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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