T h e o r e m A.6.1. Let X be a Banach space, let (~, E, #) be a a-finite and complete measure space, and let p C [1, +cr Let (Un)ncN be a sequence in LP(~, p; X ) and u" ~ --+ X . Then u C LP(~, #; X ) and l i m n _ ~ ~tn - - ~ in LP(~, #; X ) if and only if (i) limn--+cr Un - u in measure; (ii) the set {]IUnlIP ; n e N} is uniformly integrable in L I ( ~ , #; I~) ; (iii) for each c > O, there exists ~ C E, with # ( ~ ) < +cr and

ll n( )ll

--<

for each n C N.

For the proof see Dunford and Schwartz [49], Theorem 6, p. 122. The next lemma gives some sufficient conditions in order that D R - or even that fp be well-defined, either on n p(~), or on C(~).

LP(~),

L e m m a A.6.1. Assume that g " I~+ • ~ • I~ --~ IR satisfies" (gl) for each (t, x) G IR+ x ~, u ~-+ g(t, x, u) is continuous ;

ach

c R, (t,

g(t,

;

(g3) for each T > O, and r > O, the restriction of g to [0, T ] • 2 1 5 [ - r , r ] is bounded. Let p G [1,+cr and let fp " I~+ • Dp C I~+ • LP(~) -+ L I ( ~ ) be the superposition operator on L p(~) associated to g. Then

(i) L ~ ( ~ ) C Dp. I f p - +co, D ~ - L ~ ( ~ ) , and fcr is well-defined on LCr In addition, for each t G IR+, the function u ~-~ fcc(t, u) is continuous from LCr into itself and, for each u ~ LCr the function t ~+ fee(t, u) is strongly measurable. Also in this case,

314

Compactness Results i.e. p - +co, if g is jointly continuous, then fcr is well-defined on C(~) and continuous from IR+ x C(~) to C(~). (ii) If p C [1, +co), and for each T > 0 there exists aT > 0 and bT C It(, such that Ig(t,x,u)l < aTiu[ p + bT

(A.6.1)

for each (t, x, u) e [0, T] x ~ x IR, then Dp - LP(~t), and, for each t e IR+, the function u ~-+ fp(t, u) is continuous from LP(~) to Ll(~t). In addition, for each u C LP(~t), the function t F-+ fp(t, u) is strongly measurable. (iii) /f p - 1, and g satisfies (A.6.1) then, for each r C [1~ +cc], the restriction of fl to IR+ x Lr(~) coincides with fr which is welldefined on Lr(~). In addition, for each t C IR+, the function u ~-+ f l ( t , u ) is continuous from Lr(~) into itself and, for each u E Lr(ft), the function t ~ fl(t, u) is strongly measurable from IR+ to Lr(~t). Finally, for each T > O, we have [Ifx(t~u)[[L~(~ ) ~_ aT[[U[[L~(~) + [bT]#(~) 1/r

(A.6.2)

.for each t e l 0 , T], and u e L r(~), where aT and bT are given by (ii), and #(~) is the Lebesgue measure of ~. P r o o f . (i) By virtue of (g3), it follows that g maps bounded subsets in It~+ x ~ x R into bounded subsets in IR. Therefore, for each u E L~(~t) and t C IR+, x ~-+ g(t,x(u(x)) is bounded on ~. In addition, from (gl) and (g2), the function above is measurable. Thus LCC(~) C Dp for each p C [1, +co ], and f ~ is well-defined on L~(~t). If u e L ~ ( ~ ) and (Uk)keN is a sequence with limk_+~ uk -- u in L ~ ( ~ ) , then (Uk)keN is a.e. uniform convergent to u on ~t. By (gl) we have limk_+~ f(t, Uk) -- f ~ ( t , u) in n ~ ( ~ ) for each t C IR+. As g can be approximated, uniformly on bounded subsets in IR+ x gt x IR, by countably-valued functions, the strong measurability of t ~ f ~ ( t , u), for each u in L ~ ( ~ ) , follows by using standard arguments. If g is jointly continuous then, obviously fcc is well-defined on C(~), and is continuous from I~+ x C(~) to C(~). (ii) Clearly, for each (t,u) e I~+ x LP(~), x ~ g(t,x,u(x)) is measurable. On the other hand, in view of (A.6.1), the function above is in Ll(gt). So D p - LP(~). In order to prove that, for each t C IR+, the function u ~ fp(t, u) is continuous from Lp(~) to LI(~), we shall make use of Vitali's Theorem A.6.1. To this aim, let (Uk)kCl~ be a sequence which converges in LP(~) to u. For the beginning, we shall show that (g(t,. ,Uk(')))kCN converges in measure to g ( t , . , u(. )). Indeed, if we assume the contrary, then there exists c > 0 and a subsequence of ( g ( t , . , uk(" )))ke~, denoted

Problems for simplicity again by

(g(t,., uk(" )))kCN, such

315 that

({x e ~ Ig(t, x, uk (x)) - g(t, x, u(x))l _ c}) _ c, for each k E H, where # is the Lebesgue measure on ]~n. On the other hand, has at least one subsequence which converges a.e. on ~ to u. From (gl) it follows that (g(t,., uk(" )))ken has at least a subsequence convergent a.e. on ~ to g(t, . , u ( . )). Consequently, in view of Lebesgue theorem, (g(t, ",Uk(" )))k~N converges in measure on that subsequence to g(t,., u(-))~ thereby contradicting the last inequality. This contradiction can be eliminated only if (g(t,. ,Uk(" )))ken is convergent in measure on f~ to g(t, . , u ( . ) ) . Next, we shall show that {g(t, .,uk(.)); k e H} is uniformly integrable in LI(~), i.e. for each c > 0 there exists 5 > 0 such that

(Uk)kEN

L Ig(t, x, Uk(x))ld~ _< for each measurable subset w in ~t with #(w) < (~ and uniformly for k E H. To this aim, let T > 0 and and let us observe that, by virtue of (A.6.1), we have f

~

tr~

./,, [g(t,x, Uk(X))[dx <_aT ./,, [uk(x)lPdx + [bTIp(w) for each k C H, and each measurable subset w in f~. As (uk)kCN is convergent in LP(a) to u, from Theorem A.6.1, we know that the set {[uklP; k C H} is uniformly integrable in L 1(~). Then, from the last inequality, it follows that {g(t,., uk(" )); k C H} is uniformly integrable too. Consequently, by virtue of Theorem A.6.1, we conclude that (g(t,., Uk(" )))ken is convergent in Ll(f~) to g(t, .,u(.)). (iii) The proof of the first part of (iii) follows the same lines as those in the proof of (ii), and therefore we shall omit it. In order to check (A.6.2), it suffices to observe that (A.6.1) with p = 1, along with Minkowski's inequality, leads to

(/~

Ig(t,x,

)l/r ___a~ (/FtI~(~)l~d~ )l/r + Ib~l (/~ ) ~(~)l~dx

for each t C [0, T] and

1/r

dx

u C L r (~t),

relation which completes the proof.

[3

Problems.

A family 32 in co is called uniformly convergent if, for each ~ > 0, there exists n(c) C N* such that, for each x C ~ and each n >_ n(c), we have

IXnl _
316

Compactness Results

P r o b l e m A.1. Prove that a subset 2: in co is relatively compact if and only if it is bounded and uniformly convergent. Let X be a real Banach space and co(X) the space of all sequences with elements in X vanishing at c~. This, endowed with the norm I1" I1~, defined by II(Xn)ncH. IIc~ SUPnel~. IlXnll for each (Xn)nEN. E CO, is a real Banach space. -

-

P r o b l e m A.2. Using the model offered by Problem A.1, state and prove a necessary and sufficient condition in order that a subset in co(X) be relatively compact. A family 2: in lp is called uniformly p-summable if, for each c > 0, there exists n(c) C H*, such that, for each x C 2: and each n >_ n(c), we have

(Ek~=n IxklP) lip <_ C. P r o b l e m A.3. Prove that a subset :~ in Ip is relatively compact if and only if it is bounded and uniformly p-summable. Let X be a real Banach space, let p c [ 1, +oc), and let lp(X) the space of all sequences (Xn)neN* with elements in X satisfying ~-'~nC~=lIlXnll p < -~-(~:). This, endowed with the norm I1" lip, defined by II(xn)ncN. IIp - (~-~'~n~__lIXnlP)1/p for each (Xn)ncN* C lp(X), is a real Banach space. P r o b l e m A.4. Using the model offered by Problem A.3, state and prove a necessary and sufficient condition in order that a subset in lp(X) be relatively compact. P r o b l e m A.5. Prove that a family 9" in the linear space C~(R), of all continuous functions from I~ to R, having finite limits at both +c~ and -cx~, endowed with the norm I1" I[~, is relatively compact if and only if it is uniformly bounded and equicontinuous on I~. P r o b l e m A.6. Prove that a family 9" in the linear space C~(I~; X) of all continuous functions from I~ to X, having limits at both +c~ and - c o , endowed with the norm I1" I1~, is relatively compact if and only if: (i) 9" is equicontinuous on R; (ii) there exists a dense subset D in ]R such that, for each t C D, the set 9"(t) = {f(t); f C ~'} is relatively compact in X. P r o b l e m A.7. Prove that a family 9" in the linear space C~(IR; X), endowed with the norm I1" I1~, is relatively compact if and only if: (i) 9" is weakly equicontinuous on I~; (ii) for each [a,b] C 1~, 9"([a,b]) = {f(t); f C 9", t C [a,b]} is relatively compact in X.

Problems

317

P r o b l e m A.8. Prove that a family 3" in LP(~; X) is relatively compact if and only if: (i) 9" is p-equiintegrable; (ii) for each interval[a,/7] the set

f(t) dt; f C ~

is relatively

compact in X ; (iii) for each e > 0 there exists (7(e) > 0 such that

lif (t) liPdt +

(~)

Ill(t) IlPdt

<_ e,

uniformly for f E 9". P r o b l e m A.9. Prove that a family 9" in LP(R; X) is relatively compact if and only if: (j) 9" is p-equiintegrable ; (jj) for each e > 0 there exists a compact subset C~ in X such that, for each f C 9~, there exists a measurable subset E~,I in ]~ with p(E~,I) ~_ e, and such that f (t) C C~ for each t C I~ \ E~,f ; (jjj) for each e > 0, there exists/7(e) > 0 such that

Ilf (t)[IPdt +

(~)

IIf (t)liPdt

<_ e,

uniformly for f C 9". P r o b l e m A.10. Let X0, X, and X1 be three real Banach spaces with X0 compactly imbedded in X, and X continuously imbedded in X1. Let 3" be a subset in L p (I~; X0) with { ]]f liP ; f G 5"} uniformly integrable and which, in addition, satisfies: (1) 9" is p-equiintegrable in LP(I~; X I ) ; (11) for each e > 0, there exists M~ > 0 such that, for each f G 9" there exists a measurable subset E~,f in I~ with #(E~,f) _< e, and such that

IIf (t)II0 for each t C R \ E~,f ; (lll) for each e > 0 there exists (7(e) > 0 such that

IIf (t)IlPdt uniformly for f C 9".

+

(~)

IIf (t)II'dt

c,

318

Compactness Results

Prove that 9~ is relatively compact in LP(R; X). N o t e s The results in section 1 are classical. Theorem A.2.1 is an infinitedimensional version of the famous ArzelS~-Ascoli's Compactness Theorem, and Theorem A.2.2 is essentially due to Aubin [9]. Theorems A.3.1 and A.3.2 are in fact weak variants of Theorems A.2.1 and respectively A.3.2. Theorem A.4.1 is an infinite-dimensional generalization of a well-known compactness result in L p spaces and, in the form here presented, is due to Simon [115]. For a similar variant see Vrabie [127]. Theorem A.4.2 is an extension and, at the same time, a generalization of some compactness results established by Aubin [9], as well as by Brezis and Browder [30]. Theorem A.5.1 is from Gutman [66], and Theorem A.5.2 is new. Finally, Lemma A.6.1, well-known in the mathematical folklore, is undertaken in this form from Vrabie [127]. The problems included at the end of the chapter are classical.

Solutions Chapter

1

Problem 1.1. Let n C N*, let {J:)n " a - t~ < t~--. < t n - b be such that tn+l - t n - ( b - a)/n, i - 1, 2 , . . . , n - 1, and let Xn" [a, b] --+ X, defined by Xn(t) - Xn(tn+t) for t e (tn, tn+l ], i -- 1 , . . . , n - 1 and Xn(t) -- Xn(tr~) for each t E ItS, t~ ]. Obviously, for every x* C X* and every t E [ a, b], we have limn--+cc(Xn(t), X*) (x(t), x*). Therefore x is weakly measurable. On the other hand, x([ a, b ]) is included in the weak closure of the convex hull of the set {x(tin+l); n C N*, i - 0, 1 , . . . , n - l } which, by virtue of Theorem A.1.4, coincides with the strong closure of the same convex hull. As the later is obviously separable, it follows that x is almost separably valued and, thanks to Theorem 1.1.3, we deduce that x is strongly measurable. The conclusion still holds true in the general case specified, under the extra-condition that ft is separable. -

-

Problem 1.2. First, we shall prove that, for each p C [1, + ~ ] , Wt,P(O, 7r) is continuously imbedded in C([0, r~ ]). As the case p = oc is evident, we shall assume that p < +oc. Let 9" be a subset in WI,p(0, re) for which there exists M > 0 such that ]lfllLp(0,~) + I[f'llL~(0,~) ~ M

(,)

for each f E 9=. Obviously, for each f C 9", we have essinf{[f(t)l ; t C [0, Tr]} = s u p { a ; If(t)[ _> a a.e. t E [0, rr]) < M.

(**)

(Otherwise, we should have [IfilL,(0,,~) > M, inequality in contradiction with (,).) As f is continuous, there exists tf C[0, rr ] such that inf{]f(t)[; t C [0, Tr]} = [f(tf) [. In view of Leibniz-Newton formula, (,), (**), and Hblder's inequality, we have

If(t)[ <_ If (tf )] +

_< If(ts)l +

/;

If (s)[ds

If (s)lds <_ M + 7rl/qllfllL (0, ),

for each f C ~, where l i p + 1/q = 1. Hence 9" is uniformly bounded on [0, ~-], and consequently WI,P(O, 7r) is continuously imbedded in C([ 0, ~]). 319

Solutions

320

Let now t, s C [ 0, 7r ], and p C (1, +co). Again by Leibniz-Newton formula, Hhlder's inequality, and (,), we have

i f ( t ) - f(s)l _<

If(7-)l d~- <_ MIt -

sl l/q,

for each f E 9". Hence 9", which is uniformly bounded, is equicontinuous on [0, 7r ]. By virtue of Theorem A.2.1 it follows that it is relatively compact in 6'([0,7r]). I f p = 1, the set 9" = {fn; n C N*} with f'n = gn, and f~ (0) = 0, where n for t e [O, 1/n] gn (t) -0 for t e ( i / n , 7r ], although bounded in W1,l(0, Tr), is not relatively compact in C([0, Tr]) as it is not equicontinuous at t - 0.

Problem 1.3. We have (Au, v)L2(O,~) --

/0

u" (x)v(x) dx - u' (x)v(x) ]~ -

= -~(~)v'(x)Ig +

/0

u' (x)v' (x) dx

~(~)v"(x) ex - (~,Av>L~(o,~I

for each u , v C D(A), and therefore A is symmetric. Let us denote by / - / = L2(0, ~-). From the definition of the adjoint of an operator, we have

D(A*) = {v E H ; 3C > 0, I(Au, V>g <_ CllUIIH, Vu ~ D(A)}. Let us observe that H2(0, 7r) C D(A*) because, for each v C H2(0, 7r), we have

I(Au, v>l < IIv"IIL=(O,~)IluiiL=(O,~) for each u E D(A). In addition, from the symmetry, we have A*v = v" for each v C H2(0, ~). To conclude, it suffices to show that D(A*) C_ He(O, ~). As A* is a closed operator and D(A*) is dense in H2(0, 7r) we deduce [H2 (0,rr)

that D(A*) = H2(0, 7r). Thus A # A*.

Problem 1.4. We have D(A*) C_ H2(0, ~). Then, for each u C D(A), and each v C H 2 (0, ~), we have (Au, V>L2(0,~) --

/0

u " ( x ) v ( x ) dx - u'(x)v(x) I~ -

= -~'(0)v(0) + ~(~)v'(~) +

/0

u(x)'v'(x) dx

~(x)v"(x) dx.

If v(0) # 0, or v'(Tr) # 0, then v does not satisfy the condition in the definition of D(A*), because for each n C N* there exists U n C D ( A ) with I(Aun, V)L2(O,~)I > nllUnllL2(O,~ ). Indeed, if v(0) # 0 then each function

Chapter 2 Un e D(A), with

321

I1~11c~(0,~)-

1 and -u'n(O)v(O) > IIv"llL2(O,~) + n, enjoys the previously specified property. As the case v ~(Tr) r 0 follows analogously, the proof is complete.

Problem 1.5.

We have D(A*) C W1,2(0, Tr). For each u C D(A), and v E W1'2(0, 7r), it follows that

(Au, v} -

/0

/0

u' (x)v(x) dx - -u(Tr)v(Tr) -

u(x)v' (x) dx.

Let us observe that, v e D(A*) if and only if v e W1'2(0, 7r), and v(Tr) - 0. Hence D(A*) - {v e W1'2(0, 7r); v(Tr) - 0 } , and A * v - v' for v e D(A*).

Chapter 2 Problem 2.1. A simple calculation shows that {S(t) ; t _> 0} is a semigroup of linear bounded operators. In addition

for each t _> O, and each (Xn)n~N* C lp, which shows that the semigroup contains only nonexpansive operators. Let us observe that, for each t _> 0, each (Xn)~cN* C lp, and each rn C N*, we have m

(Xn)nCN* (x)

+

(1 - e-akt) p IXkl p

E k=m+l

oo -- kE{ 1,2,...,m}

~-~*)~ k=m+l

Let E > 0. Let us fix m - m~ > 0 such that oo

cp

I~l; <__-~, k=m+l

and let us choose 5 - 5(a) > 0 such that, for each t E (0, 5(a)), we have cP max kE{1,2,...,m}

(e " a k t - 1) p < --

2 II(X~)nCN*I1~

From the last three inequalities, we deduce that the semigroup is of class Co. Next we shall prove that

D(A) - {(Xn)nCN* e lp ; (anXn)nCN* e lp},

Solutions

322 and

A(Xn)nEN* -- --(akXk)kEN*. Indeed, let (Yn)nEN* E lp be such that 1

4

lim -i (S(t) - ,)(Xn) nEN* --(Yn)nEN* t4o P i.e.

lim t4o k_~~

1 (e_ak t -- 1) x k - - Y k

7

I --0~

P ~0.

This implies that

yn - lim -l a (e ntt$o t

- 1 ) Xn - -anXn

for each n E N*, and (anXn)nEN* C Ip. Hence

D(A) C_ {(xn)nEN* E lp; (anXn)nE5* E lp}, and

A(xn)neN* --(--akxk)kEN* for each (Xn)nEN. E D(A). Let (Xn)nEN* E lp be such that (anXn)nEN* E lp. Obviously, we have

I

-[ (S(t) - I)(X~)nEN* + (anXn)nEN*

P

1 (e_ak t - 1) + ak ~-

-

Ixk ]p +

k=l

~1

(e -ak

t

--

1) + ak

Ixkl p 9

k=m+l

Since ( 1 - e - a t ) / t ~ a for each t _> 0, and each a >_ 0, we have

1 akt ]P o0 ~-(e-- 1 ) + a k ] x~ < 2 p E

E k=m+l

P pk. akX

k=m+l

Consequently, 1

7 (S(t) - I ) ( x , ) nEN* -t--(anXn)nEN* (X) 1

< -II(Xn)nEN, Ip --

t

max

P iE{1,2,...,m}

{](e -ait-

1) + aiI p} + 2 p

k----m+l

Let e > 0. As (anXn)nEN* E lp, there exists m cx~

E 4 4 -<

k--m+l

E

s

mc > 0 with

P P

akx k.

Chapter 2 For m so fixed, there exists 5 1 -

5(e) > 0 such that _

max

323

~: iC{1,2,...,m}

1) + ai

[p

ep

} <

-- 2 II (Xn)nEN* IIp

for each t E (0,~). From the last three inequalities, we deduce that (Xn)nCN* e D(A). Finally, if (an)nCN* is bounded, it follows t h a t D(A) - lp, and A is continuous. Hence A generates a uniformly continuous semigroup. If (an)heN* is unbounded, then there exists (nk)kCN* with nk > k, and ank > k ~ for each k C N*, where a - 1 for p > 1, and a > 1 is fixed for p - 1. T h e n the sequence (Xn)ner~*, defined by

{1

Xn --

-~

0

if n

-- nk

if n 5r

n k

(V)k E N*,

is in lp, but (anXn)neN* ~ lp. Hence in this case D(A) 5r lp, and therefore A cannot generate a uniformly continuous semigroup.

Problem 2.2. Obviously, {S(t); t _> 0} is a semigroup of nonexpansive linear operators. Let (Xn)ncN*, t > 0, and m C N*. We have IlS(t)(Xn)ncN, - (Xn)nCN* I[oc

{

_< max

max

kC{1,2,...,m}

[ 1 - e-akt[ II(xn)ncN,

I1 ,

sup

k>_m

IXkl

}

9

Let e > 0. Let us fix m > 0 satisfying sup Ixk[ <_ e. k>m

This is always possible because (Xn)ncN* C co. For m fixed as before, there exists ( f - a(e) > 0, such that max

[ e - ~ k t - 1[ <

kC{1,2,...,m}

e

-- II (Xn)nEN* [lot'

and therefore the semigroup is of class Co. We shall prove next that

D(A)-

{(x,~)ncN* e co; (anXn)nCN* e C0}

and A(Xn)ncN* -- --(akxk)keN*. Indeed, let (Yn)neN* C CO be such that lim t$O 71 (~(~) -- I)(Xn)nCN* --(Yn)nCN*

-- O, CK?

i.e. 1

lim - sup [ (e -~kt t$0 t k>l

-

1)xk

-

y k I - - O.

Solutions

324 This implies

y n - - l i m l- a(e n t - --1) Xn----anXn t$o t for each n C N*, and (anXn)n~N* C CO. Hence D(A) C_ {(Xn)nEN, C CO; (anXn)nCN* C CO} and

A(zn)ncN* -(--akxk)kcN* for each (Xn)nCN* C D(A). Let (Xn)nCN* E CO with (anXn)nCN* E CO. For each rn C N*, we have 1

? (S(t)

I)(Xn) n c N * + (anXn)nEN*

-- max {M1, M2 }, OO

where

Ml --maxkc{1,2,...,m} ]-i1 [(e -ak t - 1) + ak] xkl M2 --SUPk>_ml~ [(e-akt--1) +ak]xkl. Since (1 - e -at) /t ~ a for each t >_ 0, and each a _> 0, we have

1

sup 7 [(e-akt--1)+ ak] xk < 2 sup ]akXkl.

k>m

k>m

Consequently

1

(S(t) -- I)(Xn)nEN* ~t_(anXn)nEN* OO

1 < max

max

kE{1,2,...,m} t

[(e -akt

--

1) + ak] Xk ,2 sup lakxkl

k>m

~

Let e > 0. As (anXn)nEN* C aO~ there exists m - me > 0 with

sup lakz l _<

C

k>m

For rn fixed as above, there exists 5 m&x

kc{1,2,...,m}

1

5(e) > 0, such that

-[ [(e - a k t - 1 ) + ak] Xk

C

for each t C (0,5). From the last three inequalities, we deduce that (Xn)nEN* E D(A). Finally, if (an)nEN* is bounded, we get that D(A) - co and A is continuous. So A generates a uniformly continuous semigroup.

Chapter 2

325

If ( a n ) n e N * is unbounded, then there exists (nk)keN* with nk > k, and auk > k for each k C N*. Then the sequence (Xn)nCN*, defined by

Xn - { i

if n -- nk if n 5r n k (V)k C N*,

is in co, but (anXn)nCN* ~ CO. Hence, in this case, D(A) ~ co, and therefore A cannot generate a uniformly continuous semigroup.

Problem 2.3. Clearly {S(t) ; t > 0} is a semigroup of nonexpansive, linear operators. Let us observe that limt40 S(t)f - f if and only if f is uniformly continuous on R. Since Cb(R) contains functions which are not uniformly continuous (as for instance f(x) - sin x2), the semigroup is not of class Co. A simple calculation shows that D(A) coincides with the space of all functions which are uniformly differentiable at the right on R, and whose right derivatives are continuous, and bounded on R, and A f - ft for each f C D(A). As the closure of D(A) coincides with the space of all uniformly continuous and bounded functions from R to R, and the later is a proper closed subspace in Cb(R), it follows that D(A) is not dense in X. Problem 2.4. In order to solve the problem, it suffices to check that, for each f e X, the function u" R -+ X, u(t)(x) - [ G ( t ) f ] ( x ) for each t e R, and each x C R, is differentiable on R, and satisfies { u ~ - Au ~(o) - f To this aim, let us observe that, for each f C X, each t > 0, each h > 0, and each x E IR, we have

l[u(t + h)(x) - u(t)(x)] - [du(t)](x) h

=

~-~h ~

k!

f (x - ks) - Z

k=0

oc ,~k t k

-~ Z

k---v-f(x - ks)

k=0

oc ,~k t k

k! f(x - (k + 1)~) + ~ Z

k=0

k=0

h

+~

k! f(x - k~) J

k! I ( ~ - k~) k=0

+~-~*Z k=O

(t + h) k - t k h

] Ak ktk-1 ~. f (x -- kS).

l

Solutions

326 We then have lim

l [ u ( t + h) - u(t)] - A t ( t )

h$0

h

- O. oc

As IIG(t)f[[cc - I l f l [ ~ for each f C X, it follows that {G(t) ; t C ~} is a group of isometries, and this achieves the proof.

Problem 2.5. Since { e - t A ; t C IR} is a uniformly continuous group on I~n, it follows that {G(t) ; t c R} is a group of linear bounded operators on L p (IRn). As for each f C L p (R n), {f } is p-equiintegrable, it follows that the group is of class Co. Finally, let us observe that its infinitesimal generator is defined by D ( A ) - { f C LP(Rn) ; x ~ (Ax, V f ( x ) ) belongs to LP(IR n} and

[A f](x) - - ( A x , V f ( x ) ) for each f C D(A) , and a.e 9 for x E R n" If, in addition, ~-2'n~i=1aii -- 0, from Liouville theorem (see Corduneanu [39], Theorem 4.4, p. 63) it follows that Ile-tAIIL(Lp(R~)) -- 1 for each t C I~, while from the formula of change of variable in multiple integrals, we have IIG(t)fl]L~(R~) - - [ I f l l L , ( ~ ) , which shows that G(t) is an isometry.

Problem 2.6. A simple computation shows that {G(t) ; t c JR} is a group on X. This is not a C0-group simply because limt40 lie-tAx - x[[ - 0 is not uniform for x c R n. Problem 2.7. One simply has to repeat step by step all the arguments in the proof of Theorem 2.3.1, with the remark that, in this case, the pointwise boundedness of the set { S ( t ~ ) x ; n c N*} follows from the fact that limn__~ S(tn)X - x in the weak topology on X. Problem 2.8. Let Y - {x C X ; limt40 IIS(t)x - xll - 0}. We observe that Y is nonempty (0 C Y) and closed in X. To complete the proof, it suffices to show that Y is dense in X. To this aim, let us observe that, for each x C X, the function t ~ S ( t ) x is weakly continuous from the right on R+. So, according to Problem 1.1, this function is strongly measurable on IR+. From Problem 2.7, it follows that there exists M _> 1~ and w C IR, such that IIS(t)ll~(x) <_ M e t~ for each t >_ 0. Consequently S(. ) is locally integrable on I~+. Let now x C X, and let ~" R --+ IR+ be a function of class C ~ for which there exists [a, b] C (0, +co) such that ~(t) - 0 for each t ~ [a, b]. Let us define x(~) -

~(t)S(t)xdt

Chapter 3

327

and let us observe that, by Lebesgue dominated convergence theorem, we deduce lim[S(h)x(~) - x(~)] h$0

~(t)S(t + h)x dt -

= lim h$O

~ ( t ) S ( t ) x dt

/o

lim [~(~- - h ) - ~ ( ~ - ) ] S ( ~ - ) x d~- - 0 hi0 strongly in X, because ~ is continuous, and S(- ) is measurable and bounded on the support of ~. Therefore x(~) C Y. To complete the proof, it suffices to show that {x(~) ; x c X, ~ c 9 +oc))} is dense in X. To this aim, let ~ C 9 +co)) be positive, with f ~ ~ ( t ) d t - 1, and s u p p ~ C [a,b], let e > O, and let us define ~ : R -+ (0, +ec) by -

C

s

Clearly ~6 is of class C 0r ~e(t) - 0 for t ~ [ca, eb ], and f + ~ ~ ( t ) d t - 1. Let us observe that, for each x* C X*, we have lim I ( z ( ~ ) - x x*)[ e$0 < lim

~ ( t ) ( S ( t ) x - x, x*) dt I(S(t)x-x,x*)l-

sup

0.

E$O tC[ Ea,~b]

Hence the weak closure of Y coincides with X. As Y is a convex set, in view of Mazur's Theorem A.1.4, it follows that the strong closure of Y coincides with X, and this achieves the proof. Chapter

3

Problem 3.1 The idea is to show that A satisfies (i) and (ii) in Hille-Yosida Theorem 3.1.1. Clearly D(A) is dense in 12 because the set of all sequences whose terms are 0 except for as finite number of indices is contained in D(A) and dense in 12. Next, let ~ > 0, let y = (Yn)ncN* and let us consider the equation ( ~ I - A)x = y. One may easily verify that this equation has a unique solution x which is given by 1 xn ~ + an ---

~

Yn

for each n C N*. So, Ilxlll2 <_ ~llYlll2 and therefore M - A is invertible, with continuous inverse and I 1 ( ~ I - A)-lll < ~. Consequently A satisfies (ii)in Theorem 3.1.1 and therefore A is closed. Summing up, we conclude that A

Solutions

328

satisfies (i) and (ii) in Theorem 3.1.1, and thus it generates a C0-semigroup of contractions.

Problem 3.2 The proof follows the same lines as those in the preceding proof, with the sole exception that instead of the norm of 12 we have to use the norm of co. Problem 3.3 Since C~(0, ~) is included in D(A) and is dense in L2(0, 7~), we conclude that A is densely defined. Let ~ > 0, f E L2(0, 7r) and let us consider the equation ( M - A)u = f which rewrites equivalently as A u - u" = f ~(0) = ~'(~)

in (0, Tr)

(*)

- 0.

From the variation of constants formula we conclude that this problem has a unique solution u in L2(0, 7r), and so A I - A is invertible. Multiplying both sides of the first equation in (,) by u, integrating by parts and taking into account the boundary conditions, we get ~11~112 + I1~'112 -

(f,

~),

where (.,.) and [1" II are the inner product and respectively the norm of L2(0, 7r). Thanks to the Cauchy-Schwarz inequality, this implies that Allull-< IIf]l, or equivalently II(/~I- A)-lll _< -~. Therefore Theorem 3.1.1 applies and accordingly A generates a C0-semigroup of contractions.

Problem 3.4 Since C~(0, 7r) is included in D(A) and is dense in C~(0, ~r), we conclude that A is densely defined. Let /~ > 0, f E C~(0, 7r) and let us consider the equation ()~I- A)u = f. This rewrites as Au+u'-f ~(0) -

in(0, Tr)

~(~).

By the variation of constants formula we have

~(~) - ~ - ~ ( o ) +

/0

~-~(~-~) f (y) dr.

Imposing the periodicity condition u(0) - u(~r), we get u(O) - 1 - 1e -~Tr f0 ~ e _a(~_y) f(y)dy and so ( M - A)u = f has a unique solution. invertible. A simple calculation leads to

Consequently ~ I -

1 [u(x)[ <_ ~11 e _~Xe_~ + e_~ ( 1 - e_~X)l [Jill-< ~[[fl[,

A is

Chapter 3

329

where I1" II stands for the norm of Cr(0, ~). Hence I I ( A I - A ) - l l l <_ ~. From Theorem 3.1.1, we get that A generates a C0-semigroup of contractions.

Problem 3.5 In view of (ii) in Lumer-Phillips Theorem 3.4.2, it suffices to show that I - A is surjective. To this aim let us observe that R ( I - A) is closed because A is closed and dissipative. Indeed, let (Yn)nCN* be in R(A) with l i m n _ ~ Yn = Y. By Theorem 3.4.1 we have ]lYn -Ymll < - I l x n - Xmi] for each n , m C H*, where Xn C D(A) is such that Yn = X n - Axn, for n E l~l*. So there exists limn__~ Xn = x, x C D(A) and x - Ax = y. Next, if we assume that R ( I - A) ~ X, there exists x* C X*, x* % 0 and such that ( x * , x - Ax) = 0 for each x C D(A). This implies x * - A ' x * = 0 and since A* is also dissipative, we must have x* = 0. This contradiction can be eliminated only if R ( I - A) = X. So A satisfies the hypotheses of Lumer-Phillips Theorem 3.4.2, and accordingly A generates a C0-semigroup of contractions on X. Problem 3.6 The first part, (i) has been already proved in Theorem 3.4.2. To check (ii), let x C D(A) and y - Ax. Let ((Xn, Yn))nCN* be a sequence in graph (A) such that limn--~Xn = X and l i m n _ ~ Yn = Y. By virtue of Theorem 3.4.1 we have IIAXn- Axni] >_ AIIXnll for each n C H* and > 0, and thus [ l A x - Axll < Ailxll for each A > 0 and x C D(A). By Theorem 3.4.1, it follows that A is dissipative and this proves (ii). Finally, let us assume that although D(A) is dense in X, A is not closable. Thus, there exists (Xn)ncN* with limn--~Xn = 0 and limn-~ccAxn = y, with ]]Y[I = 1. Using once again Theorem 3.4.1, we conclude that, for each t > 0, each n C H* and x C D(A) we have ( 1 )

( 1 )

1

Letting successively n -+ ~ and t --+ 0, we get [ I x - y[[ >_ [[xi[ for each x C D(A). Since IlYll = 1 this shows that y cannot be approximated with elements in D(A) thereby contradicting the density of D(A). This contradiction can be eliminated only if A is closable, and this proves (iii).

Problem 3.7 Since D(A) is a subspace in X, it remains to show that if x* C X* is such that (x*, x) = 0 for every x E D(A), then x* = 0. Recalling that I - A is surjective, to prove that x* - 0, it suffices to show that ( x * , x - Ax) = 0 for every x C D(A) which, in view of the supposition that (x*,x) = 0, is equivalent to (x*,Ax) = 0 for every x C D(A). Let x C D(A). By virtue of (i) in Problem 3.6, it follows that, for each n in H* there exists Xn E D(A) such that x = X n - (1/n)Axn. On the other hand Axn - n(Xn ~- x) C D(A) and therefore Xn C D(A 2) and

330

Solutions

or A x n - ( I - ( 1 / n ) A ) - l A x n 9 From Theorem 3.4.1 we deduce that [ [ ( I - ( 1 / n ) A ) - l [ I < 1 and consequently [[Axn[] ~ [IAxI] for every n e N*. Hence [[Xn - x[[ < (1/n) iiAxn[[ < [[Ax[[ and accordingly limn~cr Xn = x. Since X is reflexive and (Axn)nEN* is bounded, we may assume without loss of generality that there exists limu~cr A x n = y weakly in X. On the other hand, by Theorem 3.4.2 A generates a C0-semigroup of contractions and thus, by Theorem 3.1.1, it is closed. So x C D ( A ) and A x = y, because the strong x strong closure of graph (A) coincides with its weak x weak closure and thus with its strongx weak closure. See Mazur Theorem A.1.4 in Appendix. Inasmuch as (x*, z) = 0 for every z e D(A), we have (x*, AXn) = n(x*, X n - X) -- O. Letting n --+ oc we get (x*, A x ) = 0 for every x e D ( A ) . Thus x* = 0 and therefore D ( A ) is dense in X. Ax - Axn-(1/n)A2xn,

Problem 3.8 We begin by observing that, for each ~ > 0 and f E C([ 0, 1 ]), the equation ) ~ u - A u = f has a unique solution given by -

~

X

dy

(,)

and thus ) ~ I - A is surjective. Moreover, from (,) we deduce that ~lu(x)l _< ( 1 - e - ~ )

Iifl]-< IIAu-AufI

and therefore Hence A is dissipative. Nevertheless, D ( A ) = {u C C([0, 1]); u(0) = 0} which does not coincide with C([ 0, 1 ]).

Chapter 4 Problem 4.1 We shall use Lumer-Phillips Theorem 3.4.2. Since C2([0, 1 ]) is included in D ( A ) and is dense in X, A is densely defined. Moreover, let us observe that, for each )~ > 0, we have l u ( 0 ) - 0 ( 1 - O)u"(O)l >_ [u(O)l where 0 C [0, 1] is such that [u(0)l - Ilu[Ic([0,1]). Therefore,

for all u E D ( A ) and )~ > 0, and thus A is dissipative. We shall show next that I - A is surjective. To this aim, let us remark that Ul(S) - s and u2(s) = 1 - s belong to D ( A ) and, for every A > 0, we have (M-

A)ui = s

(*)

for i - 1, 2. Thus, it suffices to prove only that ) ~ I - A0 is surjective, where A0 is the part of A in the closed subspace X0 = {f E X ; f(0) = f(1) = 0}

Chapter 4

331

with D(Ao) - { f e Xo N C2(0, 1); lims-+0,1 s(1 - s ) f " ( s ) - 0}. At this point let us observe that Ao is in fact injective and its inverse is given by

[Aol f](s) -

/o 1G(s

,

t) t ( fl (t _ t)) dt,

where the kernel G is defined by s(t-1) t(s-1)

G(s,t) -

if0<s
So 0 C p(Ao) and since A0 is still dissipative, one has [0, +cr C_ p(Ao). By virtue of (,), it then follows that (0, +cr C_ p(A) and thus A satisfies the hypotheses of Lumer-Phillips Theorem 3.4.2.

Problem 4.2. The Problem is an "L2(0, 7r)" variant of Problem 2.1. From this reason, we shall use very similar arguments. More precisely, let us observe that {S(t) ; t > 0} is a semigroup of linear bounded operators. In addition

IIS(t) llL=(0, ) --- II

llL=(0, ),

for each t > 0, and each ~ C L2(0, ~), which shows that the semigroup contains only nonexpansive operators. We recall that, for each ~ E L 2 (0, ~), (X)

2 k=l

and let us observe that, for each t > 0, each ~ C L2(0, ~), and each m C N* we have L2(0,~) = E

e-

- 1

a2k(~)

k=l cO

(e -k2t - 1) 2 2

+ k=m+l

(e - k 2 t - 1)2

< --

CO

E

+

kC{1,2,...,m}

k=m+l

Let r > 0, let us fix m -

mz > 0, such that c~ E a~(~)< k=m+ 1

C2 23 ,

and let us choose (~- (~(c) > 0, such that max kE{1,2,...,m}

e-k2t- 1

C2

< 2

Solutions

332

From the last three inequalities, we deduce that the semigroup is of class Co. Next, we shall prove that

and

[ A ( ] ( x )= -

C k 2 a r ( ( )sinkx.

(**)

k=l Indeed, let q E L2(0,x ) be such that

This implies that

for n E N*, and Cp=l k4ag(() < +m. Hence (*) and (**) hold. Let ( E L2(0,x ) with CpI1k4a2(<)< +co,and let q = k2ak(()sin k x . Obviously, we have

EEl

2 L2(0,.rr)

Since (1 - e-at) / t

I a for each t 2 0 and each a 2 0 , we have

Consequently

max

(1

(epzzt - 1)

12}

+ i2 + 22

03

C

k=m+l

aia:(<).

Chapter 4

333

Let e > 0. As ~--~k=l ~ k 4 a~(~) < +co, there exists m c~

m6 > 0 with

c2 k4a~(~) < ~ .

k=m+l

For m fixed as before, there exists 5 = 5(c) > 0, such t h a t -

1 {1(_~2t)i212} max e - 1 + t iC{1,2,...,m}

c2 < -- 2 I[~112 L2(0,Tr)

for each t C (0, 5). From the last three inequalities, we get ~ C D(A). Finally, we have rl C D(A) if and only if ~ C H2(0, 7r)M H i ( 0 , 7r), and A~ = ~". In addition, if ~ E L2(0, 7r), we have S(t)~ c D(A) for each t > 0, and consequently (iii) holds.

Problem 4.3. Let S be the vector space of all functions f : R --+ C of class C ~ with the property that, for each m,k C N, the function x ~ xmf (k) is bounded on I~. Since C ~ ( ~ ) is included in S, and is dense in LP(I~), it follows t h a t S is dense in LB(]~). Let us denote by

and let us observe t h a t [S(t)f](x) = G t ( x ) , f(x). From Brezis [29], Th6or~me IV.15, p. 66, we know that, for each p C [1, +c~), and each f E LB(~), we have [[S(t)fIIL,(R) < which shows that

[IGtllLl(~)llfllL~(R),

S(t) maps Lp(]~) into itself and IIS(tDIlc(Lp(R)) <_ 1.

Since the convolution is associative, we have

S(s)S(t)f = as 9 (Gt * f) = (Gs 9 at) 9f for each f E ip(~). Let us we denote by exp {a} = e a, and let us observe that

S

1

+~ ~ exp

(Gs * Gt)(x) - 4 ~ v ~ 1 - 4~/~

1/_

= 4~V~

~

~

exp

{ exp

-

{

-

(x-y)2 4s

)

exp

-

4-t

{(tx-y)2+sy2}dy 4st (s + t)y2 - 2txy + tx2 } dy 4st

dy

Solutions

334

tx) 2 47rv~

or exp

+

-

s+t

4s

(s+t)2

1 /_+~ { (s+t)2y2}exp{ 47r~ ~ exp 4s

Y

x 2 dy

-

1 f_+~ exp { x 2 } f 27rx/s + t ~ 4(s + t)

1 =

27rv/s + t

exp

{ ~ / -

where

4(s + t)

s+t

(s+t) 4st

(~+st t)2 x2 J dy

~ exp { z 2 ~ - } dz

~ -

Gs+t(x)

Is4t

z4st y" Hence S ( s ) S ( t ) f - S(s + t)f for each f C S, and each t,s >_ O. So {S(t); t >_ 0} is a semigroup of linear bounded operators on LP(R). In addition,

][S(t)f](x)- f(x)]'dx p

1

oc exp

-

4t

[f (y) - f (x)] dy

f-o~ e x p { - 7 2 } I f ( y ) - f ( y + 2x/~7)] dy

) 1/p dx

ip )lip d7

As lim

F

t$O oc

If (y) - f (y + 2x/tT)l dy -- 0

a.e. for ~- E R, in view of Vitali's Theorem A.6.1, it follows that lim

t$O

(f+oc c~

I[S(t)f](x) - f (x)lPdx )l/p - O.

Hence the semigroup is of class Co, and the proof is complete.

Problem 4.4. The fact that u satisfies the partial differential equation in question follows from the differentiation rule of a convolution product. The initial condition u(0, x) = f(x) reflects in fact the following continuity property of the solution u lim lit(t, .) - flILp(R) -- O.

t$o

Chapter 4

335

Problem 4.5. Let us denote by

P~(x) =

t

7r(t 2 + x 2)

a n d let us observe t h a t [S(t)f](x) = P t ( x ) * f(x), w h e r e t h e c o n v o l u t i o n is considered with respect to the variable x. Moreover, from Brezis [29], Th6or~me IV.15, p. 66, we know that for each p C [1, +co), and each f C LP(]R), we have

IIS(t)flIL~(R) <_ [[PtllLl(~)llfllL~(~), w h i c h shows t h a t S(t) maps LP(I~) into itself and IIS(t)IIz(L~(R~)) <_ 1.

Using the Residues theorem to compute the Fourier transform Pt of Pt with respect to the variable x, we obtain Pt(x) -- e -tlxl.

Consequently S(t) f (x) - e - t x f (x),

which shows that {S(t); t > 0} satisfies S ( t + s) t, s >_ O. Finally, let us observe that

S(t)S(s)

for each

IIS(t)f - flIL~(R)

oc t2 + (x - y)2 [f(Y) - f(x)] dy

71"

l(f

71

Ip )lip dx

1/p oc

cr t2 + 02 If (y) - f (y + 0)]dy 1

1 + T2 If(y) - f ( y + t~-)] dy

IpdO)

Pd)

1/p

As lim t$o f + ~

1 +1 ~.2 If(Y) - f (Y + tT)l dY -- 0

a.e. for ~- C It(, in view of Vitali's Theorem A.6.1, it follows that lim I I S ( t ) f - flILp(R) -- 0 t$o

336

Solutions

for each f E LP(]R), which shows that the semigroup is of class Co. Problem 4.6. By a simple differentiation under the integral sign and by virtue of Fubini theorem, it follows that u is harmonic on the half-plane {(t, x ) ; t > 0, x C R}. The condition u(0, x) - f(x) signifies the fact that

lim lit(t, .) - fl[Lp(~) - 0. t$0 Problem 4.7. Let

xt-1

r(t)

Rt(x) -

0

for t > 0 and x C [0,1] for t > 0 and x e R \ [0,~[].

We observe that, for each t > 0 and each f C LP(O, 1), we have [S(t)f](x) - Rt(x) , f ( x ) ,

where the convolution is considered with respect to the variable x. On the other hand, from Brezis [29], Th~or~me IV.15, p. 66, it follows I]S(t)fllL~(O,~)

= F(t)l ( s 1 6 3

ip dx ) l / p _< F(t 1+

[[fl[Lp(0,1), 1) which shows that {S(t) ; t >_ 0} contains only nonexpansive operators. In addition 1 l f0 u (Zt -- V) t-l f (V) dv ) du [s(~)s(t)I](x)- r(~) ~ x (X -- tt) s-1 ( r(t) z (x -- y ) t - l f ( y ) d y

1 /0x f (~) (/v x(~ _ ~ ) , - 1 (~ _ v ) , - l d ~ )

= r(~)r(t)

, /o9(~ -

= r(~)r(t)

v)S+t-lf(v )

1 r(~)r(t) r(~)r(t) r(~ + t)

(/01 o s - l ( 1

(~_v)~+~_~

where ~

-

o)t-ldo

d~

)

dv

f(,) d~,

Xw~ X~V

~

o

Hence {S(t) ; t >_ 0} is a semigroup of linear bounded operators. Finally, again from in Brezis [29], Th6or6me IV.15, p. 66, we have 1 ] [ I S ( t ) / - fllLp(O,1) <- r ( t + 1) - 1 Ilfl[Lp(0,1),

Chapter 5

337

which shows that {S(t) ; t _> 0} is of class Co.

Problem 4.8. We remark that in both cases, the infinitesimal generator is self-adjoint. In the case of Problem 4.3 this follows by observing that the infinitesimal generator is symmetric on C ~ which is dense in L2(IR), and using Remark 6.4.1. Therefore, the duals coincide with the initial semigroups. Problem 4.9. A simple calculation shows that {a(t) ; t >__0} is a C0-group of unitary operators whose infinitesimal generator is defined by D(A ) - H 1 (]~) • H 1 (R) and A (u' v ) - ( OvOx' oxOU) for each(u, v) e D(A).

Problem 4.10. We observe that x : ( - ~ , T] --+ X is an almost everywhere solution of the integro-differential equation with delay if and only if the function u" [0, T ] - + L I ( - c ~ , 0 ; X ) , defined by u ( t ) ( O ) - x(t + 0 ) for each t E [0, T], and a.e. for 0 C (-c~, 0], is a strong solution of the problem u' = A u , u(O) = ~.

Chapter 5 Problem 5.1. Let (fn)ncN* be a sequence of Cantor functions from [0, ~] to [0, ~ ] with limn~cr fn (x) - x for each x e [0, ~ ]. As fn'(X) - 0 a.e. for x e [0, ~ ], it follows that, for each n e N*, fn e D(A), and A f n - O . Let us denote by f (x) - x, g(x) - 1, and h(x) - 0 for each x e [0, ~ ], and let us observe that both (f, g) and (f, h) are in the closure of the graph of A. Therefore this closure cannot be the graph of a single-valued function, and consequently A is not closable. Problem 5.2. (i) The operator is closable. The domain of the closure is D(A) - {(Xn)ncN* e 12; (2nxn)nCN * e /2}. (ii) The operator is closable. The domain of the closure is D ( A ) - l~ n 12. (iii) The operator is closable. The domain of the closure is D ( A ) - 12. (iv) The operator is not closable, because there exists at least one sequence (Un)nCN. convergent la 0 in L2(0, 1), and such that Un(1/2) - 1 for each hEN*. (v) The operator is closable. The domain of the closure is D(fi~) - L2(0, 1).

Problem 5.3. From Lemma 3.7.3 we know that A** is densely defined and closed. The conclusion follows from the fact that D(A) C_ D(A**), and A u - A**u for each u E D(A).

Solutions

338

Problem 5.4 Thanks to (3.2.3), (An)ncN* is pointwise convergent to A on D(A). Taking Y0 = D(A), we are in the hypotheses of Theorem 5.1.2 and the conclusion follows.

Problem 5.5 From the semigroup property it follows that An commutes with S(t) and thus (i) holds. Furthermore, since An e L(X) and --

~_ e - n t e ntllS(1/n)llz~(z) -- 1

it generates a uniformly continuous semigroup of contractions. Hence (ii) holds too. From the definition of the infinitesimal generator we get (iii). So, we are in the hypotheses of Theorem 5.1.2, and accordingly e~ tk

f(t + s) - li+m (etA~f) (s) -- li+m E -~.(Akn)(f)(s) k=O

for each f E D(A) and uniformly for s E IR and t C [0, 1 ]. Take s = 0 and let us observe that, for each f C X, there exists a sequence (ran)noN* of natural numbers with limn_+~ m n - oc and such that

m~ tlc f(t) - l i m e

~. (Akn)(f) (0). k=0

Since each term on the right-hand side is a polynomial, this is nothing else than the Weierstrass approximation theorem.

Chapter 6

Problem 6.1. Let us observe that, for each t > 0, each h C R+, and each (Xn)nCm* C lp with [[(Xn)ncN* [llp <- 1, we have (:x:)

IIS(t + h)(xn)nEN*

P

S(t)(Xn)nCN* lip

__

k=l O0

<- E I1 -

e -a2khlP e-pa~tlx k IP"

k-1

From this point we distinguish between two c a s e s : (an)ncN* bounded, or (an)n~N* unbounded. In the first case, by virtue of Problem 2.1, we know that the semigroup is uniformly continuous and thus equicontinuous. So we will consider only the second case. More than this, we may assume with no loss of generality that limn_+~ an -- +oc. Otherwise, we can split the sequence (an)nON* into two complementary subsequences, one bounded

Chapter 6

339

and the other one tending to +oc, a n d we reduce the problem to these two cases. Thus, let us assume t h a t limn__~ an - +co. We t h e n have

h)(xn)nEN* - S(t)(Xn)~EN* I[p <_ E I1 --e-a~hlP IxkiP k=l

IlS(t +

O~

-

max {]l--e-a~h }-t- E e-pa~tlxkl p. kE{1,2,"".,m} k=m+l

Let e > 0. Let us fix r n -

rn(e) E N* such t h a t

oo cp E e-Pa2kt<-- 2 k=m+l and let us choose 5 -

5(e) > 0 such t h a t max

kE{1,2,...,m}

11 -

e-a~h] p < ep -- 2

for each h E R+ with h _< 5. From the last three inequalities, it follows that IIS(t + h ) -

S(t)llc(~0) _< e

for each h E R with h _< 5, which shows t h a t the semigroup is uniformly continuous "from the right". The uniform continuity from the left follows from the simple observation that, for each t > 0 and each h E R with Ihl < t/2, we have IlS(t + h) - s(t)ll~(x) _< IlS(t/2 - h) If an -- n 2, we know form P r o b l e m 2.1 t h a t the semigroup is not uniformly continuous because its infinitesimal generator is not defined on the whole

co(t/2)ll~(x).

space

lp.

Problem 6.2. Let us observe that, for each t > 0, each h E R with Ihl < t/2, and each (Xn)nEN. E Co with II(Xn)nEN. Ilc0 ~ 1, we have

[[S(t + h)(Xn)nEN* -- S(t)(Xn)nEN* Ilcc -

sup

kEN*

]e-a2k(t+h) -- e -a~t ] Ixk

_<sup Ie-a2k(t+h) - e-a2kt I

kEN*

max {

sup

kE{1,2,...,m}

Ie -a 2(t+h) max k < max { kE{1,2,...,m}

, sup

k>m+l

le-a2k(t+h) --e-a~t[}

e -a2t k I , 2 sup

k>m+l

e

_ pa2kt

2 I o

340

Solutions

Let e > 0, let us fix m = re(c) C N*, such that sup

e

_z,a~ t

2 <-

k>m+l

--

C

2

and let us choose 3 = (~(r e (0, t / 2 ) , such that

kE{1,2,...,m} for each h C R with [hi _< 5. From the last three inequalities it follows that

Ils(t + h) - S(t)ll ( 0) _< E for each h C R with Ihl _< ~, which shows that the semigroup is of class Co. If an = n 2, we know from Problem 2.2 that the semigroup is not uniformly continuous because its infinitesimal generator is not defined on the whole space co. P r o b l e m 6.3. Let us observe that if the semigroup is compact, then it cannot be uniformly continuous, because lp is infinite-dimensional. (Otherwise, one should have that S(0) = I is a compact operator). From Problem 2.1, it follows that limn~cc an = oc. Conversely, if limn~oc an = oc, then, according to Problem 6.1 the semigroup is equicontinuous. Moreover, for each )~ > 0, ( A I - A) -1 is a compact operator. Indeed, (Xn)nCN.

-- ()~I --

A) -1

(Yk)kCN*

if and only if 1 Xn = )~ + a2n yn

for each n C N*. On the other hand, if ~ is a bounded subset in lp, from the relation above and from the condition l i m n ~ an = oc, we deduce t h a t ( / ~ I - A ) - I ~ is bounded and uniformly p-summable in Ip. In view of Problem A.2, it follows that ( A I - A) -1 is a compact operator, and from Theorem 6.2.1 we deduce that the semigroup generated by A is compact. P r o b l e m 6.4. Let us observe that if the semigroup is compact then it cannot be uniformly continuous because co is infinite-dimensional. (Otherwise, one should have that S(0) - I is a compact operator). From Problem 2.2 it follows t h a t limn__~ an = ~ . Conversely, if limn__~ an = ~ , then, tanks to Problem 6.2, the semigroup is equicontinuous. In addition, for each A > 0, ( , k I - A) -1 is a compact operator. Indeed, (Xn)nCN*

--

(AI-

A)-l(yk)k~N,

Chapter 6

341

if and only if 1

Xn -- ~ -+-a2n y~ for each n C N*. On the other hand, if t9 is a bounded subset in co, from the relation above and from the condition limn-+cc an = oc, we deduce that ( A I - A ) - I ~ is bounded and uniformly convergent in co. From Problem A.1 it follows that ( A I - A) -1 is a compact operator, and from Theorem 6.2.1, we deduce that the semigroup generated by A is compact.

Problem 6.5. For each t _> 0 we have S(t) - T(t)G(t) - G(t)T(t), where T(t)(Xn)ncN* - (e-k2txkl

kEN*

and

G(t)(Xn)ncN*- (e iek4

t Xk) kEN*

From Problem 6.3, we know that {T(t) ; t _> 0} is a compact semigroup. As {G(t) ; t > 0} comes from a C0-group, we deduce that {S(t) ; t >_ 0} is a C0-compact semigroup. Its infinitesimal generator is defined by

D(A)- {(Xk)kcl~, ~ ((--~2Avick4)xk)kEN.

C/2},

and

(A(Xn)nCH* )kEH* - (( - k 2 + iek4)Xk) kcN. for each

(Xn)ncN* e D(A). On the other hand (AS(t)(Xn)ncN* )kcN. - ((--k2 + iek4)e-k2t+iek4tXk) kcN*

for each (Xn)neN* C D(A). observe that, for t :> 0,

In order to conclude the proof it suffices to

((--k2 + iek4)e--k2t+iek4t~)

~12, kEN*

1 although (n)

noN* E 12. Hence the semigroup is not differentiable.

Problem 6.6. Let us observe that each Hilbert-Schmidt operator is the uniform limit on the unit ball of operators with finite-dimensional range, and thus compact. Indeed, if T is a Hilbert-Schmidt operator, then lim TnX - Tx n--~ oo uniformly for x C B(0, 1), where, for each n C N*,

Tn is defined by

Solutions

342 for each x C H, x -

E ~ - I (x, ek)ek.

Problem 6.7. Let {en ; n C N*} be the canonic orthonormal system in 12, i.e. the system defined by en - (e~)kcI~*, where n [ 0 fornCk ek-- ~ 1 forn-k. Then, for each t > 0, we have OO

OO

2 n=l

< +oc, n=l

which proves that S(t) is a Hilbert-Schmidt operator. proof with the help of Problem 6.6.

We complete the

Problem 6.8. Let g - {ek; k C N*}, where for each k C N*, sinkx a.e. for x C (0, 7r). One knows that g is a complete orthonormal system in L2(0, 7r). Let us observe that, for each t > 0, we have C~

OO

IIS(t) sinnxll L2(0, 2 7r) -- E n=l

e-n2t <

+oc,

n=l

which shows that S(t) is a Hilbert-Schmidt operator. proof with the help of Problem 6.6.

We complete the

Problem 6.9. If a C0-group is uniformly continuous then, its infinitesimal generator is in L ( X ) , and therefore it is uniformly differentiable, and hence differentiable. See Theorem 2.2.1. Conversely, if a C0-group {G(t) ; t E R} is differentiable, then it is differentiable at t = 0 because, for each x C X, we have lim h,o -s1 ( a ( h ) x

- x) -

a-l(t)1~~ ~1 ( a ( t

+ h)x - a(t)x)

,

where t > 0 is arbitrary. Therefore D(A) - X , and, according to the closed graph theorem, A is continuous. From Theorem 2.2.1 it follows that the group is uniformly continuous, and hence uniformly differentiable.

Chapter 7 Problem 7.1. If a group has an analytic extension to a sector in the complex plane then it is uniformly differentiable and, according to Problem 6.9, it is

Chapter 7

343

uniformly continuous. In view of Theorem 2.2.1, its infinitesimal generator is bounded. Problem 7.2. The eigenvalues of A are Ak = k 2 with k E N* and the corresponding set of eigenfunctions, which is a complete orthonormal basis in L2(0, 7r), is

So, according to Example 7.6.1, we have D ( A ~) -

u E L 2(0, 7r) ; u(x) - Z

k2auk s i n k x

,

k-1

where uk -

u(y) sin ky dy

for k E N*. Therefore, if c~ E [89 1], D ( A ) - H2c~(O, 7r) A H~(O, 7r).

Problem

7.3. Let f C C~(0, +oc). We have

(A_C~f)(s) _

_-

1

r(~)

t~-~[S(t)f](s)dt

1

r(~)

1 t a - l f ( t + s ) d t - F(a)

(u - s) a-1 f ( u ) du - ( K ~ f ) ( s ) .

We notice that, for f C C~(0, +co), the integral from 0 to + ~ , which defines A -a, is convergent for each c~ > 0. Finally, let us observe that, for each f, g E C~(O, +cx:)), and each o~ > O, we have

/o

/o

+~ f ( s )

f ( s ) ( K a g ) ( s ) ds -

( l f ro( ~ 0) - f o ~g(~)

~

(

1 F(a)

(u-- 8)~-1f(8)d8

(u - 8)c~-lg(~t)dzt

)/o du -

)

d8

( J a f ) ( u ) g ( u ) du.

Problem 7.4 Let us recall that D ( A ) C_ D(A1/2). See (ii) in Theorem 7.6.1. Let f E D(A), and let {$1/2(t); t>__ 0} be the semigroup generated by A~/2. Let us observe that the function u(t) - S1/2(t)f satisfies

{

f = -Af ~(o)

-

.

Solutions

344

Recalling that A is the Laplace operator on LP(I~) (see Problem 4.4), we deduce that u is the solution of the elliptic equation U t t -~- U x x - - 0

u(0, x) - f(x). An appeal to Problem 4.6 shows that {S1/2(t); t > 0} coincides with the Poisson semigroup. Finally, the analytic expression of A1/2f follows directly from Definition 2.1.2 of the infinitesimal generator.

Chapter 8

Problem 8.1. We begin by observing that D(A) is dense in ~ because D(A) is dense in X, and WI,~(0, + c e ; X ) is dense in L~(0, + c e ; X ) . Moreover, the operator A is closed because both A, and the differentiation operator are closed. Let ~ > 0. It is easy to see that ()~J - A)(~, f) = (~, g) if and only if

//0/+

(,)

The second equation has the general solution given by /(t) - I(0)~ ~ -

/0

~ ( ~ - ~ ) g ( ~ ) a~,

where f(0) E X. In order that f belong to LI(0, +oc; X), we must have lim inf IIf (t)II = O. t--++cx~

Let us observe that this condition is satisfied if and only if

I(0)

-

/0

~-~g(~)

a~.

As A is the infinitesimal generator of a C0-semigroup of contractions, from Theorem 3.1.1, it follows that, for f(0) as above, the first equation in (,) has a unique solution ~. In view of Theorem 3.1.1, to conclude the proof, it suffices to show that, for each (r/, g) C ~, we have

1 11(~, f)llx ~_ ~11(~,g)llx.

(**)

To this aim, let us observe that

IlfllLl(o,+~;x) <

e ~t

e-x~llg(s)[ [ds

dt

Chapter 8

- ~1 e)~t

345

e_;~s lig(~) ii d~i 7 + ~1

=

1

1

f0

IIg(t) ii dt

~llf(0)ll +-fiigIIL~(O,+~;x).

On the other hand, again from Theorem 3.1.1, we deduce that 1

1

II~ll _< xIIf(0)ll + ~11~11. From the last two inequalities, we deduce 1

I1~11+ IIIIIL~(O,+~;x) <-- -f (ll~ll + IlgllLl(O,+~;x)) , inequality which is equivalent to (**).

Problem 8.2. Theorem 4.4.1 implies that A ' D ( A ) defined by

C_ LP(]R3) ~ LP(]~3)~

D(A) - {u C X; a. y u E X } Ou for each u c D ( A ) , Au--a. Vu=-~ai~ i=1

generates a C0-group of isometries {G(t) ; t E R}, [G(t)f](x) - f ( x - ta) for each f E Lp(R3), each t C R, and a.e. x C R a. Therefore, the unique C~ of the problem considered is given by

Problem 8.3. In view of Problem 2.5, the unique C~ problem considered is given by t

of the Cauchy

(t-s)A x

fo

Problem 8.4. We have ()~I- A ) - l f - u if and only if u, u' e L~(I~), and A u - u ' = f. This means that

~ ( t / - ~(0/~ ~ -

~(~-~)/(~/d~.

From the periodicity condition, u ( 0 ) = u(27r), it follows

e2~a 1 fo 2~ e-a~ f (s ) ds. u(O) -- e2~)~_ Hence, if 9" is bounded in L~(I~), the set { [ ( A I - A ) - I / ] (0); f e 9"}

Solutions

346

is bounded in IR. Consequently ( A I - A ) - 1 ( 9 ") is bounded in W1'2(0, 2~). From Theorem 1.5.4 it follows t h a t ( A / - A ) - ~ ( 9 ") is relatively compact in L22~(]~). Let us observe that, for each n E N*, Qfn - t. sin(n(t + x)), where fn - sin(n(t + x)). On the other hand, one can easily see that the family {t. sin(n(t + x)); n e N*} is not relatively compact in C([0, 1 ]; L~(I~)), because it is not equicontinuous on [0, 1] except for t - 0.

Problem 8.5 First, let us observe that, in view of the resolvent equation (1.7.3), R(A; A) - ( I I - A ) we have

-1 is compact for all 1 > 0. On the other hand,

~ /o ~ +

~-~(s(~)~(t) - ~(t)) d~

d~ + A/0 ~

§ ~)

**(t)II d~.

But IIS(7)u(t) - u ( t + ",-)ll < ft+-,-

(o)II dO

dt

and lim Ilu(t + w) - u(t)ll - 0 7-40 uniformly for t C IR+. So, we have lim AR(A; A)u(t) - u(t) A---+cxD

uniformly for t C R+. The conclusion follows from Corollary A.1.2. Chapter

9

Problem 9.1 Let 7/C (Cub(R))* and let us observe that (f (s), [S(t)*~](s)) - ( [ S ( t ) f ] ( s ) , r/(s)) - / R

f(s-t)d~(s)

#

- ]~ f(~)dv(~ + t) - (f(~), v(~ + t)). Thus, [S(t)*~](s) - ~(s + t), i.e. the translation of the Radon measure ~7Consequently, X ~ coincides with the space of all Radon measures on R whose right translations are continuous with respect to the variation, i.e. the space of all ~e for which limt40 Var ((~e (. + t) - ~Te( 9)), ]~) - 0. We note that the Lebesgue measure is in X ~ but the Dirac delta is not.

Problem 9.2 Let A 9a - to < t l . - " < tn - t and let ~i E [ti-l,ti] for i - 1, 2 , . . . n. If either a <_ t < 0, or 0 < a < t, the Riemann-Stieltjes sum

Chapter 9

347

over [a, t] of s ~ S ( t - s) with respect to g is 0. So, it suffices to analyze the remaining cases. If a < 0 = t, we have Cr[a,t](/k, S , g, 7i) -- S ( t -- T n ) ( g ( t n )

-- g ( t n - 1 ) ) ,

simply because, on any other interval of A, g is constant. Therefore

ff[a,t](A, S , and hence, in this case f ta S ( t 0 = a < t follow similarly.

g, Ti) -- S ( t -- T n ) X

s)dg(s) - x. The cases a < 0 < t and

Problem 9.3 Let A : a = to < t l ' " < tn = t and let ~-i E [ t i - l , t i ] for - ~-~.inl(h(ti)- h ( t i - 1 ) ) S ( t - Ti)x. i - 1 , 2 , . . . n . We have C~[a,t](/k,S,g,~-i) But the later sum is the Riemann-Stieltjes sum over [ a , t ] of the vector valued continuous function s ~ S ( t - s)x with respect to g. Since, by classical arguments, it follows t h a t s ~ S ( t - s)x is Riemann-Stieltjes integrable on [a, t] with respect to g, we easily conclude t h a t there exists fa S(t

--

--

S(t

--

Problem 9.4 The conclusion follows from Problem 9.3 by a simple induction argument.

Problem 9.5 Let (gn)ncN* be a sequence in BVA([ a, b ] ; X ) such t h a t lim Var(gn - g, [a, b]) = 0. n----~ cx3

Then lim II~[a,t](A, S, gn, Ti) n--+(x)

O[a,t](A, ~, g,

- o

uniformly with respect to A E 9 a, t] and 7i C [ti-1, ti ]. See Section 9.1 for the notations above. But this relation shows t h a t whenever s ~ S ( t - s ) is Riemann-Stieltjes integrable on [a, t] with respect to gn in the norm of X for n E 1~*, then s ~+ S ( t - s) is Riemann-Stieltjes integrable on [a, t] with respect to g too, of course, in the norm of X.

Problem 9.6 Since g e B V ( [ a , b ] ; X ) satisfies Var(g,[c,d]) < g ( d - c ) for each [c, d] C [a, b], it follows t h a t g is continuous on [a, b]. Hence, in view of Theorem 9.2.1, for each ~ E X , the unique Lee-solution u = Q(~, g) is continuous on [a, b]. From now on, we adapt the proof of T h e o r e m 8.4.1 to the case here considered. T h a n k s to Theorem A.2.1, the necessity is obvious. In order to prove the sufficiency, we shall make use again of Theorem A.2.1. To this aim, we shall show t h a t Q( 9 S) is equicontinuous on (a, b]. Let e > 0. Since the function g in the definition of the equiabsolute continuity is continuous at t = 0 and g(0) = 0, for each e > 0

Solutions

348

there exists

Y(E)

> 0 such that

<

e(r) E for each r E [ O , Y ( E ) SO, for each [c, d ] & [a,b ] with d - c V"'(97 [ c , d l ) 5

< Y(E),we have

E

uniformly for g E 9. Let t E (a, b ] and let us fix X = A(&) > 0 such that ) t - X E D. This is always possible because D t - X > a , 2X 5 y ( ~ and is dense in [ a , b]. Inasmuch as Q ( 9 , 9))(t - A) is relatively compact in X, for each E > 0, there exists a finite family {(tl, gl), (t2, g2),. . . (tk, gk(s))) i n 9 x 9 such that, for each (J,g) E 9 x 9, there exists i E {1,2, . . . k ( ~ ) ) with the property

is On the other hand7 the family {Q(tl)gl))Q(t2,g2),...Q(tk(E),~k(E))) equicontinuous at t , being a finite family of continuous functions on [ a , b]. Therefore, for the very same E > 0, there exists 6(&)E (0, A ] , such that for each i = 1,2, . . . k(&),and each h E R with 1 hl that Q(7, g)(t + h) = S(X + h)Q(1),g)(t -

+

/

t+h

t-X

< 6 ( ~ ) Let . us observe S(t+ h

- s ) dg(s),

For the sake of simplicity, let us denote u = Q(<,g ) and ui = Q (<,gi). Since

+

[/C:h

S(t

I

+ h - d g ( ~ )+ [S(h + h)ui(t - A) - ui(t + h)] +[ui(t + h) - ui (t)]+ [ui(t) - ~ ( t ) ] S)

7

we have

< I(u(t- A) - ui(t - X)II + Var (g, [ t - A, t + h ] ) + Var (gi, [ t - A, t + h ] ) + I l ~ i ( t + h ) - ~ i ( tIl+ll~i(t-X)-~(t-X) ) 11 +Var (gi, [t-A, t ])+Var (g, [ t-A, t I)

Chapter 10

349

for each (~,g) C 9 x 9 and each h C (-5(e), 5(e)). From this inequality and from the manner of choice of both ~ > 0 and 5(e) > 0, it follows

IIQ(~, g)(t + h) - Q(~, g)(t)l I < 7e for each (~, g) E 9 x ~ and each h E IR with Ihl _< 5(e). Therefore Q( 9 ~) is equicontinuous on (a, b ], and, by virtue of Theorem A.2.1, it is relatively is compact in C([c,b];X) for each c C (a,b). If a E D, then Q( 9 equicontinuous at a as well, and this achieves the proof.

Problem 9.7 Thanks to Problem 9.6, it suffices to check that, for each is relatively compact in X. Let t E (a, b) and let t C (a, b), Q(9 )~>0witht-A>a. We have Q(~, f)(t) - S ( t - a)~ + S(A) fa t-x s ( t -

+

S(t-

Since S(~) is compact, it follows that the operator P a ' Q ( 9 defined by

PxQ(~,g)(t) - S ( t - a)~ + S()~) fa t-x

s(t

-

-

S)(t) -+ X,

f

maps the set Q( 9 9)(t) into a relatively compact subset in X. In addition, from the preceding relation and from the equi-absolute continuity of 9, it follows that

limllP~Q(~,g)(t ) ~40

- Q(~,g)(t)l

I

<_ lim supVar ;%0

(g, [t - ,k, t]) -

0

uniformly for g E 9. From Lemma A.1.2 it follows that Q( 9 is relatively compact in X for each t C [a,b]. An appeal to Problem 9.5 completes the proof. C h a p t e r 10

Problem 10.1. We have only to repeat the proof of Theorem 10.1.1, noticing that, first of all we have to state and prove an appropriate extension of Lemma A.1.3 from the case f continuous, to the case f integrable. In order to obtain the latter result, one approximates the functions f in Ll(a, b;X) by continuous functions (in fact of class C ~) taking values in the closed convex hull of the range of f (see Theorem 1.3.3), and then one uses Mazur's

Solutions

350

Theorem A.1.3. For passing to the limit in the approximating equations one uses Lebesgue dominated convergence theorem.

Problem 10.2. We have only to repeat the proof Theorem 10.2.1, using the extension of Lemma A.1.3 suggested in the proof of Problem 10.1. We notice that, here, we need the following "local" variant of Lemma 10.2.2: L e m m a Let g: I[x X --+ X be a function satisfying (i), (ii) in Problem 10.1, and (iv) in Problem 10.2. Then, for each a E IT, and each ~ C X , there exists 6 > 0 such that [ a, a + 6 ] C I[, and, for each h in L1 (a, a + 6; X ) , the problem u ' - g ( t , u ) + h(t) u(a) - ~, (*) has a unique solution defined on [a, a + 6 ]. In addition, for each a, ~, and 6 fixed as above, the mapping h ~-~ S(h)-the unique solution of the problem (.) corresponding to h-satisfies ]iS(hi)

-

S(h2)[ic([a,a+6];x) <_ e fa+5 i(t)dtiiH1 - H211C([a,a+~];x),

where ~ is given by (iv), and Hi(t) -

hi(s) ds

for i = 1, 2, and t E [a, a + 6 ]. In order to prove this lemma, let us choose 6 > O, sufficiently small, such that

~a a+5 q-

g(t) d t < 1.

(**)

Using mathematical induction, we show that, for each n E H*, and each u, v C C([ a, a + 5 ]; X), we have (with the same notations as in the proof of Lamina 10.2.2)

[]Qntt -- ~nviic([a,a+6];X) ~ qniiu -- viIc([a,a+6];X).

(* * *)

C h a p t e r 11

Problem 11.1. We have to repeat the proof Theorem 11.2.1, with the special mention that, in the process of passing to the limit one has to use Lebesgue dominated convergence theorem. Problem 11.2. Consider first ft = X. We choose 6 > 0 satisfying (**) in the proof of Problem 10.2, and we define Q : C ( [ a , a + 6 ] ; X ) -4 C ( [ a , a + 6 ] ; X ) by (Qu)(t) - S(t - a)~ +

S(t - s ) f ( s , u ( s ) ) d s

Chapter 12

351

for each u C C([a, a + 5]; X). We then show that this operator satisfies (***) in the proof of Problem 10.2 and therefore, according to a consequence of Banach fixed point theorem, Q has a unique fixed point. For the general case, in which ~ is strictly included in X, we use the method of proof of Theorem 11.2.1. Problem 11.3. We apply Problem 11.2 and then, using Theorem 11.3.2, we show that each local solution can be extended up to one defined on I~+. Problem 11.4. We apply Problem 8.3 combined with Problem 11.2, and with the suggestion made for the preceding proof. Problem 11.5. Let us observe that the problem in question can be rewritten as an abstract ordinary differential equation of the form u'-

Au + f ( t , u ) -

in the space X = Cub(I~ • [--1, 1]), where A : D ( A ) C_ X --+ X is defined by D ( A ) = {u e X ; yUx e X}, Au = -yUx

for each u C X, and f :R+ • X --+ X is defined by f (t, u)(x, y)

-

~ ( x ) / . 1 u(x,z) dz 2 1

for each (t, u) e I~+ • X , and each (x, y) e I~ • [ - 1 , 1]. We can easily see that we are in the hypotheses of Problem 11.2, with g = c.

Chapter 12 0

Problem 12.1 We have to repeat the very same routine as that in the proof Theorem 12.1.1 with the sole special mention that here, the existence of the operator Q is ensured by the definition of B V A ( [ a , b ] ; X ) , and not by Theorem 9.1.1 which might not be applicable in this case. Problem 12.2 We observe that the problem above rewrites equivalently as du - { A u } d t + dg

where X - LI(I~), A " D ( A ) C_ X --+ X is defined by D(A)-{ueX; A u - u'

u'eX} for u C D ( A ) ,

Solutions

352 and g : N -+ X is given by

-O h

g(t) lh

if t < to if t - to ift >0.

Repeating the arguments in Problem 9.1, we easily see that g E BVA (R; X) and thus the conclusion follows from Problem 1.2.1.

Problem 12.3 Let (a,~) E R • X and let c > a. We define the operator Q: L ~ ( a , c ; X ) ~ L ~ ( a , c ; X ) by Q(v)(t) - S(t - a)~ +

/a

S(t - s ) f (s, v(s)) ds +

/a

S(t - s)dg(s)

for each v E L c r and t E [a,b]. Since f ( R > X) is compact, there exists M > 0 such that Iif(t,~)ll -~ M for all (t,r]) E R x X. Let r = M + Var (g, [a, c ]) and l e t

K-{vEL~(a,c;X);

[]v(t)l I _ < r a . e . f o r t E [ a , c ] } .

In fact K is the closed ball of radius r and centered at 0 in L ~ ( a , c ; X ) . We consider K as a subset of L1 and we notice that it is nonempty, closed, convex and bounded. Moreover it is easy to see that Q maps K into itself. So, in order to apply Schauder fixed point theorem A.1.5, we have merely to show that Q is continuous from K to K in the norm topology of LX(a, c ; X ) and Q(K) is relatively compact in the same topology. Since for every v, w E K we have

(a, c; X)

IiQ(v)(t) - Q(w)(t)]] <_

/a

]If(s, v(s)) - f (s, w(s))l] ds

and f is continuous and bounded, by Lebesgue dominated convergence theorem, it follows that Q is continuous from K to K when both domain and range are endowed with the norm topology of L l(a, c ; X ) . To check the compactness of Q(K), let t E [a, b] and let us observe that, inasmuch as (s, ~) ~+ S ( t - s)~ is continuous from [a, t] • f ( R > X) to X and the former is relatively compact, { S ( t - s ) f ( s , v ( s ) ) ; v E K, s E [a,t]} is relatively compact in X. From this remark and Lemma A.1.3, we readily conclude that, for each t E [a, c],

{~at

}

S(t - s)f(s, v(s))ds ; v E K is relatively compact in X. Let us denote by G(t) - f ta S ( t - s)dg(s) and let us observe that, by virtue of Theorem 8.4.1, we conclude that the set { Q ( v ) - G; v E K} is relatively compact in C ( [ a , c ] ; X ) . On the other

Chapter 12

353

hand Q(K) = {Q(v) - G;v e K} + G, and thus it is relatively compact in L 1(a, c; X) (in fact, it is relatively compact even in the uniform convergence topology on [a, c]). By Schauder fixed point theorem A.1.5, Q has at least one fixed point in K which obviously is an L~-solution of the problem considered on [a, c l, as claimed.

Problem 12.4 Let us observe first that, if g E BVA(I~; X) and h E C(I~; X), then t ~ ~(t) - f ta h(s) ds + g(t) belongs to BVA (~ ; X) too. From now on, we shall mimic the proof of Theorem 12.2.2. Namely, let (a, ~) C R x X with (g(a + O)- g(a)+ ~) C 9 and let us denote by r / = g(a + O)- g(a)+ ~. Inasmuch as 9 is open and f is b-compact, there exist c > a and r > 0, such that [ a, c ] x B(r/, r) C 9 and K = f ([ a, c ] x B(~, r)) is relatively compact in X. In particular, there exists M > 0 such that Ill (t, )II

M

for each (t, u) e [ a , c] • B(r/, r). Let us define p : X -+ X by y

for y C B(V, r)

p(y) IlY

~lI ( y - 7 7 ) + ~

foryeX\B(~,r).

We may easily see that p maps X to B(~, r) and is continuous Now, let us define the function f~ : R x X --+ X by

f~(t,u) -

I f(a,p(u)) f(t,p(u)) f(c,p(u))

if t E ( - ~ , a ] if t C (a,c) if t C [ c , + ~ )

anduEX and u E X and u C X.

Clearly fr is continuous and f~(R x X) is relatively compact. By virtue of Problem 12.3, it follows that the Catchy problem

du - { A t + f~(t,u)}dt +dg x(a) has at least one L~-solution, u : [a,c] --+ X. Since, u(a) = ~ and, by (9.2.1) in Theorem 9.2.1, u(a + O) = g(a + O) - g ( a ) + u(a) = r/, there exists b e (a,c] such that, for each t e (a,b], u(t) e B ( ~ , r ) . But in this case p(u(t)) = u(t) for each t e (a,b], and consequently fr(S,u(s)) must coincide with f(s,u(s)) for each s e (a,b]. Since u : [a,b] --+ X is piecewise continuous and (t,u(t)) e (a,c] x B(~?,r) C 9 it follows that (t,u(t + 0)) e [a,c] x B(rl, r) C 9 and thus, by Remark 12.1.3, u is an L~-solution of the problem (12.2.2).

Problem 12.5 Set H - H~ (~t) x L 2 (f~)

Solutions

354

which, endowed with the inner product (.,->, defined by

((u, v), ((t, ~)> - ~ u'(x)(t'(x) dx + / v(x)~(x) dx for each (u, v), (~, ~) E H, is a real Hilbert space. We define the operator A : D(A) C H ~ H by . D ( A ) - (H2(~)M Hol (Ft)) • H~(~t) A(u, v) - ( v , A t ) for each (u, v) E D(A). Furthermore, let us define f :I~+ • D(f) ~ H by {(u,v) E H ; b(t, . , u ( ) ) ) E L2(~t) for each t E [ 0 , + ~ ) } f (t, (u, v))(x) = (0, b(t, x, u(x)~j

D(f)-

for each t E [ 0 , + ~ ) , each (u,v) E D(f) and a.e. for x E ~t. At this point, let us observe that the problem considered can be rewritten under the equivalent form

dz - {Az + f(t,z)}dt + dg(t) z(a)

--

where A and f are as above, z(t)(x) = (u(t,x),v(t,x)) a.e. for (t,x) in (0, T) • f~, ~ = (u0, v0) and g : I ~ • H --+ H is defined by

g(t)(x)

-(0, 89 (0 0) (O'89

-

if t < t 0 anda.e, f o r x E f ~ if t - t 0 anda.e, f o r x E if t < t 0 a n d a . e f o r x E f ~ 9

.

In order to prove that A, f and g satisfy the conditions in Problem 12.4, let us remark first that, by virtue of Theorem 4.6.2, A generates a C0group of isometries. As concerns f, from Theorem 1.5.4 and Lemma A.6.1, it follows that D(f) = H and f is continuous on ]~ • H. Finally, again from Theorem 1.5.4, we know that H~(~t) is compactly imbedded in L2(f~) and thus f is compact. Using similar arguments as those in the proof of Problem 9.2, we show that g E B VA(]R;H) and therefore we are in the hypotheses of Problem 12.4.

Problem 12.6 We rewrite the problem in the abstract form du-{Au}dt+dgu -

To this aim, take X = Ll(~t) and A : D(A) C_X ~ X, defined by D(A)-

{u E WI'I(~t); Au E LI(~)}

A u - Au

for u E D(A),

Appendix

355

and let us recall that A generates an analytic and compact C0-semigroup of contractions. See Theorem 7.2.7. We also recall that in this case X ~ = C0(~) and XA = (C0(~))*, i.e. the space of all Radon measures concentrated in f~. See Example 9.5.1. Next, take T > 0 with to E (0, T) and let us define G" Lc~ (0, T ; X) --+ B V ([ 0, T ] ; (C0 (f~))*) by -}5(m-mu)

G(u)(t) -

0 l a ( x - Xu)

fort to.

Since u ~ Xu takes values in a compact subset gtK in ~, there exists r > 0 such that, for each x E ~K, B(x,r) C ~. Next, take e E (0, r) and let us define Ge " L ~ ( O , T ;LI(ft)) -+ BV([O,T] ;LI(ft)) by t < to

-

0

Ge(u)(t) where

~ (x - xu) -

l a e ( x - xu)

{

1

,B(x~,~) 0

for t - to for t > to,

if x E B(xu, e) if x E ~ \ B(x~, e),

p(B(x~,e)) being the Lebesgue measure of B(x~,e). We may easily verify that {G~;e E (0, r)} is of equibounded variation in B V ( [ O , T ] ; L I ( ~ ) ) . Moreover, for each e E (0, r), Ge satisfies the conditions in Theorem 12.2.1 because it is bounded and pointwise continuous in the a ( L ~ ( ~ ) , C 0 ( ~ ) ) topology. The boundedness was already been proved. From Theorem 12.2.1 it then follows that the e-approximating problem obtained from the initial one by substituting 5 ( t - to) x 5 ( x - x~) by 5 ~ ( t - t~) • 5 ~ ( x - Xu), has at least one L~-solution uE. By virtue of Theorem 9.4.2, it follows that {ue ; c E (0, r)} is relatively compact in LI(0, T ; LI(f~)). Moreover, by the Helly-Bray Theorem 1.4.6, we conclude that lim Ge(ue)(t) = g(t) e$0 pointwise in the a((Co(-~))*,Ll(~))-topology. Again by the continuity of the mapping z ~ z~, it follows that g = G(u), and this completes the proof.

Appendix Problem A.1. Necessity. Let 27 be a relatively compact family in co, and let e > 0. In view of Hausdorff's Theorem A.I.1, X is precompact. So, it has a finite e/2-net XE - { ( X i ) n E N * ; i - 1, 2 , . . . , k(e)}. Since X~ is finite, it is

Solutions

356

uniformly convergent, and thus, for the same e > 0, there exists n(e) E N* such that i

C

for each i - 1, 2 , . . . , k(e), and each n C N*, n >_ n(e). In order to conclude the proof of the necessity, let us observe that, for each (xk)kcN* C X, and each n E N*, n >_ n(e), we have i

C

C

Ixn] _< ]Xn - Xnl + IXinl <_ -~ + -~ -- C.

Sufficiency. Let ~ be a bounded and uniformly convergent family, and let c > 0. Let n(c) in N* such that, for each (Xn)neN*, and each n E N*, n > n(e), we have E

Ixnl < - . -2 As {(Xl,X2,... ,xn(s)) ; (Xn)nCr~* e X} is bounded i n R n(E), it is precompact and thus it has a finite e/2-net { (x~, x~,.. .,Xn(~) i ); i - l , 2 , . . . , k ( e ) us observe that the set {(Xin)n~, ; i X, because max Ix~ - x n I -< max

hEN*

< sup --

i

max IXn -- X hI,

n<_n(e)

max n
1, 2 , . . . , k(e)} is a finite e-net for

Ixn --

1 xnl'

sup

n>n(~) i /

sup ]Xnl + sup IXnl

n>n(E)

} . Let

n>n(E)

IXn -- Xnl

<

E

-- -~

+

C -2

J -- e

"

Problem A.2. By analogy with the case X = R, we introduce the notion of uniformly convergent family in co(X). Using similar arguments as those in the proof of Problem A.1 (one replaces I" [by I1" II), we obtain the following compactness criterion: A family ~ in co(X) is relatively compact if and only if it is uniformly convergent and, for each i E N*, the section of X at i, {xi ; ( X n ) n E N * C ~C}, is relatively compact in X . Problem A.3. Necessity. Let X C lp be relatively compact, and let e > 0. In view of Hausdorff's Theorem A.I.1, X is precompact, and thus it has a finite e/2-net 3C~ - {(Xi)n~N, ; i - 1, 2 , . . . , k(e)}. Since %~ is finite, it is uniformly p-summable, and therefore, for the very same e > 0, there exists n(e) C N* such that Ix l

< -2

Appendix

357

for each i - 1, 2 , . . . , k(a), and each n E N*, n > n(a). In order to conclude and the proof of the necessity, let us observe that, for each each n E N*, n > n(c), we have

(Xk)kEN,E~

I~l ~

<_

I ~ - x~l ~

+

Ix~l ~

<_ ~ + ~ -

~.

Sufficiency. Let E be a bounded and uniformly p-summable family, and let c > 0. Let n(c) E N* be such that, for each (Xk)kEN* and each n E N* with n >_ n(e), we have

k=n

< ~_.

Ixkl p

-4

Inasmuch as {(Xl,X2,... ,Xn(4) ; (Xn)nEN* E E} is bounded in R n(~), it is precompact and therefore it has a finite a/2-net (with respect to any one of the equivalent norms on I~n(*)), {(xil,xi2,...

, X ni ( ~ ) ) ;

g --

1,2,...,k(e) }

.

Let us observe that the set { (Xn)nEN, i ; i -- 1, 2 , . . . , k(e)} is a finite e-net for 2:, because

k=l

k=n(e)+l

I~

II/P(~__~ II/Pl ~

k - n (E)

\ k = n (r + 1 C C E

- +

+

~

-2

/ I lip

k---n (r + 1

-- c.

~

Problem A.4. We mimic the case X - I~, and we define the concept of uniformly p-summable family in lp(X). Similar arguments as those used in the proof of Problem A.3 (one replaces l" I bY II" II), lead to the following compactness criterion" A family ~ C lp(X) is relatively compact if and only if it is uniformly psummable and, for each i E N*, the section of ~C at i, {xi; (Xn)n~* E %}, is relatively compact in X . 7F Problem A.5. The function g" IR U { - ~ , +co} --+ [ - y7F, y]

g(t) -

7r 2 arctan t

for t -

~2

fort-

-oc

for t E IR +oc

Solutions

358

defines a homeomorphism ~B" Ct(IR; R) --+ C([ - ~, ~ ]; R), by lim f(~-)

fort--oc

T---+-- (~

[:B(f)](t) -

f ( t a n t) lim f(~-)

for t e R for t - +oc.

Since this homeomorphism preserves the property of being equicontinuous of any subset in Ce(]R), the conclusion follows from Theorem A.2.1.

Problem A.6. The proof follows the same lines as those used in solving Problem A.5. Problem A.7. The proof follows the same lines as those used in solving Problem A.5, with the mention that, instead of Theorem A.2.1, we have to use Corollary A.2.3. Problem A.8. Necessity. Let 9" be a relatively compact family in LP(R; X ) . Obviously, 9" is bounded. Let c > 0, and let 9"E - { f l , f 2 , . . . , fn(e)} be a finite r As 9:E is finite, it is p-integrable. Accordingly, there exists r/(c) > 0 such that

(f-roe cxD I]fi(t + h) -

fi(t)ll p dt

)I/P C --< -4

for each i = 1, 2 , . . . , n(c), and each h e (0, r/(c)). We have then Ilf (t + h) - f (t)II p dt

<

I]f(t + h)

+

+

f~(t + h)[I p dt

IIf~(t + h) - fi(t)ll p dt

(f+cx~ec ] l f i ( t ) - f ( t ) l l

pdt

) 1/pEEE < -+

+

- E

for each f C 9" and h C (0, ~(c)). So the family is p-equiintegrable. [a, fl] be an arbitrary compact interval and let ( f 2 fn (t)dr) .%

sequence i n / f ~ z f ( t ) d t ;

nEN*

Let be a

f c 9=~. As 9" is relatively compact, the sequence J

(fn)ncN* has at least one subsequence, convergent to some element f in

359

Appendix

Lp(R; X ) . We denote for simplicity this subsequence again by (fn)nEN*, and we observe that

5 (8 - a ) for each n E

(1: 11 f n

1IP

(t) - f (t)I

I P ~ ~ )2

N*.From this inequality, we deduce that

and consequently (ii) holds. Finally, for the very same S(E)> 0 such that

E

> 0, there exists

. for i = 1 , 2 , . . . ,n ( ~ ) Consequently

I

:

[5 (

,

f (t) - fi(t)IPdt

(El

llfi(t)IlPdt +

+

+

;;

1;;

1IP

If (t) - h(tlllp dt)

llfi(t)IlPdt)

l P

5

E

5+

E

=&

Therefore (iii) holds, and this completes the proof of the necessity. Suficiency. Let 3 be a family satisfying (i), (ii) and (iii), let E > 0, and ) given by (iii). From Theorem A.4.1, it follows that the let S = 6 ( ~ / 4 be restriction of the family 3to LP(-S, S; X ) is relatively compact in this space. Therefore, for the very same E > 0, it has a finite ~ / 2 - n e t ,{ f l , f2, . . . , fn(,)). We have

360

Solutions

~ - -Ck C + - C -4 4 2' which proves that {fl, f 2 , . . . , fn(a)} is a finite c-net for Z. Therefore Z is precompact and the proof is complete. Problem A.9. The proof follows the same lines as those used in solving Problem A.8, with the mention that, instead of Theorem A.4.1, one has to use Theorem A.5.1. Problem A.10. The proof follows the same lines as those used in solving Problem A.8, with the mention that, instead of Theorem A.4.1, one has to use Theorem A.5.2.

Bibliography [1]

[2] [3] [4] [5] [6] [7]

[8] [9] [10] [11]

[12] [13] [14] [15]

[16] [17]

Adams, R. A. Sobolev spaces, Academic Press, Boston San Diego New York London Sidney Tokyo Toronto, 1978. Ahmed, N. U. Semigroup theory with applications to systems and control, Pitman Research Notes in Mathematics Series 246, Longman Scientific and Technical, 1991. Ahmed, N. U. Some remarks on the dynamics of impulsive systems in Banach spaces, Journal of Dynamics of Continuous, Discrete and Impulsive Systems, Set. A Math. Anal., 8(2001), 261-274. Aizicovici, S. On a nonlinear integrodifferential equation, J. Math. Anal. Appl. 63(1978), 385-395. Amann, H. Linear and quasilinear parabolic problems, Volume I Abstract theory, Birch~user Verlag, Basel Boston Berlin, 1995. Amann, H. Linear parabolic problems involving measures, RA CSAM, Rev. R. Acad. Cien. Serie A. Mat., 95(2001), 85-119. Anita, S. Analysis and control of age-dependent population dynamics, Mathematical Modelling. Theory and Applications, 11, Kluwer Academic Publishers, Dordrecht / Boston / London, 2000. Arendt, W., Vector-valued Laplace transforms and Cauchy problems, Israel J. Math., 59(1987), 327-351. Aubin, J. P. Un th6or~me de compacit~, C. R. Acad. Sci. Paris, 256(1963), 50425044. Balakrishnan, A. V. An operator calculus for infinitesimal generators of semi-groups, Trans. Amer. Math. Soc., 91(1959), 330-353. Balakrishnan, A. V. Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10(1960), 419-437. Balakrishnan, A. V. Applied functional analysis, Applications of Mathematics Vol. 3, Springer Verlag, New York, 1976. Baras, P. Compacit~ de l'op~rateur definissant la solution d'une 6quation d'~volution non lin6aire (du/dt) + Au ~ f, C. R. Acad. Sci. Sdr. I Math., 286(1978), 1113-1116. Baras, P., Hassan, J. C. and Veron, L. Compacit~ de l'op~rateur definissant la solution d'une ~quation d'~volution non homog~ne, C. R. Acad. Sci. Paris Sdr. I Math., 284(1977), 779-802. Barbu, V. Sur un probl~me aux limites pour une classe d'~quations diff~rentielles nonlineaires abstraites du deuxi~me ordre en t, C. R. Acad. Sci. Paris, 274(1972), 459-462. Barbu, V. A class of boundary problems for second-order abstract differential equations, J. Fac. Sci. Univ. Tokyo Sect. I., 19(1972), 295-319. Barbu, V. Nonlinear semigroups and differential equations in Banach spaces, Noordhoff, Leyden, 1976. 361

362

[18] [19]

[20] [21] [22] [23]

[24] [25] [26] [27] [28] [29] [3o] [31] [32] [33] [34] [35] [36] [37] [38]

Bibliography Barbu, V. Constrained control problems with convex cost in Hilbert spaces, J. Math. Anal. Appl., 56(1976), 502-528. Barbu, V. A semigroup approach to an infinite delay equation in Hilbert spaces, A bstract Cauchy problems and functional differential equations, Kappel, F. and Schappacker, W. Editors, Research Notes in Mathematics 48, Pitman Boston, London, Melbourne, 1981. Barbu, V. Analysis and control of nonlinear infinite dimensional systems, Academic Press, 1993. Barbu, V. Boundary value problems .for partial differential equations, Editura Academiei Roms Bucure~ti, 1993 (Romanian). Barbu, V. and Precupanu, Th. Convexity and optimization in Banach spaces, Second Edition, Editura Academiei Bucure~ti, D. Reidel Publishing Company, 1986. Bejenaru, I., Diaz, J. I. and Vrabie, I. I. An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions, Electronic Journal of Differential Equations, Electronic Journal of Differential Equations, Vol. 2001(2001), No. 50, 1-19. Belleni-Morante, A. Applied semigroups and evolution equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1979. Bochner, S. Diffusion equations and sctochastic processes, Proc. Nat. Acad. Sci. U.S.A., 35(1949), 368-370. Bochner, S. and Taylor, A. E. Linear functionals on certain spaces of abstractlyvalued functions, Ann. of Math. 39(1938), 913-944. Bogachev, V. I. Deterministic and stochastic differential equations in infinite dimensional spaces, Acta Appl. Math., 20(1995), 25-93. Brezis, H. Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies 5, North-Holland/American Elsevier, 1973. Brezis, H. Analyse fonctionelle Thdorie et applications, Masson Paris Milan Barcelone Bonn 1992. Brezis, H. and Browder, F. E. Strongly nonlinear parabolic initial-boundary value problems, Proc. Nat. Acad. Sci. USA, 76(1979), 38-40. Brezis, H. and Cazenave, T. Linear semigroups of contractions, the Hille Yosida theory and some applications, Publications du Laboratoire d'Analyse Numerique, Universit~ Pierre et Marie Curie, A92004, 1993. Butzer, P. L. and Berens, H. Semi-groups of operators and approximation, Springer Verlag, New York, 1967. Cs O. Some methods of nonlinear analysis, MATRIXROM, Bucure~ti, 2003 (Romartian). Cs O. and Vrabie, I. I. Some new viability results for semilinear differential inclusions, NoDEA, Nonlinear Differential Equations Appl., 4, (1997), 401-424. Casas, E. Pontryagin's principle for state-constraint boundary control; problems of semilinear parabolic equations, SIAM J. Control Optim., 35(1997), 1297-1327. Cazenave, T. and Haraux, A. An introduction to semilinear evolution equations, Clarendon Press. Oxford, 1998. Chernoff, P. R. Note on product formuals for operator semigrous, J. Funct. Anal., 2(1968), 238-242. Constantin, P. and Foia~, C. Navier-Stokes equations, The University of Chicago Press, Chicago and London, 1988.

Bibliography [39]

363

Corduneanu, C., Principles of differential and integral equations, Chelsea Publishing Company The Bronx, New York, Second Edition, 1977. [4o] Corduneanu, C. Integral equations and applications, Cambridge University Press Cambridge New York Port Chester Melbourne Sydney, 1991. [41] Courant, R. and Hilbert, D. Methods of mathematical physics, vol. 2, Interscience, John Wiley, 1962. [42] Crandall, M. G., Londen, S.-O. and Nohel, J. A. An abstract nonlinear Volterra integrodifferential equation, J. Math. Anal. Appl. 64(1978), 701-735. [43] Crandall, M. G., Pazy, A. and Tartar, L. Remarks on generators of analytic semigroups, Israel J. Math., 32(1979), 363-374. [44] Dafermos, C. Contraction semigroups and trend to equilibrium in continuum mechanics, Applications of methods of functional analysis to problems in mechanics, Germain, P. and Nayroles, B. editors, Lecture Notes in Mathematics 503, Springer Verlag, Berlin 9 Heidelberg 9 New York, 1976, 295-306. [45] Davies, E. B. One-parameter semigroups, Academic Press, London, 1980. [46] Diestel, J. and Uhl, J. J. Jr, Vector measures, American Mathematical Society, Mathematical Surveys Number 15, 1977. [47] Dinculeanu, N. Measure theory and real functions, Editura Didactic~ ~i Pedagogic~, Bucure~ti, 1964 (Romanian). [4s] Dinculeanu, N. Vector measures, Pergamon Press, New York, 1967. [49] Dunford, N. and Schwartz, J. T. Linear operators part I: general theory, Interscience Publishers, Inc. New York, Interscience Publishers Ltd. London, 1958. [5o] Edwards, R. E. Functional analysis theory and applications, Holt, Rinehart and Winston, New York 9 Chicago 9 San Francisco 9 Toronto 9 London, 1965. [~1] Engel, K.-J and Nagel, R. et al One-parameter semigroups for linear evolution equations, Graduate Text in Mathematics, 194, Springer 2000. [52] Fattorini, H. O. Boundary control systems, SIAM J. Control Optimization, 6(1968), 349-385. [53] Feller, W. The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55(1952), no. 2, 468-519. [54] Feller, W. Semi-groups of transformations in general weak topologies, Ann. of Math., 57(1953), 287-308. [55] Feller, W. On the generation of unbounded semi-gorups of bounded linear operators, Ann. of Math., 58(1953), 166-174. [56] Feynman, R. P. Space-time approach to non-relativistic quantum mechanics, Rev. Modern Phys., 20(1948), 367-387. [57] Frigon, M. and O'Reagan, D. Existence results for initial value problems in Banach spaces, Differential Equations and Dynamical Systems, 2(1994), 41-48. [ss] Fujita, H. and Kato, T. On the Navier-Stokes initial-value problem, Arch. Rational Mech. Anal. 16(1964), 269-315. [59] Gilbarg, D. and Trudinger, N. S. Elliptic partial differential equations of second order, Springer Verlag, Berlin, 1983. [60] Goldstein, J. A. Semigroup theoretic proofs of the central limit theorem and other theorems of analysis, Semigroup Forum, 12(1976), 189-206, 388. [61] Goldstein, J. A. Semi-groups of operators and abstract Cauchy problems, Cambridge University Press, 1985. [62] Goldstein, J. A., Radin, C. and Showalter, R. E. Convergence rates of ergodic limits for semigroups and cosine functions, Semigroup Forum, 16(1978), 89-95.

364

[63]

Bibliography

Graves, L. M. The theory of functions of real variable, McGraw-Hill Company, New York, 1946. [64] Grisvard, P. Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24, Pitman Advanced Publishing Program, Boston London Melbourne, Pitman Publishing Inc. XIV 1985. [65] Grosu, G Existence results for semilinear evolution equations involving measures, submitted. [66] Gutman, S. Compact perturbations of m-accretive operators in general Banach spaces, SIAM J. Math. Anal. 13(1982), 789-800. [67] Hale, J. Functional differential equations, Applied Mathematical Sciences 3, Springer Verlag New 9 York 9 Heidelberg 9 Berlin, 1971. [68] Haraux, A. Linear semi-groups in Banach spaces, Pitman Research Notes in Mathematics, 152, Longman, 1986, 93-135. [69] Hille, E. Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., 31, New York, 1948. [7o] Hille, E. and Phillips, R. S. Functional analysis and semi-groups, Amer. Math. Soc. Colloquium Publications, 31, Fourth Printing of Revised Edition, 1981. [71] Hrusa, W. J. A nonlinear functional-differential equation in Banach space with application to materials with fading memory, Arch. Rational Mech. Anal. 84(1984), 99-137. [72] Kato, T. Remarks on pseudo-resolvents and infinitesimal generators of semigroups, Proc. Japan Acad., 35(1959), 467-468. [73] Kellermann, H. and Hieber, M., Integrated semigroups, J. Funct. Anal., 84(1989), 160-180. [74] Kis:~sinski, J. A proof of the Trotter-Kato theorem on the approximation of semigroups, Colloq. Math., 18(1967), 181-184. [75] Komatsu, H. Fractional powers of operators, Pacific J. Math., 19(1966), 285-346. [76] Komatsu, H. Fractional powers of operators II Interpolation spaces, Pacific J. Math., 21(1967), [77] Komatsu, H. Fractional powers of operators III Negative powers, J. Math. Soc. Japan, 21(1966), 205-220. [78] Komatsu, H. Fractional powers of operators IV Potential operators, J. Math. Soc. Japan, 21(1969), 221-228. [79] Komatsu, H. Fractional powers of operators V Dual operators, J. Fac. Sci. Univ. of Tokyo, 17(1970), 373-396. [8o] Krasnoselskii, M., A. Two remarks on the method of successive approximations, Uspehi Mat. Nauk, 10(1955), 123-127 (Russian). [81] Laha, R. G. and Rohatgi, V. K. Probability theory, John Wiley & Sons, New York Chichester Brisbane Toronto, 1979. [82] Lasiecka, I. Boundary control of parabolic systems: regularity of optimal solutions, Appl. Math. Optim., 4(1978), 301-327. [83] Lasiecka, I. State constrained control problems for parabolic systems: regularity of optimal solutions, Appl. Math. Optim., 6(1980), 1-29. [84] Lasiecka, I. and Triggiani, R. Control theory for partial differential equations: continuous and approximation theories I abstract parabolic systems, Encyclopedia of Mathematics and its Applications 74, Cambridge University Press, 2000. [85] Lions, J. L. and Magenes, E. Probl~mes aux limites non homog~nes et applications, Volume 1, Dunod Paris, 1968.

Bibliography

365

[86] Liu, K. A characterization of strongly continuous groups of linear operators on a Hilbert space, Bull. London. Math. Soc, 32(2000), 54-62. [87] Lumer, G. and Phillips, R. S. Dissipative operators in a Banach space, Pacific J. Math., 11(1961), 679-698. [88] Maz'ya, V. G. Some estimates for solutions of elliptic second-order equations, Doklady Acad. Nauk SSSR, 137(1961), 1057-1059 (Russian). [89] McBride, A. C. Semigroups of linear operators: an introduction, Pitman Research Notes in Mathematics Series, 156, Longman Scientific&Technical, 1987. [90] Mitidieri, E. and Vrabie, I. I. Existence for nonlinear functional differential equations, Hiroshima Math. J., 17(1987), 627-649. [91] Miyadera, I. Generation of strongly continuous semi-groups of operators, Thhoku Math. J., 4(1952), 109-114. [92] Nagumo, M. Einige analytische Untersuchungen in liniaren metrische Ringen, Japan J. Math., 13(1936), 61-80. [93] Nathan, D. S. One parameter groups of transformations in abstract vector spaces, Duke Math. J., 1(1935), 518-526. [94] Ne~as, J. Les mdthodes directes dans la thdorie des dquations elliptiques, Academia, I~ditions de l'Academie Tch(~coslovaque des Sciences, Prague, 1967. [95] Neveu, J. Th(~orie des semi-groupes de Markov, Univ. Calif. Publ. Statistics, 2(1958), 319-394. [96] Nikolsky, S. M. A Course of mathematical analysis, Volume 1, Mir Publishers Moscow, 1981. [97] Pavel, N. H. Evolution equations and nonlinear semigroups, Lecture Notes in Mathematics, 1260, Springer Verlag, 1987. [98] Pazy, A. On the differentiability and compactness of semi-groups of linear operators, J. Math. and Mech., 17(1968), 1131-1141. [99] Pazy, A. A class of semi-linear equations of evolution, Israel J. Math., 20(1975), 23-36. [100] Pazy, A. A perturbation theorem for linear m-dissipative operators, Memorias de Matematica da U.F.R.J., 131(1981). [101] Pazy, A. Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer Verlag, New York Berlin Heidelberg Tokyo, 1983. [102] Peano, G. Integrazione per le serie delle equazioni differenziali lineari, Atti Reale Acc. Sci. Torino, 22(1887), 293-302. [103] Peano, G. Integration par s(~ries des (~quations diff~rentielles line,aires, Math. Ann., 32 (1888), 450-456. [104] Phillips, R. S. On the generation of semi-groups of linear operators, Pacific J. Math., 2(1952), 343-369. [105] Phillips, R. S. The adjoint semi-group, Pacific J. Math., 5(1955), 269-283. [106] Phillips, R. S. Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90(1959), 193-254. [107] Raymond, J. P. Nonlinear boundary control of semilinear parabolic equations with pointwise state constraints, Disc. Cont. Dyn. Syst., 3(1997), 341-370. [108] Raymond, J. P. and Trhltzsch, F. Second order sufficient optimality conditions for nonlinear parabolic problems with state constraints, Discrete Contin. Dynam. Systems, 6(2000), 431-450.

366

Bibliography

[109] Raymond, J. P. and Zidani, H. Time-optimal problems with boundary controls, Differential Integral Equations, 13(2000), 1039-1072. [110] Richtmyer, R. D. Principles of advanced mathematical physics, Volume I, Texts and Monographs in Physics, Springer Verlag, New York Heidelberg Berlin, 1978. [111] Segal, I. Nonlinear semi-groups, Ann. of Math., 78(1963), 339-364. [112] Showalter, R. E. Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs Volume 49, American Mathematical Society, 1997. [113] Showalter, R. E. and Ting, T. W. Pseudoparabolic partial differential equations, S I A M J. Math. Anal., 1(1970), 1-26. [114] Simon, B. Functional integration and quantum physics, Academic Press, New York, 1980. [115] Simon, J. Compact sets in the space LP(0, T; X), Ann. Mat. Pura. Appl. 146(1987), 65-96. [116] Stein, E. M. and Weiss, G. Introduction to Fourier analysis on euclidean spaces, Princeton University Press, Princeton, New Jersey, 1971. [117] Stone, M. H. On one-parameter groups of unitary operators in Hilbert spaces, Ann. of Math., 33(1932), 643-648. [118] Sulem, C. and Sulem, P.-L. The nonlinear SchrSdinger equation Self-focusing and wave collapse, Applied Mathematical Sciences 139, Springer, 1999. [119] Taylor, M. S. Partial differential equations basic theory, Text in Applied Mathematics, Springer, 1996. [120] Thieme, H. R., "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152(1990), 416-447. [121] Thieme, H. R., Positive perturbation of operator semigroups : growth bounds, essential compactness, and asynchronous exponential growth, Discrete and Continuous Dynamical Systems, 4(1998), 735-764. [122] Trotter, H. F. Approximation of semi-groups of operators, Pacific J. Math., 8(1958), 887-919. [123] Trotter, H. F. On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10(1959), 545-551. [124] Vrabie, I. I. The nonlinear version of Pazy's local existence theorem, Israel J. Math., 32, (1979), 225-235. [125] Vrabie, I. I. A compactness criterion in C(0, T; X) for subsets of solutions of nonlinear evolution equations governed by accretive operators, Rend. Sere. Mat. Univ. e Politec. Torino, 45(1985), 149-157. [126] Vrabie, I. I. A new characterization of generators of linear compact semigroups, An ~tiint. Univ. A1. L Cuza Ia~i, Sect. I a Mat. 35(1989), 145-151. [127] Vrabie, I. I. Compactness methods for nonlinear evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, Second Edition, 75, Addison Wesley and Longman, 1995. [128] Vrabie, I. I. Compactness in L p of the set of solutions to a nonlinear evolution equation, Qualitative problems for differential equations and control theory, Corduneanu, C. Editor, World Scientific, 1995, 91-101. [129] Vrabie, I. I. Differential equations, MATRIXROM, Bucharest 1999 (Romanian). [130] Vrabie, I. I. A new characterization of generators of differentiable semigroups, RA CSAM, Rev. R. Acad. Cien. Serie A. Mat., 95(2001), 297-302.

Bibliography

367

[131] Vrabie, I. I. Compactness of the solution operator for a linear evolution equation with distributed measures, Trans. Amer. Math. Soc., 354(2002), 3181-3205. [132] Webb, G. Theory of nonlinear age-dependent population dynamics, Marcel Dekker, New York 1985. [133] Widder, D. V. The Laplace transform, Princeton University Press, London: Humphrey Milford Oxford University Press, 1946. [134] Yosida, K. On the group embedded in the metrical complete ring, Japan J. Math., 13(1936), 7-26, 459-472. [135] Yosida, K. On the differentiability and representation of one parameter semi-groups of linear operators, J. Math. Soc. Japan, 1(1948), 15-21. [136] Yosida, K. Functional analysis, Third Edition, Springer Verlag Berlin Heidelberg New York, 1971.

List of Symbols A k'p (a, b;X)

A

~

a

9~

BV([a,b]) BV([a,b];X)

BV( ) BV( t; X) BVA ([ a, b ]; X)

BVA

; X)

C(a,b) C([a,b]) C([a,b];X) C([a,b];Xw) Ck(a,b) Ck([a,b]) CO

o(X) c3o( ) Cub(D)

the set of all u : [a, b] ~ X, whose ruth-order derivatives u (m) are absolutely continuous on [ a, b] for m = O, 1 , . . . , k - 1, and belong to LP(a,b;X), for m = O, 1 , . . . , k the adjoint of the operator A : D(A) C_X --+ X the directional derivative of u along a isomorphic the space of functions of bounded variation from [a, b] to the space of functions of bounded variation from [a, b] to X the space of functions f : I~ --+ I~ whose restrictions on any interval [ a, b ] are of bounded variation the space of functions f : I~ -~ X whose restrictions on any interval [ a, b] are of bounded variation the space of all g C BV([ a, b l; X) such that, for each t e [a, b], there exists the integral fo S ( t - s)dg(s) in the norm topology of X the space of all functions g C BV(I~;X) with the property that, for each a < b, g e BVA ([a, b]; X) the space of continuous functions from (a, b) to I~ the space of continuous functions from [a, b] to the space of continuous functions from [a, b] to X the space of continuous functions from [a, b] to Xw the space of functions of class C k from (a, b) to R the space of functions of class C k from [a, b] to the space of sequences of real numbers tending to 0 for n --+ cc the space of sequences in X tending to 0 for n -+ oc the space of indefinite differentiable functions from f~ to I~, with compact support in f~, i.e. 9 the space of uniformly continuous and bounded functions from

D C ~n to lI~ D(A) A 9 f(k)

the domain of A the Laplace operator b) to X the set of linear continuous operators from 9 the kth-order derivative of f in the sense of X-valued distributions on (a, b) 368

List of Symbols

369

the space of indefinite differentiable functions from ft to I~, with compact support in f~, i.e C~(f~) the derivative of order a of u in the sense of distributions on

9

] graph (A)

Hm(fl) Hm,p(f~)

H-l(f ) H (r)

H~(f~) HS(I~ ~) Im ker(A) lp

LP(a,b;X)

n (a)

LP(f~,# ; X)

l (x) N N* V

/]

I (A)

the space of distributions on 9 i.e. of linear continuous functionals from 9 to R i(~,w) f(w) dw the Fourier transform of f, i.e. ] - (2,~/2 1 fRthe graph of A the space W m'2 (f~) - H m'2 (f~) the completion of the space {u c cm(f~); IlU[]m,p < +CO} with respect to the dual of H~ (f~) the space of all functions u C L2(F) with the property that ui C Hs(I~ n - l ) for each i C :J, where {Xi; i C J} is a family of local charts on F, {0i; i C :l} a subordinated finite partition of the unity and ui - Xi o Oiu the set of restrictions to f~ of all elements in Hs(I~ n) is the space of all u c L2(I~ ~) with (1 + [1~II2)s/2~(~) C Le(I~ ~) the imaginary part of the complex number the kernel of A, i.e. ker(A) - {x C D(A); Ax - 0 } the space of all linear bounded operators from X to X the space of sequences (xn)~cN* in I~ with En~ ]x~IP < + ~ . the space LP((a, b), y ; X ) with # the Lebesgue measure on

(a,b) the set of all functions f " ft -+ I~ with the property that f is strongly measurable on ft C_ I~n and Ilfll p is integrable on f~ with respect to the Lebesgue measure the set of all functions f 9 fl -+ X with the property that f is strongly measurable on f~ and IIfll p is integrable on f~ with respect to y the quotient space LP(f~, y ; X ) / , , ~ , ,,~ being the almost everywhere p-equality on ft the space of sequences (Xn)nEN* in X with Enc~=l IIxn[Ip < +co. the space of measurable functions f 9 I~+ --+ I~ satisfying f E L l(0,b) f o r e a c h b > 0 the set of nonnegative integers the set of positive integers the nabla operator, i.e. V~ - grad ~, V . F - div F and V • F - curl F the unit exterior normal at F at the current point the norm on D(A), [IX]ID(A) --Ilxll + IIAxI[ for x e D(A) the norm on D(A), IXlD(A) - - ] I x - Axl] for x e D(A) the operator norm on L ( X ) , i.e. IIUIl~(x) - supllxll
List of Symbols

370

the norm on wm'p(f~), defined by

II" IIm,p

E IlUllm'P -

o_
g~ R R* I~+

R(A) Re R(s A) p(A) r~(A) a(A) supp~ {S(t)|

t >__O } -

Ut

X

V

wm'p(a, b ; X )

Wo'"(a) XA Xe X1 @22 X* Xt X* Xw

if 1 < p < +c~ if p -

oc

a partition of the interval [a, t] the set of all partitions of the interval [a, t] the set of real numbers the set of real numbers excepting 0 the set of nonnegative real numbers the set of positive real numbers the range of A, i.e. R(A) - A(D) the real part of the complex number the operator ( M - A) -1 the resolvent set of the operator A" D(A) C_ X ~ X the spectral radius of the operator A the spectrum of the operator A" D(A) C X ~ X equivalence relation the set {x e f~; ~(x) r O} the sun dual of the semigroup {S(t) ; t >_ O} the normal derivative of u on F, i.e. u~ - u. Vu the partial derivative of u with respect to t the second (third) order partial derivative of u with respect to

U

)lip

x

the vector product of u by v the set of all X-valued distributions f on (a, b) with fm in L p ( a , b ; X ) for each m - O, 1 , . . . , k the set of all u e LP(a) with 9 e LP(a) for 0 _< I~l < m the closure of 9 in Wm'P(f~) the set of all distributions u E 9 (f~), u - )--~l~l_<m 9 with f~ C Lq(f~), the dual of w m ' p ( ~ ) the space (X | the sun dual of the space X with respect to the operator A " D(A) C_ X -~ X , i.e., X G - D(A*) the direct sum of the spaces X1, X2 the topological dual of X, i.e. the space of linear continuous functionals from X to I~, or C the function x t ' [ - r , O] --+ ~n defined by xt(O) - x(t + 0), where x ' [ - r , +oo) ~ It{n the space X endowed with the weak topology

Subject Index

complex

Hilbert

s p a c e , 22

complex

inner product,

f u n c t i o n

22

-

almost separably-valued, 2 b - c o m p a c t , 228 Bochner integrable, 4 bounded and weakly pointwise c o n t i n u o u s , 273 - c o m p a c t , 228 - countably-valued, 1 - countably valued Bochner integrable, 5 - ~-mollified of a, 11 - g l o b a l l y L i p s c h i t z w i t h r e s p e c t to its last a r g u m e n t , 250 - h e r e d i t a r y , 270 limit p o i n t of a, 238 - l o c a l l y Lipschitz, 270 - locally L i p s c h i t z w i t h r e s p e c t to b o t h a r g u m e n t s , 250 - locally L i p s c h i t z w i t h r e s p e c t to its last a r g u m e n t , 231, 250 mollifier, 11 - of b o u n d e d v a r i a t i o n , 13 piecewise c o n t i n u o u s , 14 - r e p r e s e n t a t i o n of a c o u n t a b l y valued, 4 -scalarly measurable, 2 a - f i n i t e r e p r e s e n t a t i o n of a, 4 - strongly measurable, 2 - s u p p o r t of a, 11, 15 - resolvent, 24 v a r i a t i o n of a, 13 v a r i a t i o n w i t h r e s p e c t to a p a r t i t i o n :P, 12 - wave f u n c t i o n , 117 - weakly measurable, 2

16

d i s t r i b u t i o n ,

n o r m a l of m e a n 0 a n d d i s p e r s i o n r > 0, 114 - t h e derivative of a f u n c t i o n in sense of d i s t r i b u t i o n s , 16 - X - v a l u e d d i s t r i b u t i o n s on (a, b), 12 -

d u a l i t y

m

a

p

p

i

n

g

,

58

equation(s) - heat, or diffusion, 156 - K l e i n - G o r d o n , 95 n o n h o m o g e n e o u s t r a n s p o r t , 203 - of linear t h e r m o e l a s t i c i t y , 96 - of linear viscoelasticity, 98 - s e m i l i n e a r K l e i n - G o r d o n , 243 - S i n e - G o r d o n , 245 - Stokes in R 3, 163 - Stokes in ~ , 165 - resolvent, 25 -

-

-

family

-

e q u i c o n t i n u o u s , 295 - e q u i - a b s o l u t e l y c o n t i n u o u s , 224 p - e q u i i n t e g r a b l e , 304 - u n i f o r m l y c o n v e r g e n t , 315 u n i f o r m l y i n t e g r a b l e , 308 - u n i f o r m l y p - s u m m a b l e , 316 - w e a k l y e q u i c o n t i n u o u s , 299 -

-

-

-

-

formula

-

- C h e r n o f f p r o d u c t , 110 - F e y n m a n , 119 Hille's e x p o n e n t i a l , 112

-

371

Subject Index

372

group

of linear

operators,

38

- u n i f o r m l y c o n t i n u o u s , 38 inequality - F r i e d r i c h s , 19 - H a u s d o r f f - Y o u n g , 120 - S t r i c h a r t z , 181

-

integral -

B o c h n e r of a r e p r e s e n t a t i o n , 5 B o c h n e r of a f u n c t i o n , 6 D u n f o r d , 28 of a c t i o n , 119 W e y l f r a c t i o n a l of o r d e r a, 180

measure - complete, 1 a-finite, 1

-

-

operator -

-

-

linear, 20 - M a x w e l l , 85 - p a r t of an, 70 - p o w e r of e x p o n e n t - a of an, 171 r a n g e of an, 20 - S c h r 5 d i n g e r in H - I ( F t ; C), 90 - S c h r 5 d i n g e r in L 2 ( ~ ; C), 91 - s e l f - a d j o i n t , 21 - s k e w - a d j o i n t , 21 - s k e w - s y m m e t r i c , 21 - s o l u t i o n o p e r a t o r , 193 - s p e c t r u m of an, 24 - s p e c t r a l r a d i u s of an, 25 - s p e c t r u m of an, 28 - S t o k e s in ~t, 164 - S t o k e s in R a, 162 s u p e r p o s i t i o n , 313 - s y m m e t r i c , 21 - r e g u l a r values of an, 24 - u n b o u n d e d , 20 u n i t a r y , 72 - w a v e in L2(~t) x H - l ( ~ t ) , 92 - w a v e in H I ( ~ t ) x n 2 ( ~ ) , 93 -

a d j o i n t , 21 A i r y , 95 b o u n d e d , 20 closed, 8 closable, 112 c l o s u r e of an, 112 c o m p a c t , 293 d i r e c t i o n a l d e r i v a t i v e , 88 d i s s i p a t i v e , 59 d o m a i n of an, 20 g r a p h of an, 20 H i l b e r t - S c h m i d t , 150 k e r n e l of an, 20 K l e i n - G o r d o n in L 2 ( ~ ) x H - l ( ~ t ) , 94

- K l e i n - G o r d o n in H I (~t) x L 2 (~t), 94

- well-defined, 313 - Y o s i d a a p p r o x i m a t i o n of an, 54 operator

norm,

21

partition - r e f i n e m e n t of a, 8 semigroup - a n a l y t i c , 151 - analytic and uniformly b o u n d e d , 151 - c o m p a c t , 133 Co, 41 - Co i n t e g r a t e d , 76 - Co of c o n t r a c t i o n s , 42 - Co of n o n e x p a n s i v e o p e r a t o r s , 42 - C o of t y p e ( M , aJ), 42 - d i f f e r e n t i a b l e , 137 e q u i c o n t i n u o u s , 129 - e v e n t u a l l y d i f f e r e n t i a b l e , 137 - eventually uniformly d i f f e r e n t i a b l e , 138 -

- Laplace with Dirichlet boundary c o n d i t i o n in H - I ( ~ ) , 78 - Laplace with Dirichlet c o n d i t i o n in L2(~t), 79 - Laplace with Dirichlet c o n d i t i o n in L ; ( ~ t ) , 80 - Laplace with Dirichlet c o n d i t i o n in C o ( ~ ) , 82 - Laplace with Neumann c o n d i t i o n in [ H I (~t)]*,

-

boundary boundary

-

boundary

I

boundary 83

- Laplace with Neumann boundary c o n d i t i o n in L2(~t), 84

- G a u s s - W e i e r s t r a s s , 101 - H i l b e r t - S c h m i d t , 150 - of linear o p e r a t o r s , 35

Subject Index - of class Co, 41 Poisson, 102 - t h e d u a l of a, 67 - t h e infinitesimal g e n e r a t o r of a, 36 - t h e sun d u a l of a, 71 - u n i f o r m l y c o n t i n u o u s , 36 - u n i f o r m l y differentiable, 137 sequence - c o n v e r g e n t in 9 11 - c o n v e r g e n t in 9 16 - inferior limit of a sequence of sets, 107 - of e q u i b o u n d e d variation, 14 set - of e q u i b o u n d e d variation, 14 - resolvent set, 24

373

s o l u t i o n

- a b s o l u t e l y c o n t i n u o u s , or strong, 184 - a s y m p t o t i c a l l y stable, 261 blowing up in finite time, 241 - c l a s s i c a l , or C 1, 183 - c o n t i n u a b l e , 236, 255 - C ~ or mild, 185, 249, 255 - g l o b a l , 236, 255 - L ~ - s o l u t i o n , 270, 271 - s a t u r a t e d , 236, 255, 277 space - of admissible m e a s u r e s , 222 - Sobolev, 17 - w i t h R a d o n - N i c o d : ~ m p r o p e r t y , 189 uniform variation

operator

topology,

of constants

21

formula,

184

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