Cazenave & Haraux - An Introduction to Semilinear Evolution

8/18/2019 Cazenave & Haraux - An Introduction to Semilinear Evolution

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n Introduction to

Sem ilinear Evolution E quations

n z 1 N I V E R S I D A D C O M P L U T E N S E

Revised edition

. ^ f ?

I I I I I ^ ^ ^ ^ I I I I ^ I I ^ ^ ^ I ^ I I I I ^ I I I I I ^ I I ^ I I I

3 9 45

Thierry C azenave

CN R S and Univ ersity of Paris V I France

n

lain H araux

CN R S and Univ ersity of Paris V I France

Translated by

Yvan Martel

Univ ersity of Cergy-Pontoise France

k

0R 5/ ??0

CL RENDON PRESS • OXFORD

99


2/198

Oxford University Press, Great Clarendon Street, Oxford 012 6DP

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Ox ford is a trade m ark o f O xf ord University Press

Published in the U nited States

by O xf ord University Press Inc., N ew Y ork

Introduction aux problem es dev olution sem i-lineaires

© E dition M arketing SA , 1990

First published by Ellipses

Translation © Ox ford University Press, 1998

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Preface

This book is an expa nded version of a post-gradua te course taught for severa l

yea rs at the La boratoire d'Ana lyse Nume rique of the Universite Pierre e t Marie

Curie in Pa ris. The purp ose of this course w as to give a self-contained pre sen-

tation of some rece nt results concerning the fundam ental properties of solutions

of semilinear e volution partial differe ntial equations, with special em phasis on

the asym ptotic be haviour of the solutions.

We begin with a brief description of the abstract theory of sem ilinear evolu-

tion equa tions, in orde r to provide the re ade r with a sufficient bac kground. In

par ticula r, we rec all the ba sic results of vector integra tion (Chapter 1) and lin-

ear semigroup theory

i n

Bana ch spaces (Chapters 2 and 3). Chapter 4 c oncerns

the local e xistence, uniqueness, and re gularity of solutions of abstract sem ilinear

problems.

In Nature, many propagat ion phenome na are described by evolut ion equa-

tions or evolution systems wh ich ma y include non-linea r intera ction or self-

intera ct ion term s. In Cha pters 5, 6, and 7, we a pply some gene ral me thods

to the following thre e problem s.

(1) The he at equa tion

ut

=Au 01

which models the therma l energy transfer in a hom ogeneous me dium, is the

simplest exam ple of a di f fusion equ ation. This equa tion, as we ll as the sel f -

interac tion problem

Ut

= Au

+f u,

0.2)

can be considered on the entire space

RN

or on various doma ins S1 (bounded

o r

not) of RN . In the case in which

c i

# RN,

we ne ed to speci fy a boundary

condition on I' = 852. It ca n be, for exam ple, a hom ogeneous D irichlet condition

u=0on, 03

o r a

homogeneous Neu ma nn condition

au _

an

0 our,

0.4)


4/198

vi

reface

Chapter 5 studies in detail the properties of the solutions of (0.2)—(0.3) when Sl is

bounde d. In this problem, the m aximu m principle plays an important role. This

is the rea son for studying equa tion (0.2) in the spa ce of continuous functions.

Vector-valued generalizations of the form

aui

c

czAui

+ fi(ul,

... ,

uk),

= 1 , ... , k,

0.5)

ca lled re ac tion— diffusion system s, often arise in chem istry and biology. One of

the m ain tools in the study of these system s (and in pa rt icular o f their non-

nega tive solutions) is the ma ximum principle, which gives

a priori

estimates

in

L

0°

(5l) k

for the trajectories. We thus develop Co methods rather than L

2

me thods, which are ea sier but less suitable in this fram ework.

(2 )

The wa ve equa tion (also ca lled the Klein— Gordon equa tion)

t

A u — m u ,

0.6)

with m > 0, mode ls the propa gation of different kinds of wave s (for exam ple light

wave s) in homogene ous media. Non-linea r m odels of conservative type ar ise in

quantum m echa nics, where as variants of the form

Utt

= Au —

f (u,

Ut)

0.7)

appe ar in the study of vibrat ing system s with or without dam ping, and with

or without forcing terms. Other perturbations of the wave equation arise in

electronics (the telegraph e qua tion, semi-conductors, etc.).

The ba sic me thod for studying

(0.6)

with suitable bounda ry c onditions (for

exa mple (0.3)) consists of introducing the a ssociated isome try group in the e n-

ergy space Hl

x L

2

. Loca l existenc e a nd uniqueness of solutions is established in

this spac e. Howe ver, in genera l, the solutions are differe ntiable only in the sense

of the larger spac e

L 2 x H-1.

These local questions are considered in Chapter

6.

(3 )

The S chrodinger e quation

iUt

= Au,

0.8)

possesses a c ombination of the pr operties described in (1) and (2). Pr imarily a

simplified mode l for some problem s of optics, this equa tion also arises in qua n-

tum field theory, possibly coupled w ith the Klein— Gordon e qua tion. Various

non-linear pe rturbations of (0.7) have appe are d rec ently in the study of lase r

beam s when the cha racter istics of the medium depend upon the tempera ture;

for example, focusing phenomena in some solids (where the medium c an brea k

down if the temperature reaches a critical point) and contrastingly, defocusing


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P r e f a ce

ii

from the source.

close examination of sharp properties of solutions of the

non-linear Schrodinger equation is delicate, since this problem has a mixed or

degenerate nature (neither parabolic nor hyperbolic).

n Chapter 7, which is

devoted to Schrodinger's equation, it becomes clear that even the local theory

requires very elaborate techniques.

The choice of these three problems as model examples is somewhat arbitrary.

This selection was motivated by the limited experience of the authors, as well as

by the desire to present the easiest models (in particular, semilinear models) for

a first approach to the theory of evolution equations. We do not address several

other equally worthy problems, such as transport equations, vibrating plates,

and fundam ental equations of fluid mechan ics (such as Bo ltzmann's equation, the

Navier—Stokes equation, etc.). Such complicated systems require many specific

methods which could not be covered or even approached in a work of this kind.

Chapters 8, 9, and 10 are devoted to some techniques and results concerning

the global behaviour of solutions of semilinear evolution problems as the time

variable converges to infinity. In Chapter 8, we establish that, for several kinds

of evolution equations, the solutions either blow up in finite time in the original

space or they are uniformly bounded in this space for all t >_ 0.

his is the

case for the heat equation and the Klein—Gordon equation with attractive non-

linearity,

as

well as for non-autonomous problems with dissipation.

o such

alternative is presently known for Schrodinger's equation. Chapter 9 is devoted

to some basic notions of the theory of dynamical systems and its application to

models (1) and (2) in an open, bounded domain of R

N

.

We restrict ourselves

to the basic properties, and we give an extensive bibliography for the interested

reader. In Chapter 10, we study the asymptotic stability of equilibria. We also

discuss the connection between stability and positivity in the case of the heat

equation.

Finally, in the notes at the end of each chapter there are various bibliograph-

ical comments which provide the reader with a larger overview of the theories

discussed. Moreover, the limited character of the examples studied is compen-

sated for by a rather detailed bibliography that refers to similar works.

e

hope that this bibliography will serve our goal of a sufficient yet comprehensible

introduction to the available theory of evolution problems. At the time of pub-

lication, new results will have made some parts of this book obsolete. However,

we think that the methods presented are, and will continue to be for some years,

an indispensable basis for anyone wanting a global view of evolution problems.

Pa r is

. C.

1998

. H .

U


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Contents

Notation .

.

.

.

. . . . .

iii

1. Preliminary results . . . . . . .. . . . . . . . . . . . . . . . 1

1.1.

Some abstrac t tools

.

.

1.2.

The e xponential of a linea r continuous operator

.

1.3.

Sobolev spaces

.

.

1.4. Vector-valued functions

.

1.4.1.

easurable functions

.

.

1.4.2. ntegrable functions

..

.

1.4.3.

he spaces

LP(I,X)

.

1.4.4.

ec tor-valued distributions

.

0

1.4.5.

he spaces W 1, P(I,X)

.

3

2. m-

dissipative operators

. . . . . . . . . . . . . . . . . . .

8

2.1.

Unbounded operators in Ba nach spaces

.

.

8

2.2.

De finition a nd m ain prope rties of m-dissipative ope rators . .

9

2.3.

Extrapolation

.

1

2.4.

Unbounde d operators in Hilbert spac es .

2

2.5.

Complex Hilbert space s

.

. 5

2.6.

Exam ples in the theory of pa rtial differential equations

.

6

2.6:1.

he La plac ian in an open subset of R

N:

2 theory

.

.

6

2.6.2.

he La plac ian in an open subset of R

N

:

o theory . .

7

2.6.3.

he wa ve operator (or the Klein— Gordon operator)

in Ha (1l)

x

L

2 (1l)

.

9

2.6.4.

he wa ve operator (or the Klein— Gordon operator)

in L

2

(1) x H

—

'(Il)

.

0

2.6.5.

he Schrodinger operator

.

.

3. The Hille—Yosida—Phillips Theorem and applications . . . . 33

3

The sem igroup gene rated by an m -dissipative operator

.

3

3.2.

Two important special cases

.

.

5


7/198

x Contents

1 The heat equation. . . . . . . . . . . . . . . . .

2

3.5.2. The wave equation (or the Klein—Gordon equation) . . . 47

3.5.3. The Schrodinger equation . . . . . . . . . . . . . .

7

3.5.4. The Schrodinger equation in Rr` . . . . . . . . . . .

8

4.

Inhomogeneous equations and abstract semilinear

problems .

.

.

. . . . . . .

. . .

.

.

0

4.1. Inhomogeneous equations . . . . . . . . . . . . . . . . .

0

4.2. Gronwall's lemma . . . . . . . . . . . . . . . . . . . .

4

4.3. Semilinear problems . . . . . . . . . . . . . . . . . . .

5

4.3.1. A result of local existence . . . . . . . . . . . . . .

6

4.3.2. Continuous dependence on initial data . . . . . . . .

9

4.3.3. Regularity

. . . . . . . . . . . . . . . . . . .

0

4.4. Isometry groups . . . . . . . . . . . . . . . . . . . . .

1

5.

The

heat equation . . . . . . . . . . . . . . . . . . . . .

2

5.1.

Preliminaries

.

2

5.2.

Local existence

.

.

4

5.3.

Global existenc e

.

5

5.4.

Blow-up in finite time

.

.

2

5.5.

Applica t ion to a m odel case

.

.

6

6.

The Klein—Gordon equation . . . . . . . . . . . . . . . .

8

6.1.

Preliminaries

.

8

6.1.1.

n abstrac t result

.

8

6.1.2.

unctionals on Ho (S2)

.

9

6.2.

Local e xistence .

.

2

6.3.

Global existence

.

4

6.4.

Blow-up in finite time

.

.

7

6.5.

Applica t ion to a m odel case

.

.

9

7.

The Schrodinger equation

. . . . . . . . . . . . . . . . .

1

7.1. Preliminaries

. . . . . . . . . . . . . . . . . . . . .

1

7.2. A gene ral result

. . . . . . . . . . . . . . . . . .

2

7.3. The linear Schrodinger equation in RN . . . . . . . . . . .

5

7.4. The non-linea r Sc hrodinger equa tion in R

N:

local existence . . 100

7.4.1. Some estimates . . . . . . . . . . . . . . . . . . 101

7.4.2. Proof of Theorem 7.4.1 . . . . . . . . . . . . . . . 106

7.5. The non-linea r Sc hrodinger equa tion in R

N

:

global existence . . 112


8/198

Contents xi

7.6.

The non-linea r Sc hrodinger e quation in R

N

:

low up

I

n finite time

.

14

7.7.

A rem ark c oncer ning behaviour at infinity

.

.

20

7.8.

Applica tion to a model ca se

.

2 1

Bounds on global solutions

.

24

8.1.

The he at equa t ion

.

24

8.1.1.

singular G ronwa ll's lemm a: a pplica tion to the he at

equation

.

25

8.1.2.

niform estima tes

.

29

8.2.

The Klein—G ordon equation

.

30

8.3.

The non-autonomous heat equation

.

.

34

8.3.1.

he Ca uchy problem for the non-autonomous heat

equation

.

34

8.3.2.

priori

estimates

.

3 5

8.4.

The dissipative non-autonomous Klein— Gordon e quation

.

37

The

invariance principle and some applications

.

4 2

9.1.

Abstract dynamica l system s

.

.

4 2

9.2.

Liapunov functions and the invarianc e principle

.

4 3

9.3.

A dynam ical system associated with a semilinear evolution

equation

.

4 5

9.4.

Application to the non-linea r hea t equation

.

46

9.5.

Application to a dissipative Klein— Gordon equa tion

. .

49

. Stability

of

stationary solutions

.

54

10.1. Definitions and simple exam ples

.

54

10.2. A simple genera l result

10.3. Exponentially stable system s govern ed by PD E

.

5 6

.

5 8

10.4. S tability a nd positivity

.

64

10.4.1. The one-dimensional case

.

.

6 5

10.4.2. The m ultidimensional ca se

.

67

.

6 9

.

.

8 5

9/198

the space of linear, continuous mappings from

X to

Y

the space of linear, continuous mappings from

X to

X

the topological dual of the vector space X

the Banach space

(D(A)

,

( I

I I D (A) )

with II

u I ID ( A ) = I I

uiI + IIAuII,

when A is a linear operator with a closed graph

the space of C°° (real-valued or complex valued) functions with

compact support in S2

= C°°(^) = D( l)

the space of continuous functions with compact support in S2

the space of functions of C(S2) w hich are zero on 011

the space of distributions on 11

the space of measurable functions on 11 such that

I u I P is integrable

(1 < p< oo)

= (f

n

u ) , for u E

Lp(1)

the space of measurable functions u on SZ such that there exists C

such that

I u(x ) I < C f or

almost every x E 1 1

= Inf{C > 0, Iu(x)I < C almost everywhere}, for u E L(1)

the conjugate exponent of

p,

i.e. p' = p/(p — 1)

for 1 < p < oo

0

kI

=

= (ai, ... , aN),

IaI = E a^

_ { f E

LP (St),

Da f E L P (S2)

for all

aE N' such that

I al < m }

_ .Ijcj

10/198

xiv Notation

H

Wo 2

(fl)

D(I,

X)

he space of C°° functions with compact support from

I

to

X

u

'

u

t

=

du/dt, for u

E D'(I, X)

C,(I,X)

the space of continuous functions with compact support from

I to

X

Cb(I, X)

the space of continuous and bounded functions from

I

to

X

Cb,

w

(I,X)

the space of uniformly continuous and bounded functions from

I

to X

LP(I,X)

he space of measurable functions u on

I

with values in

X

and such

that

I I u I I P

is integrable (1 < p < oo)

I I U I I L P

( I i IuIP)1

/P,

for u

E LP(I,X)

L(I,X

he space of measurable functions

u

on

I

such that there exists C

such that

I I u ( x ) I I

< C for almost every x E I

I I u I I L ° °

Inf{C

> 0,

Iu(x)I

< C almost everywhere}, for

u

E LO°(I, X)

W

1 'P(I,X) = {u

E LP(I,X), u' E LP(I,X),

in the sense of

D'(I,X)}

I I u I I w

i,P

I IU I I L P +

I I n ' I I L P

for u

E W1,P(I,X)

11/198

u

1

Preliminary results

1.1. Some abstract tools

We r eca ll here some classical theorem s of functional ana lysis that are nece ssary

for the study of semilinear evolution equations. The proofs can be found in

Brezis [2].

Theorem 1.1.1. (The Banach Fixed Point Theorem)

Let

(E,

d)

be a

c o m -

pletemetric

space and let f

: E –+E

be a mapping

such

that there

exists

k E [0, 1)

satisfying

d(f (x), f (y))

<

kd(x,

y) for all (x,

y) E E x

E. Then there

exists a unique point

10

E

E

such that f (xo) = xp.

Theorem 1.1.2.

(The Closed Graph Theorem)

Let X and Y

be Banach

spaces

and let A: X — Y be a linear mapping. Then A E

L(X,Y) if and only

if the graph of A

is a c losed subspace

of X x Y.

Remark 1.1.3. We recall that the graph of

A is G(A) = {(

x, y) E

X x Y; y =

Ax}.

Theorem 1.1.4. (The Lax—Milgram Theorem)

Let H be a Hilbert space

and let

a : H x H – IR

be a

bilinear functional.

Assume that there e xist two

constants C <

oo,

a

> 0 such that:

(i ) Ia(u,v)I

ClIull IMI

for all (u, v)

E

H x H (continuity);

(ii)

a(u, u)

>

a I I u I I 2

for all u

E

H (coerc iveness) .

Then,

f o r every

f

E

H* (the

dual space of H), there exists a unique

u

E

H such

that

a(u, v) = (f, v) for ally E H.

1.2. The expon ential of a linear continuous operator

Let

X

be a Ba nach space a nd let

A

E

C(X).

Definition 1.2.1. We denote by e A

the sum of the series E

n

An.

n> 0

It is clear tha t the ser ies is norm c onver gent in

C(X)

an d that Ile

A ll <

e

l I A 1 I

.


12/198

2

reliminary resu lts

In ad dition, for f ixed

A,

the func tion

t

'-4

e tA belongs to C°° (R, £(X))

and we

have

dt

e

tA = e1AA = Ae

IA

for a ll t

E

R. Fina lly, we have the following classica l result.

Proposition 1.2.2.

Let A

E

£(X).

For a ll

T> 0 and all x

E

X,

there

exists

a unique solution u E

C

1

([0,

T I,

X) of the following problem:

u (t) = Au(t), for alit

E [0,T];

u(0) = x.

This

solution is

given by u(t) = e

tA

x, for all t E [0, T].

Proof. It is clea r that e

tA

x

is a solution there fore, we ne ed only show unique-

ness. Let v be anothe r solution a nd let z(t)

= e-tAV(t). We have

z

'(

t ) = e

-tA

(Av(t)) - A(e

-tA

V(t)) = 0.

Therefore,

z(t) __ z(0) = x;

and so v(t) = ex.

1 3 Sobolev spaces

We refer to Adams [1] for the proofs of the results given below. Consider an

open subset S2 of R

N

. A distribution

T

E

D'(S2)

is said to belong to LP(11)

(1

13/198

Sobolev spaces 3

Then Hm(1l) is a Hilbert space with the scalar produc t

u )Hm

=

D

a

uD

a

v dx

IIm

If S2 is bounded, there e xists a c onstant C(S2) such that

u I I L 2 < _ C ( c l )I IV U I IL 2 ,

for all u E Ho

(S2) (this is Poincar e's inequality). It may be more convenient to

equip Ho (1l)

with the following sca lar produc t

(u, v) =

in

Vu • Vv dx,

which defines an e quivalent norm to

n the c losed space HH(SZ). The

following two resu lts ar e e ssential in the theory of pa rtial differe ntial equations.

Theorem

1 3 1

If S2

is open and

has a Lipschitz continuous

boundary, then:

(i) if 1 N, then W r'r(cl) - - - >

L(c) fl C°' 1 1

(52),

where

a = (p — N)/p.

Theorem 1.3.2.

In

addition, if 11

is bounded, embeddings (ii) and (iii)

of

Theorem 1.3.1

are com pact . Embedding (

i) is

compact for q

E [p,p*).

Remark

1.3.3.

The c onclusions of Theorem s 1.3.1 a nd 1.3.2 re ma in valid

without any smoothness assum ption on

5 2 ,

i f one replac es

W

1

'P(1l)

by WW'P(c)

We also reca ll the following result (see F riedman [1], Theorem 9 .3, p. 24).

Theorem 1.3.4.

Let q,

r

be such that 1

< q,

r

< oo,

and let j, m be integers,

0 < j < m.

Leta

E

[j/m,1]

(a _ 0), and

let p

be

given by

p

1/

Then there

exists C(q, r,

j, m, a, n)

such that

a

DauIIL1

C

I D a u I I L -

I u I I L 9 a ,

IcI=7

k I = ' m


14/198

4 Preliminary results

Fina lly, we r ec all the following com position rule (see M ar cus a nd Mizel [11).

Proposition 1.3.5.

Let F

: R - IR be a Lipsch itz continuous func tion, and

let 1

15/198

Vector-valued functions 5

I \En .

Let k(n) be such tha t II

fn,k(n)

fnII < 1/n on

I \E

n

and let g

n

= fk(n).

Take F = E U (f u E

n

) then IFI = 0. Let

t E I \ F.

We have

f, (t) —* f (t);

m>O n>m

on the other ha nd, for

n

large enough,

t E I \E

n

.

It follows that IIg

n

— fn

I

f (t)

and so f

is masurabe.

Remark 1.4.4. If

f : I -- X

and

cp : I

-4 R are mea surable, then

cp f : I --> X

is measurable.

Remark 1.4.5. If

(xn)n>o

is a family of elements of

X

and if (w

n ) >

0

is

a family of measurable subsets of

I

uch that

w i

n

w^

_ 0 for

i j,

hen

i

>o

x1

s measurable.

Proposition 1.4.6. (Pettis' Theorem)

Consider f : I —i X. Then f is

mea surable i f

and

only i f the fo l lowing two co nditions

are satisfied:

(i )

f is weakly

me asura ble (i.e.

for every x' E X*, the function t H (x', f (t))

is measurable);

(ii)

th ere

exists a set

N C I of measure 0

such that

f (I \ N) is sep a ra ble.

Proof.

First, since

f

is mea surable, it is clear that

f

is weakly measurable.

Now let (fn)n>o

C C

c

(I,X)

be a sequence such that

fn -i f on

I\N

as n --> oo,

where INI = 0. It is clear that

f

n

(I \ N)

is separa ble, and then so is

f (I \ N).

Conversely, we ma y assume that

f (I)

is separa ble, so that

X

is sepa rable

(by possibly re placing

X

by the smallest closed subspace of

X

containing

f (I)).

We nee d the following lem ma (see Yosida [1], p. 132).

Lemma 1.4.7.

Let X be

a separable Banach space,

let X* be its dual, and

let S*

be the unit

ball of X. There exists a sequence (x)

>

of

f S* such

that,

for every x' E S*, there exists

a subsequence (x'

nk )k> o

of (x)> withith

xnk

(x)—>x'(x) for all x c X.

Proof.

Let

(xn)n>o

be dense in

X.

For a ll

n> 0,

define Fn

: S* -* 2

2

(n), by

Fn(x

) = (x

(x1)

...,x

'

1xn)),

for a ll

x' E X*. Since

t

2

(n)

is separ able, there exists a seque nce (xn

k )k>o of

S *

such that F

n

((xn

k

)k>

o

) is dense in F

n

(S*).

In particular , for all

x'

E X*, there

exists xn E S* such that

x'

( x.7 ) —

x

lk(,. (x

7 ) I


16/198

6 Preliminary

results

for 1

oo, for all

j

E N. Since

(xn )

n

>o is dense in

X ,

we deduce easily that

x n

k(n)

(x) —* x (x)

as

n --f

oo, for

all x E X. The result follows.

End

of the

proof of Proposition 1.4.6.

Let

x E X.

Then

t H

f(t) — xli

is

measurable. Indeed, for all a > 0,

{ t ,

IIf(t)—x11

17/198


1.4.2.

ntegrable functions

Definition 1.4.10. A measurable function

f : I -> X

is integra ble if there

exists a sequenc e ( f

n

)

n

>o C C

c

(I, X) such tha t

f

Ilfn(t)

— f ( t ) I I d t - - , 0

asn ->oo.

Remark 1.4.11. II

I

f

- f

II is non-negative and measura ble, and so

f

IIfn

- f I

makes sense.

Proposition 1.4.12.

Let f : I - X

be integrable.

Ther e

exists

x

E X such

that

if a sequence (f

n

)

n

> o

C C,(I,

X) satisfies f

I

IIfn

- f

II —4

0, as n -

then

one

has

f

fn —* x as n -* oo.

Proof..

We have

f fn—^f

P

) < f

IIfn—fII+jIIf—ftI.

Therefore,

f

fn

is a Cauc hy sequence that converges to some element

x E X.

Consider a nother sequenc e

( 9 n

)

n >o

that satisfies

f

II9n

- f

II — > 0 as

n --> oo.

We have

I I J7

9 n

—

x l

I 9 n— A +LII fn— fA I + I

X.

Therefore,

fl g n , x as

n - .

oo.

Definition 1.4.13 The element

x

constructed a bove is denoted

by f

f,

or

f

I f.

If

I = (a, b),

it is also denote d

by fa f.

As for rea l-valued func tions, it is

convenient to set

l

a

f= - f

a

b f .

Proposition 1.4.14. (Bochner's Theorem)

Let f : I -4 X be measurable.

Then f is integrable i f and only

if 111 11 is integrable.

In addition,

we have

If

_ f I I . .

Proof.

Assume that

f

is integrable and c onsider a seque nce (fn)n>o C C^(I,

X)

such that

f

If

— I

ll --> 0. We have II

f

1I < IlfniI + Ilfn — f

1 I ; a n d s o I I

f

I 1 i s

i n t e g r a b l e .

18/198

8 Preliminary results

Conversely, suppose that

I I

f

I

is integrable. Let g

n

E

C,(I,

R) be a sequence

such that g

n — J ^f 1I in

L'(I)

and such that g, o

C C,(I, X)

be a sequence such that f

n

—* f

almost

everywhere. Finally, let

un

f' I n +

n fn

We have (u

n

I1

o be

a sequence

of

integrable functions

I --> X,

let g

: I

— > IR be an

integrable

function,

and

let f : I — X. If

for a l l n E

N, 1

f

n

II < g, almost

everywhere on I ,

f(t)—f(t) as n —> oo, for almost all t E I,

then

f is integrable and f1f =

lim

f

I f

n

.

n o0

1.4.3.

he spaces

LP(I,X)

Definition 1.4.16.

et

p E

[1,J. One denotes

by LP(I,X)

the set of

(equivalenc e classes of ) me asura ble functions

f : I —* X

such that

t -->

I ff

( t ) I I

belongs to

LP(I).

For

f E LP(I, X),

one defines

If (IL

P = NsSUPtEIII

A0111

^I f ( t )jjP dt) ,

f

p <

oo;

ifp=

cc.

Proposition 1.4.17.

(LT'(I,X)j

I • ^^LP) is a B ana ch space.

If p <

oo,

then

D(I,X)

is dense in

LP(I,X).

Proof.

The proof is similar to that of the real-valued case (in particular, the

den sity of

D(I, X)

is obtained by truncation and convolution).

Remark 1.4.18. Let

f E LP(I,X)

and let g

E LP (I,X*).

Then

t

g(t),f(t))x•,x

19/198


is integrable and

J

I (g(t), f (t))X*,X I

I f I I L p I I J I I L p

The following result is related to the preceding remark. The proof is much

more difficult than that for real-valued functions.

Theorem 1.4.19.

I f 1 <

p < oo

and if X is reflexive or if X*

is separable,

then

(LP(I,X))*

Lp (I, X*). In addition, if 1

o be a bounded sequence of

LP(I,X)

and let f : I —; X

be such that

f(t) — f (t)

weakly in

X as n -->

for almost alit El.

Then

f E LP(I,X),

and I

f

L P

< lim of 1lfn11LN .

Pro o f .

y Corollary 1.4.9,

f

is measurable. We define g,,, and g by

g(t)

= inf

fk(t)I)

k>n

g(t) = lim

g(t)

g(t)

= l im inf I f , , , ( t )

I I

lmost everywhere.

I


20/198

10

Preliminary results

Since

g

n

(t)

21/198

Vector-valued functions

11

t}h

Th(t)

IIPdt <

f Jt

1f(s)IIIdsdt

h

Ilf(t)IIIdtds<

f

I l f ( t ) I I

P

d t .

E C(LP(R,X))

and that IlThjI i C D(R, X)

be a sequence such that

fn --> f

of -> ac in

L P

(R,

X) and on IR \ E, w ith IE I = 0 (such a sequence exists by

roposition 1.4.17). Let t E R \ E. W e have

I A h f ( t ) I I < I I A h ( f( t )

— f^(t))II + I J A h f n ( t) I I

1

/ ^

t + h

I 1 1 ( t ) — f(t)II + h J

If(s) — fn(s)IJ d s +

I IA h fn ( t ) [I

t

n large enough, one has II f (t) - f,(t)II < E/4.

On the other hand, since II f O -

fn(•)II

E L o^(R ), by the theory of Lebesgue

ints (see D unford and Schwartz [1], p. 217, Theorem 8 ) w e know that

1

If(s)

— f(s)IId s— >

I l f ( t )

n ( t ) I I ,

t

E R, as

h -> 0.

Therefore, for almost all t,

n

being fixed so that

f (t) - f(t)II

< E /4, and if

h

is small enough, we have

1 f

t}h

J l f(s)

- fn(s)II

ds 1

I I A hf I I LP(R,X)

< _ 2 1 1 f

— fn I I LP(R,X) + I [A hf^.I l LP(R,X) '

a ny n,

it is well known that

I I A h f

fl I I

r

P

R

x)

-' 0 as

h -*0;

it follow s that

A

h

f

I I L

„

(i,x) -+ 0, wh ich comp letes the proof.

Let f E Li ;(I, X) be such

that f = 0 in D'(I, X). Then

almost everywhere.

22/198

12 Preliminary

results

Proof.

First, we rem ark that if J

is

a bounded subinterval

of I,

we have

f,

f

0. Indeed , let

(n)n>1

C D(I), V,,

< 1, an d

c o n -->

1j almost everywhere. W e

have

f

f =

lim

f

fw =

lim(f,

ca) = 0.

J

Then fix a bounde d subinterva l

J C I

and consider

f

E

L i ( I R ,

X),

defined by

{

J(t)

(t) = f (t),

if t

E J,

J(t)=0

ift¢J.

It follows that

T h f

= 0 for all

h > 0. By

Proposition 1.4.29, w e obtain

f = 0

a lmost everywhere . There fore ,

f

= 0 almost everywhere on J. Since J is

arbitrary, we have

f

= 0 almost everywhere.

Corollary 1.4.31.

L et g E

L (I,

X), to E I a nd

let

f E C(I, X) be given by

1(t)

= f t.

9(s) ds.

Then:

(i )

f` = gin

D'(I, X);

(ii)

f is

differentiable a lmost everywhere

and f' = g

almost

everywhere.

Proof.

Rea soning as be fore, we m ay re strict ourselves to the case I = IR and

gEL

1

(R,X). We

have

Thg(t)

=

f(t+h)

-

(t)

h

By P roposition 1.4.29 ,

we

deduce (ii).

Now

consider

c p

E D(R). W e have

(f',

w) _ -(f, co ) = -

f

f

(t)w'(t) d t.

Furthermore,

p(

t

+ h) — P(t)

p'(t)

uniformly on R, a s

h — . 0 .

Therefore,

` ,

—

l im

(t)

c(t + h) — co(t)

h

= —

li

o

f

co(t)

f (t(t)

_

o

J R

T hgco = (g,

c o ) ,

by P roposition 1.4.29 ; henc e (i).

23/198


Proposition 1.4.32.

Let T E D'(I, X)

be such that

T' = 0. Then there exists

z,o E

X

such that T = xo, i.e.

(T, ^) = xof,

I

for all tpED(I).

Proof. Let 0

E D(I) be such that

f6 =

1, and let xo =

(T, 6).

' Let (a, b) be

the support of 0

and let to E I, to < a. Now consider cp E D(I). We define

D(I)by

)

(V(s)

- 6(s) f

da) ds,

t o

for alltEI. We have

\ I

).

H e n c e

0

= (T,') _ (T,) — xo

J

.

It follows tha t

(T,) _ xo

J

;̂

I

hence the result.

1.4.5.

he spaces W' P(I, X)

Definition 1.4.33. Let 1

24/198

14

Preliminary results

(iv)

f is absolutely continuous, differentiable almost everywhere, and f' belongs

to LP(I,X);

(v )

f is weakly absolutely continuous,

weakly

differentiable

almost

everywhere,

and f' belongs to LP(I, X).

Proof.

(i) .(ii)` Let t

o

E I . For a ny

t E

I, set

w ( t ) _ . f ( t ) — . f ( to ) — L

t

. f ( s )

ds.

We have w E

C(I,X)

and

w'

= 0 in

D'(I, X)

by Corollary 1.4.31. There -

fore, there exists X

o

E X

such that

w

= xo in

D'(I„X)

(Proposition 1.4.32).

Since w (0) = 0, it follows from Corollary 1.4.30 that w = 0 a lmost everywher e;

henc e (ii).

(ii)(iii) is immediate.

( ii i)^(iv). By possibly modifying f on a se t of mea sure 0, we m ay a ssume

that

1(t) = xo + f

g(s)

d s ,

t o

for all t E I, and we apply Corollary 1 .4.31.

(iv)(v) is immediate.

(v)=(i). Let g be the weak derivative of

f.

Let t

o

E I,

and set

t

(t) = f(t) — f(to) - f

g(s) ds,

t

o

for a ll

t E I.

By C orollary 1.4.31, c p

is differentiable almost everywher e and its

derivative is 0 almost everywhere .

Let x' E X* and let zb be defined

by ib(t) = (x', p(t)). Ali

is absolutely

continuous, differentiable almost everywher e a nd

,b'(t)

= 0 a lmost everywhere.

Since 0(to) = 0, we obtain

0(t) - 0.

Since

x'

is arbitrary, we conclude that

co(t) - 0.

Hence, (i) follows from Corollary 1.4.31.

Corollary1436

Let 1

< p < oo.

Then W I"P(I, X )

b, v, (I, X ).

Proof.

We have

I l f(t) - f(s) I1 s f t

1 1 f (o ,

) I I d a ,

s

I

or a ll

s, t E I.

Hence we have uniform continuity.

Fu rtherm ore, if we se t h(.) = IIf (

)II, we have

h(t) — h(s)I : Ilf(t)

— f(s)II .

25/198

Vector-valued functions

15

It follow s that

^ t

I h(t) – h(s)I <

J

I If ( ^ ) II d a m ,

3

for a ll s, t E I. Ther efore , h is absolutely continuous a nd we have Ih' < II f'II E

L 5

(I)

almost everywhere . We obtain

h E W

1

P(I) ^-+ L°O(I),

which completes

the proof.

Corollary 1.4.37.

III is bounded, then C°° (7, X) is dense in W 1' ' (I, X ).

Proof.

Let

f E W ''r'(I,X)

an d let (g n ),,>1 C

D(I,X)

be such that

g, ---> f' in

LP(I,X).

Let to el, and set

fn(t) = .f (to)

+ f

t .

t g n ( s ) d s .

It is now ea sy to verify that

fn

E C

°O

(I,X)

and that

f,, —f

in

W

1

'T'(I,X),

a s

T1 -4

00.

Corollary 1.4.38.

Let 1

 C°'

a

(I,X), with

a = ( p – 1

) / p

Proof.

By Holder's inequa lity, we ha ve

p

IIf(t+h) -f (t)jj

26/198

16

Preliminary

results

Proof.

It is clear that this condition is necessary (for example, take

p() _

Converse ly, since X is complete, we ea sily verify that we ca n modify f in a set

of me asure 0 so that (1.1) holds for all t -r E I. In particular,

f

is continuous and

we m ay consider only the case in which

I = R. Since

f

is continuous, it is clea r

that f (R) is sepa rable. By possibly considering the sma llest closed subspa ce of

X,

we re strict ourselves to the ca se in which

X

is ref lexive and separa ble, and

so X''

is sepa ra ble. For all

h > 0,

se t

fh(t) = f(t + h) - f(t)

h

If

p =

oc, it is clear that

fh

is bounded in

LP(IE€,X).

If

p <

oe, by Holder's

inequality, we h ave

t+h

l ifh(t)l ip < T I

s)I'ds.

Integrating on R, we find that

r

rt+h

r r5

]

fh(t) 1

we have (fh(t), x')

--> n(t), as h -* 0.

Let F be the complement of the set

of Lebesgue's points of (we know Fl = 0 ). By (1.1), for all t

E H \ F,

we have

fh(t)M < K(t)

< oo, if Ih is small enough. We claim that for all

t E hR \ (E U F), there exists w(t)

e X such that

fh(t) — w(t)

as

h -> 0 .

Indeed,

^ l fh(t)

( is bounded, and since X

is reflexive, there exists a sequence

h„ — 0 and

w(t) E

X, such that fh„ (t) — w(t)

weakly in

X as n -- x.

In particular, we have (w(t), x

t

n) = V'(t),

for all

n

E N. Since the se-

quence (x)>i is de nse in

X*, w(t) does not depe nd on the sequence

h

n

and so fh(t) — w(t).

By Proposition 1.4.24, we have

w E LP(R,X),

and

(W ( LP ( I.X

) < 'p(

LP(I,&)

. By Theorem 1.4.35(v) and Corollary 1.4.31, we have

f

E W"P(R. X) and w

27/198


From this, we imm ediately deduce the fol lowing result

Corollary 1.4.41. Assume that

X is reflexive. Let f : I —^ X be bounded,

Lipschitz continuous

with Lipschitz

constant

L. Then, f E W(I,X)

and

< L .

orollary 1.4.42. Assume that

X is reflexive

and that 1 o be a bounded sequenc e in

W"r(I, X) and let f : I —^ X

be such that

fn

(t) — f (t)

as n

— > oc,

for almost

every t E I. Then f E

W

i

^P(I, X), and

M.f'M Lr(r, .x)

< liiminf (f,'ALF

,

(1,X).

oo.

y P roposit ion 1.4.24, we have

f E LP(I, X).

Let, E be a set of measure

0 such that fn (t) — f (t)

as n

for all

t E I \ E. For all

t, r C I \ E,

we

h a v e

) J (

t

) - f() < 1

im

of

))fn(t) - fn(

T ) 1 1

1 and

p E LP(I)

such that :p nk — ^ p

weakly in LP(I) (weak-* if p

= oc)

as k — oo, and litn inf ^^ ^p,^

k ^1 L n — lim inf

In particular, we have

—

--oo

J

4

(s) ds —>

J p(s) ds,

as

k

--^ oo,

1.4)

for all t, r E I,

and

((^F))Lv < liminf))fn(ILr,.

1.5)

n—+oo

It follows from (1.3) and (1.4) that

§f t)—f T)11

28/198

2

m-dissipative operators

Throughout this chapter, X is a Banach space, endowed with the norm

I I • I I

2 1 Unbounded operators in Banach spaces

Definition 2.1.1.

A linear unbounded operator in

X

is a pair

(D, A),

where

D

is a l inear subspa ce of

X

and

A

is a l inear ma pping

D — X.

We say that A

is bounded if there exists c > 0 suc h that

IIAu

I I < _

c,

for all

u

E

{x E

D.

I I r I I

< 1}. Otherwise,

A

is not bounded

Remark 2.1.2. Note that a linear unbounded operator can be either bounded

or not hounded. This somewha t strange terminology is in gene ral use and should

not lead to misunder standing in our a pplica tions.

Remark 2.1.3. If

A

is bounded,

A

is the restriction to

D of

an operator

A

EY, X),

where

Y

is a closed l inea r subspac e of

X,

such that

D

C

Y. If A

is not bounded, there exists no operator A

E

G(Y,X)

with Y

closed in

X and

D C Y,

such that

AID = A.

Definition 2.1.4. Let

(D, A)

be a linear operator in

X.

The graph

G(A) of

A

and the range

R(A) of

A are defined by

G(A)={(u,f)EXxX;uED and f=Au},

R(A) = A(D).

G(A)

is a linear subspac e

of X x X,

and

R(A)

is a l inear subspac e

of X

Remark 2.1.5. In this chapter, a linear unbounded operator is just called an

opera tor where there is no risk of confusion. As usual, we de note the pair

(D,

A)

by

A

with

D(A) = D,

me aning the doma in of

A is D.

Note, however , that when

one de fines an ope rator, it is absolutely nece ssary to define its doma in.

Remark 2.1.6.

When

D(A) = X,

it follows from Theorem 1.1.2 that

A E

G(X) if

and only

if G(A)

is closed in

X

More generally, for not bounded

opera tors, it is very use ful to know w hether or not the grap h is closed.

29/198

Definition and main properties of m-dissipative operators 19

2. Definition and main properties of m

dissipative operators

I ^ u - A A u l l > I lu l l ,

ion 2.2.2. An operator

A in

X

is m-dissipative if

i ) A is dissipative;

for all A > 0 and all

f

E X, there exists u E

D(A)

such that u — A A u = f .

emark 2.2.3. If

A

is in-dissipative in

X,

i t is clea r, from De finitions 2.2.1

f

E X and all A > 0, there exists a unique solution

u of

tion

u — AAu = f.

In add ition, one ha s I ull 0. For

f

E X, we denote by

Jaj or by

(I — AA)

-f

he solution a of the equa tion

emark 2.2.5. By Remark 2.2.3, one has Ja

E L(X) and IJaj I

c

( x ) < 1.

roposition 2.2.6. Let A

be a dissipa tive

operator in X. The following prop-

are

equivalent.

i )

A is m-dissipative in X;

there

exists Ao > 0 such tha t for

all f E X, there e xists a

solution u E D(A)

ofu—A0Au= f.

It is clear that (i)^(ii). Let us show that (ii)^(i). Let A > 0. Note

equation

u — AAu = f is equivalent to

U-A O Au=A°f+(1- A

)u.

A is dissipative an d

R(I — \o A) = X,

the operator J a

0

= (I — A0A)

-1 can

in Definition 2.2.4. This operator is a contraction on X. Next,

ced ing equa tion is also equivalent to

u=J50 L0

o

f+(1_ )u)

.

0

, this last equation is u =

F(u),

where F

is Lipschitz continuous

with a Lipschitz constant

k =

I(A — A0)/AI < 1. Applying Theo-

u

of

u — AAu = f,

for all A E (A0/2,).


30/198

2


Iterating this argument

n

time s, ther e e xists a solution for all A

E (2-̀Ao, oo),

n > 1.

Since

n is

arbitra ry there exists a solution for all A > 0.

Proposition 2.2.7.

If A

is m-

dissipative, then

G(A) is

closed in X.

Proof.

Since J 1

E

£(X),

G(J1) is closed. It follows that

G(I - A )

is closed,

and so G(A) is closed.

Corollary

2.2.8.

Let A

be an

m-

dissipative operator. For every u

E

D(A),

let

I IL I ID(A)

= I I u l I +

IIAull.

Then (D(A),

I I

- I I

D(A))

is a Ba nach space , and

A E

12(D(A ), X ).

Rem ark 2 .2.9 . In wha t fol lows, and in part icular in Cha pters 3 and 4,

D ( A )

mea ns the B anac h space (D(A),

I I -

IID(A))•

Proposition 2.2.10.

If A

is

m-dissipative,

then li

o I I

Jau - ull = 0 for all

uED(A).

Proof.

We have IIJ) -

I I I

0, we denote

by A ) ,

the operator defined by

Aa=AJ)=

Ja-I

We have Aa

E G(X)

and IIAa IIc(x) 5 2/A.

Proposition 2.2.12.

If A is m-dissipative

and

if D(A) = X,

then

A

),u ---> A u

as A 0 for all u

E

D(A).

Proof.

Let

u E D(A).

By Proposition 2.2.10, one has

JA A u-Au— *0

asAJ0.

On the other ha nd, it follows ea sily from D efinition 2.2.11 that

Aau = J)Au.

Thus,

IIAau - AuII = IIJ)Au -

AuII --30

as A j 0;

hence the result.

31/198

Extrapolation

21

2.3. Extrapolation

In this section, we show tha t, given an m -dissipative opera tor

A on X

with dense

doma in, one c an extend it to an m-dissipative opera tor

A

on a larger space

X.

This result will be ver y useful for char ac terizing the we ak solutions in Chapter s 3

and 4.

Proposition 2.3.1.

Let A

be an

m-dissipative operator in X with

dense do-

main.

There

exists

a Banach space X,

and an

m-dissipative

operator A

in X ,

s u c h t h a t

(i)

X '—

ith

dense em bedding;

(ii)

for

all u E X,

the norm of u in X is equal to

liJlulI;

(iii)

D(A) = X,

with

equivalent

norms;

(iv)

Au = Au, for all u E D(A).

In addition, X

and

A satisfying (i)-(iv) are unique,

up to

isomorphism.

Proof.

For

u E

X, we define IMu IIt = I^J luII. It is clea r that III - III is a norm on

X.

Let X

be the com plet ion of

X

for the norm .

X

is unique, up to an

isomorphism, and X '— X, with dense em bedding. On the other han d, observe

t h a t

J1A u = Jlu — u, du E D(A ).

Thus,

II IA u I II < - IM u I II + h u h < - 2 1 1 u 1 1 , V u E D ( A ) .

He nce, A can be extended to an operator

A E £(X, Y).

We de fine the linea r

operator

A on X

by

D(4)=X ,

A u=A u, VuED(A ).

It is clea r tha t A satisfies (iii) and (iv). Now, let us show tha t A is dissipa tive.

Take X> 0. Let

u E D(A)

and let

v = J

lu.

One has

v — .XA v = Ji(u —

,Au)

Since A is dissipative, it follows tha t

I I I u —)

AuIII = liv — AAvII > IIvli = IIIuIIl

By continuity of A, we de duce that

32/198


and so

A is

dissipative. Fina lly, let

f

E

X, and

(f>o C

X, with f ,, , - - - > f

in

X

as n ,

oo. Set u

n =

J1

f,.

Since (fn ) n,>

o is a Ca uchy sequence in

X,

(un,)n

>o

is also a Ca uchy sequence in

X;

and so there exists u

E

X,

such that u

n p

u

inXasn --*oo. We have

f

= u

n

— Au

n

=

un

— Au

n

Since

A

E

£(X,Y);

it follows that

f

= u — Au = u — Au.

Hence

A is m-

dissipative. The u niqueness of

A

follows from the uniqueness of

A.

Corollary

2.3.2.

Ifs E

X is such that

Ax

E

X,

then x

E

D(A) and Ax = Ax.

Proof.

Let

f = x — Ax

E

X.

Since

A is m-

dissipative, there exists

y E

D(A)

such that y — Ay =

f.

By P roposition 2.3.1(iii), we ha ve

(x — y) — A(x — y) = 0,

and since

A is

dissipative, w e obtain

x=y

2.4. Unboun ded operators in Hilbert spaces

Throughout this section, we a ssume tha t X is a Hilbert space , and we de note by

(•, •) its scala r produc t. If

A is

a linear operator in

X

with dense doma in, then

G(A*) = {(v, go)

E

X x X; (cp, u) = (v, f)

for a ll

(u, f)

E

G(A)},

defines a l inear operator

A*

(the a djoint of

A).

The doma in of

A* is

D(A*) = {v

E

X, 3C < o o , ((Au,

v)I <

u

E

D(A)},

and A*

satisfies

(A*v, u) = (v, Au), `d

E

D(A),

Indeed, the linea r ma pping u

H

v, Au),

defined on

D(A)

for a ll

v

E

D(A*),

can

be extended to a unique linea r ma pping

c p

E

X' X,

denoted by

cp

= A*v.

It

is clea r that

G(A*) is

systematically closed.

Fina lly, it follows e as ily tha t if

B

E

£(X),

then

(A + B)* = A* + B

*.

1

Proposition 2.4.1.

(R(A)) = {v E

D(A*); A*v = 0}.

Proof.

One has

v

E

(R(A))

v,Au) = 0,Vu

E

D(A) 0,v)

E

G(A*).

This last property is equivalen t to

v

E

D(A*)

and A*v = 0;

henc e the re sult.

Proposition 2.4.2.

A is dissipa tive in X

if and

only if (Au, u) < 0, for

a ll

33/198

Unbounded operators in Hilbert spaces 23

Proof.

If A is dissipative, one has

— 2A(A u, u)

+ A 2

I I A u 1 I

2

= I Iu

— AAuII

2 — I I U I I 2

> 0, VA > 0, Vu

E

D(A).

Dividing by A an d letting A j 0, we obtain

(Au, u) < 0, for all

u

E

D(A).

Conve rsely, if the last prope rty is satisfied, then for a ll A > 0 a nd u

E

D(A)

we

have

I I u

— AAuII

2

= I I u I I

2

— 2A(Au, u) + A

2

I I A u I I 2

> I Iu l l

2

and then A is dissipative.

Corollary 2.4.3. If

A is m-dissipative in X,

then D(A) is dense in X.

Proof. Let

z E

(D(A))

l

and let u = Jjz

E

D(A).

We have

0 = (z, u) = (u —

Au, u).

Hence,

I I u I I 2 = ( A u ,

u) < 0.

It follows that u = z = 0;

and so D (A) is dense in

X.

Corollary

2.4.4. If A is

m-dissipative in X, then

J,u -- u as A j 0,

for all u

E

X and

AAu --* Au as A j 0,

for all u

E

D(A).

Proof.

We apply Corollary 2.2.3 and Propositions 2.2.10 and 2.2.12.

Theorem

2.4.5. Let A be a linear dissipative operator

in

X

with dense do-

main.

Then A

is

m-dissipative

if and

only if A*

is dissipative and G(A) is

closed.

Proof.

If

A

is m-dissipative, then

G(A)

is closed, by P roposition 2.2.7. Let us

show that

A *

is dissipa tive. Let v

E

D(A*).

We have

(A*v, Jay) _ (v, AJav) = (v, AAv)

_ (v,J v—v)=

{(v,Jav)—

I I v I I 2 }

34/198

24

m -dissipative operators

Since

(A*v, Jav) -->(A*v, v)

as A J 0, it follows that A* is dissipative.

Conversely, since

A

is dissipative a nd

G(A)

is closed, it is clear that

R(I — A)

is closed in X. On the other ha nd, by Proposition 2.4.1, one ha s

(R(I — A)) = {v

E

D(A*); v — A*v = 0} = {0},

since A*

is dissipative. Therefore

R(I — A) = X,

and

A

is m-

dissipative, by

Proposition 2 .2.6.

Definition 2.4.6.

Let

A

be a l inear opera tor in

X

with dense dom ain. We

say that

A

is self-adjoint (resp ec tively ske w-adjoint) if

A*

= A (respectively

A* = —A).

Remark 2.4.7. The equality A* = +A has to be taken in the sense of opera-

tors. It me ans that

D(A) = D(A*)

and

A*u = ±Au,

for a ll

u

E

D(A).

Corollary 2.4.8.

If A

is a self-adjoint

operator in X,

and if

A < 0 (i.e.

(Au, u)


35/198

Complex Hilbert spaces 25

Corollary 2.4.11.

Let A be a linear operator in X with

dense domain.

Then

and

—A are m-dissipa t ive

if an d only if A is sk ew-a djoint.

P r o o f .

pplying Corollary 2.4.9 , it suff ices to show that if A a nd — A are

m-

dissipative, then A is skew -adjoint. Applying Pr oposition 2.4.2 to A

and —A,

we obtain

(Au, u) = 0, u E D(A).

For a ll

u, v E D(A),

we obtain

(Au, v) + (Av, u) = (A(u + v), u + v) — (Au, u) — (Av, v) = 0.

*) .

)here fore G(—A) C G ( A

It rema ins to show that

G(AC G(—A). Consider

(u, f) E G(A*) and let

g = u — A*u = u — f.

Since — A

is m-dissipative, there

exists

v E

D(A) such that g = v + Av, and since G(—A) C G(A*),

we have

v E D(A*)

and

f = v — A*v.

Hence (v — u) — A*(v — u)

= 0 a nd since

—A*

is

dissipative (Theore m 2.4.5), we obtain u = v.

Therefore,

(u, f) E G(A*); and so

A = —A*.

I

2.5.

omplex Hilbert spaces

In this sec t ion, we assum e that X is a com plex Hilbert space . Reca ll that by

definition X is a c omplex H ilbert spac e provided tha t ther e e xists a c ontinuous

R-bilinear mapping

b: X x X — > C satisfying the following properties:

b(iu, v) = ib(u, v),

(u,

v) E X x X;

1

b(v, u) = b(u, v),

(u, v) E

X x X;

b(u, u) =

^ l u l l

2 ,

u E X.

In that ca se

(u, v) = R e(b(u,_

v)) def ines a (rea l) sca lar produc t on X. Equipped

with this scalar product, X is a rea l Hilbert space . In wha t follows, we c onsider

X as a

real Hilbert space.

Let

A

be a linear opera tor on the real Hilbert space

X. I f A is C-linea r, we

ca n define iA as a l inear opera tor on the real Hilbert space X.

Proposition 2.5.1.

ssume that

D(A)

is dense a nd that A is C-linea r . Then

A*

is

C-linear, and (iA)* = —iA*.

P r o o f .

et

v E D(A), f = A*v

and let z E C . For al l

u E D(A),

we have

I

(zf, u) = (f, u) = (v, A(u)) = (v, zAu) = (zv, Au).

There fore zv

E D(A*)

and zf = A(zv).

Hence

A* is C-linear. In addition, I

36/198

6


for a ll

(v,

f) E

G(A*)

and a ll

u

E

D(A);

and so

G(—iA*) C G ((iA)*).

Applying

this result to iA, we obtain

G(—i(iA)*) C

G ((i •

iA)*) = G(—A*).

It follows tha t

G ((iA)*)

C

G(—iA*),

and so G ((iA)*) = G(—iA*).

Corollary

2.5.2.

If

A is

self-adjoint, then

iA is

skew-adjoint.

Proof. (iA)* = —iA* = —iA.

2.6. Exam ples in the theory of partial differential equations

.6.1.

The

Laplacian in an open subset of

RN: L

2

theory

Let Sl be any open subset of

R

N

, and let

Y = L 2

(S2). We c an consider either rea l-

valued functions or complex-valued functions, but in both cases, Y is considered

as a rea l Hilbert space (see §2.5). We define the linea r opera tor

B

in

Y by

{

D(B) = {u

E Ho(S2);

Au

E L

2

( c l ) } ;

Bu = Au,

d u

E

D(B).

Proposition

2.6.1.

B is

m-

dissipative

with dense doma in.

More prec ise ly, B

is se l f -a djo int a nd B < 0.

We need the following lemma .

Lemma

2.6.2.

W e

have

s

f

vAudx=—

I

n

Vu•Vvdx.

2.1)

for alluED(B) andvEHo(5l).

oo.

2.1) is satisfied by

v

E D(l).

The lem ma follows by density, since both

term s of (2.1) are continuous in v on

Ho (S2).

Proof of Proposition 2.6.1.

First,

D(S2) C

D(B), and so

D(B) is

dense in

Y.

Let

u

E

D(B).

Applying (2.1) with

v

= u, we obtain

(Bu, u) < 0,

so that

B is

dissipative (Pr oposition 2 .4.2). The bilinear continuous ma pping

b(u, v)

(uv

+ Vu • Vv)dx


37/198

Examples in the theory of partial differential equations 27

is coe rcive in

Ho(f ).

It fol lows from The orem 1.1.4 that, for all

f

E L

2 (Sl),

there e xists u E H o (Sl) such that

J

(uv+Du•Vv)dx =

J

fvdx,

by E H o(Q).

We obtain

u — Du= f,

in the sense of distributions. Since , in addition

u E

Ho (S2), we obtain u E

D(B)

and

u — Bu = 1.

Therefore

B

is m-dissipative. Fina lly, for a ll u, v E

D(B),

we

have, by (2.1),

(Bu, v) = (u, Bv).

Therefore G(B) C G(B*),

an d by C orollar y 2.4.10, it follows that

B is self-

adjoint.

Remark 2.6.3. If Sl has a bounded boundary of class C

2 , then D(B) =

H2 (1)

f l Ho(c l), with equiva lent norms (see B rezis [2], Theorem IX.25, p. 187,

or Fr iedma n [1], Theore m 17.2, p. 67).

2.6.2. The Laplacian in an open subset of RN: Co theory

Let Sl be a bounded open subset of IR

N

, and let Z = L°°(Sl). We define the

l inea r opera tor C in Z by

D(C)

= {u E Ho (cl) n Z, Au E Z},

Cu = Au, b'u E D(C).

Proposition 2.6.4.

C is m-dissipative in Z.

Proof.

First, let us show that C is dissipative. Let ,A > 0, f E Z, and let

M = I f ^I L ^ .

Let u

E H o (f l) be a solution of

u

—

AAu= f,

in D'(S2). In particula r, this equa tion is satisfied in L

2

(S2), and we ha ve

(u— M)— A A (u— M)=f— M,

in L

2

(S1). On the other ha nd,

v = (u — M)+

E Ho(1l), with Vv = 1

{1uI>M}Du

(Corollary 1.3.6). Applying Lem ma 2.6.2, we obtain

J

v

2

dx+w

DuI2dx=

J(f— M)vdx_

38/198

8


Therefore

f v 2

dx < 0, and so

v = 0.

We conclude that

u < M

a.e. on

Q. Similarly, we show that

u >_ — M

a.e. on Q. Hence u

E

L°°(f ), and

U

L°°

< I I f 1 I

L —

• It follows tha t C is dissipative. Now let

f

E

L°°(1) C L

2 ( c

).

By §2.6.1, there e xists u

E Ho(Sl),

with Du E

L

2

(1l), a solution of u — Au

= f ,

in

L

2

(S2). We a lready know that u

E

L(1l),

so that

u

E

D(C),

and

u—Cu

= f .

Thereore Cis mdsspaive.

Lemma 2.6.5.

If S2 has a L ipschitz continuous

boundary,

then

D(C) C C0(l) = {u

E

C(S2); u

an =

0}.

Proof.

The p roof is dif f icult, an d uses the notion of a ba rrier func tion (see

Gilbarg and Trudinger [1], Theorem 8.30, p. 206).

Remark 2.6.6.

It follows from Le mm a 2.6.5 that in gene ral the dom ain of C

is not dense in Z. The fa ct that the doma in is dense will turn out to be very

important (see C hapter 3). This is the rea son why we a re led to consider another

example.

We now set X = C

o

( l) , an d we def ine the opera tor

A as follows:

(D(A)=

{uEXnH.()),AuEX} ,

S

I A u = Au, Vu E

D(A).

Proposition 2.6.7. Assume that S2 has a Lipschitz

continuous boundary.

Then A is m-dissipative,

with dense domain.

Proof. D(Sl)

is dense in

X, and

D(SZ) C D(A );

and so

D(A) is

dense in

X .

On

the other hand,

X

is equipped with the norm of

L°°(S2),

and so

X — +

Z and

G(A) C G(C).

Since C is dissipative,

A is

also dissipative. Now let f

E

X y

L°° (f ). Since C is

m -

dissipative, there exists u E

D(C),

such that u — Au

= 1.

By Lemm a 2.6.5, we ha ve

u

E

X ,

and so Au

E

X. Therefore, u

E

D(A)

and

u — A u

= f.

Hence A is m-

dsspave

Rem ark 2.6.8. In the three exa mples of §2.6.1 a nd §2.6.2, note that the sa me

formula (the Laplac ian), corresponds to severa l operators that enjoy dif ferent

properties (since they are defined in dif fere nt domains). In pa rticular, the e x-

pression the opera tor A has a mea ning only if we spe cify the spac e in which this


39/198


The wave operator (or

the Klein—Gordon operator) in Hp

(S2) x

2 (Q)

any open subset o f RN, and let X

= Ho (fl) x L

2

(1). We deal either

tions or with complex-valued func tions, but in both cases

dere d as a rea l Hilbert space (see §2.5). Let

A = inf

{11VU1IL2,u

E

Ho(S2), I I u I I

L2 = 1}.

2.2)

( In the c ase in which 5l is bounded, we rec all that A is the f irst eigenva lue of

— ,L in

Ho (S2),

and that A> 0). Let

m> —A.

Then

X

ca n be equipped with

((u, v), (w, z)) _

/ (Du • Vv + m aw + vz) dx.

t defines a norm on X which is equivalent to the usual norm.

f ine the linea r opera tor

A in X

by

D(A) = {(u,v)

E

X, Au

E

L

2

(S2),v

E Ho(S2)};

A(u, v) = (v, Au — mu), V(u, v)

E

D(A).

2.6.9.

A is skew-adjo int ,

an d in

particular A

and —A are m-

with dense do ma ins.

D(1) x V(l)

C

D(A)

and so

D(A) is

dense in X.

On the other hand,

l l

((u, v), (w, z))

E

D(A)z,

and by (2.1), we have

(A(u, v), (w, z))

= J

(Ov . Ow + mvw + (Du — mu)z) dx

= —

J

(vu

•Vz+muz+ (Aw — mw)v) dx

_ —((u, v), A(w, z))•

2.3)

(u, v) _

(w, z), it follows tha t

(A(u, v), (u, v)) = 0.

A is

dissipative (Proposition 2.4.2). Now let

(f,

g) E

X.

The e quation

is equivalent to the following system:

r2u—iu

=f+g;

2.4)

Slv=u— f.

2.5)

u E

H

o

(Sl)

of (2.4), satisfying L u

E

2

(1). Next, we solve (2.5) and we obtain v

E

Ho

(52). Therefore

(u, v)

E

D(A)

(u, v) — A(u, v) = ( f , g), so

that

A is

m-dissipative. Similarly, we show

—A is

m-dissipative. By (2.3), we have

G(A)

C G(—A*).

Corollary 2.4.11

A is

skewadon.

1

1


40/198

30


2.6.4. The wave operator (or the Klein-Gordon operator) in L

2

(Sl) x

H(2)

Let S2 and m be as in §2.6.3. We recall that Ho (Q) ' L

2

(Q) '-+ (Ho (Q))' _

-1(Q) with dense embeddings. We equip Ho (1) with the scalar product de-

fined in §2.6.3. Theorem 1.1.4 shows that

H

-

(r) = {u E

D (),

5

Ho (ŝ ), o^

u

— m

= u in D'(sf)},

2.6)

and that we can equip H

-

(Q) with the scalar product

(u,v)-i =

J (v

0

.V +

Y = L2

(Q)

x H

-

'(St).

We deal either with real-valued functions or with

complex-valued functions, but in both cases X is considered as a real Hilbert

space (see X2.5). We define the linear operator

B

in

Y by

{

D(B) = Ho(l) x L

2

( Q ) ^

B(u, v) = (v, Au - inn) E Y,

u, v) E

D(B).

Proposition 2.6.10.

B

is

skew-adjoint.

In particular,

B and -B are m-

dissipative with dense domains.

Proof.

D(S2 ) x D(S2)

C D(B)

and so

D(B)

is den se in

Y.

Let

((u, v), (w, z)) E

D(B)2,

and consider cp„ and

z

defined by (2.6). Since v, z E L

2

(t1), we have

E L

2

(Q). Applying (2.1), we obtain

), (w, Z))L2

X

H_ =

f

vwdx+(Au

-

mu,

z ) - i

=

J

vw dx +

J

(Du • V(p

2

+ mu c)

z

) dx

=

J

vw dx -

f

u(A

z

- m) dx

=

J

vw dx -

J

uz dx.

Similarly, we have

((u,v),B(w,z))

t

, 2 XH

-1 =

J

zudx -

J

wvdx.

Therefore,

2.7)


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Applying (2.7) with

(u, v) = (w , z),

it follows that

(B(u, u), (u, v)) = 0.

Thus, B

is dissipative (Proposition 2.4.2). Now let

(f, g) E Y.

The equation

(u, v) — B(u, v) = (f, g)

is equivalent to the system (2.4)—(2.5) of §2.6.3. By

Theorem 1.1.4 (see the proof of Proposition 2.6.1), there exists a solution u E

H o (f2) of (2.4). Next, we solve (2.5) and w e obta in

v

E L

2

(fl). Therefore

(u, v) E

D(B)

and

(u, v) — B(u, v) = (f, g);

henc e B is m-dissipative. Similarly, we show

that —B

is m-dissipative. By (2.7), we have

G(B) C G(—B`). Corollary 2.4.11

proves tha Bis skewadoint.

Proposition 2.6.11.

We use the same notation as in §2.6.4.

Then Y

and

B

are the extensions of X and A given by Proposition 2.3.1.

Proof.

Properties (i), (iii), and (iv) are clearly satisfied. We need only show (ii),

i.e.

IIU)r

§(I — A)—'UMMx, VU E X.

Let U

e X and

V E D(A)

be such that U = (I — A)V.

We show that §(I

A)Vy Vf^x. Indeed, since

B

is skew -adjoint, we ha ve

II(I - A)V

I I Y

= ((I - B)V, (I - B)V)' -

II^IIY

+ II BV(Y.

Let

V = (u, v).

We have

J J B V 1 1 2 =

w 1 1

§§AU -

mu)

- §V L2 + IIuI1

2 , = § § V I 1

2 .

henc e the result.

2.6.5.

he Schrodinger operator

Let f be any open subset of Rh', and let Y = L 2

(52,C).

Y is considered as a

real Hilbert space (see §2.5). We define the linear operator B

in Y

by

D(B) = {u E

Ho (1l. C), L u E Y};

{ By = i^u, Vu E D(B).

In what follows, we w rite L

2

(4) and H) instead of L

2

(cl, C) and Ho (St, C).

Proposition 2.6.12.

B is skew-adjoint, and in

particular B

and -B are m-

dissipative with dense domains.

42/198

32 m-dissipative operators

Re ma rk 2.6.13. As in §2.6.1, if 11 ha s a bounde d boundar y of class C

2 , then

D(B) = H

2

(S2) fl

Ho(fl),

with equivalent norms.

We now set

X

= H-

(1, C) and, given u

E X,

we denote by cp

v

, E Ho (52, C)

the solution of

-^cp

v, +

u

in

X.

We equip X

with the scalar produc t

(u, v)-i = (^P .., ^Pv)H1

=Re J

x,

for u, v E

X.

We de fine the linea r opera tor

C in X by

D C) = Ho

1

l);

Cu=Lu,

VuED(C).

Proposition 2.6.14.

C is self-a djoint < 0.

Proof.

We have D(Sl, C) C

D(C)

so that

D(C)

is dense in X.

Furthermore,

for a ll u, v E

D(C),

(Cu,v)_i = (Cu —u,v)_1 +(u,v)_1 = (u,cp„)Hi +(u,V)_1

=

-

u,v)L2 + (u,v)_1.

2.8)

Ta king u = v, it follows that

(Cu,u)-1= —IIkIIL2 + IIUIIH-1 < 0,

and so C is dissipative. Theore m 1.1.4 proves tha t C is m-dissipative. By (2.8),

we have

Cu,v)_1 = u,Cv)_1,

for all u,v E

D C).

It follows that

G C)

C G(C*), and so C is self-adjoint

(Corollary 2.4.10).

Finally, consider the ope rator

A in X

given by

J D(A) = Ho (l);

Au=i/u, VuED(A).

Applying Proposition 2.6.14 and Corollary 2.5.2, we obtain the following result.

Corollary 2.6.15.

A

is skew-adjoint, an d in pa rticular

A and -A are m-

dissipa tive with de nse

domains.

Notes.

For more information about §2.6, see Brezis [2), Courant and Hilbert [1],


43/198

44/198

34 The Hille–Yosida–Phillips Theorem and applications

In particular,

Ilua t II

Ixil,

3.8)

for a l l A > 0 and a l l t > 0.

Step 2.

Assume that

x E D(A).

I t is c lear by construction that AA a nd A.

comm ute, for a ll A, p > 0. In particular, for all s, t > 0, we ha ve

ds

{Ta(st)T, (t – st)} = tTA(st)T

N

,(t – st)(A A – A

µ

) .

It follows that

Ilua t)

–u

µ (t)II = IIT. (t)x–T

µ (t)xII

i

<

s

{T(st)T(t – st)x}

dsl

< tIlA ax – A

µxII.

We deduc e (Proposition 2.2.12) that uA is a Ca uchy seque nce in

C([0,T],X),

for all

T > 0.

Let

u E

C([0, oo),

X) be its limit.

Step 3. Set

u(t) = T(t)x.

By (3.8), we ha ve

(IT(t)xII

Ixil,

for a ll t > 0 , x E

D(A); and so T(t)

ca n be extended to a unique operator

T(t) E

£(X)

satisfying IIT(t)II < 1, for all

t >_ 0.

Take x E X, and (x

n

,)>o C D(A),

such that

x,,, — x

as

n

–^ oo. We have

IITA (t)x –T (t)xII < IITA (t)x –T A (t)x

f

hj + IITA (t)x

n

–T(t)xIj

+ IIT(t)xn

–

(t)x I

<

2

11x. – xjI

+ IITa(t)xn

–

T(t)x,,Il ;

and so TA (t)x --> T ( t)x

as A j 0 uniformly on [0, T] for all

T > 0. Proper-

ties (3.1) and (3.2) follow. To show (3.3), it suffice s to rem ark that

TA(t)TA(s) _

TA (t + s), and so

IIT(t)T(s)x – T(t

+ s)xlI <

IIT(t)T(s)x – T(t)TA (s)xjI

+ (IT(t)TA (s)x – TA (t)TA

(s)xjj

+ IITA(t + s)x

– T(t + s)xI I.

It follows tha t

IIT(t)T(s)x – T(t +s)xII —;0

as A J 0.

Step 4. Returning to the case in which u E

D(A), set va(t) = AATA(t)x =

TA(t)Aax = u'(t). We have

45/198

Two important special cases

35

Hence,

v, --> T (t)Ax

as A .. 0, uniform ly on [0, T] for all

T > 0.

Taking

ua (t) = x +

J O

t

va (s) ds,

and letting .\

1 0, it follows tha t

f o

t

u(t) = x +

T(s)Ax dx.

Thus u

E

C l

([0, oo), X), an d

u (t) = T(t)Ax,

3.9)

for a ll

t > 0.

Finally, we ha ve

v(t) = A(JATA(t)x),

and

IJATT(t)x —T(t)xH) < IITA,(t)x —T(t)xjj + IIJaT(t)x — T(t)xjj.

Therefore, (J

a

T

a

(t)x, A(J

a T

A

(t)x)) --3(T(t)x,T(t)Ax) in

X x X a s A 1

0. Since

G(A)

is closed, it follows that

T(t)x

E

D(A) for all

t > 0, and

AT(t)x = T(t)Ax,

hence (3.7). We conclude that

u

E

C([0, oo), D(A)).

P utting together (3.7)

and (3.9 ), we obta in (3.5).

Step 5.

Uniquene ss of the solution of (3.4)— (3.6). Let

u be a solution, and let

T

> 0. Set

v(t) = T(r — t)u(t),

for

t

E

[0,r]. We have v

E

C([0,t],D(A))

nC

l

( [

0,t],X),

and

v'(t) = —AT(r — t)u(t) +T(rr — t)u'(t) = T(r — t)[u (t) — Au(t)) = 0,

for a ll

t

E

[0, r ]. H enc e, v(r ) = v(0), and so

u(r) =

T(T)x.

r >_ 0 be ing arbitrary,

the proof is complete.

3 .2. Two im portant special cases

W e a ssume in this section that X is a rea l Hilbert spac e. The following result

sharpens the conclusions of Theorem 3.1.1.

Theorem 3.2.1.

Assume that

A

is self-adjoint < 0.

Let x

E

X, and let

u(t) = T(t)x. Then

u is

the unique solution

of the fo l lowing problem:

u E

C([0, oo),

X) n

C((0, oo),

D(A))

n C

1 ((0, oo), X);

3.10)

u'(t) = Au(t), Vt > 0;

3.11)

u(0) = x.

46/198

36 The Hille-Yosida-Phillips Theorem and applications

In addition, we have

I I

Au(t)

I I

< - 1

1 x I I ;

3.13)

- (Au(t), u(t)) <

2t

I I x I I 2 •

3.14)

Finally,

Au(t)

1 1 2 <

2t

1

(Ax, x),

3.15)

if x

E

D(A).

Proof.

We easily verify that A. is self-adjoint < 0, for all A > 0. If

u(t) =

T.\(t)x,

the func tions J jua(t)II and IIua(t)II are non-incre asing with respe ct to t.

In addition, we ha ve

d

ua(t)11 2

= 2(Aaua(t),ua(t)),

3.16)

dt(A,\u),(t),ua(t)) =

2

(Aaua(t),ua(t))

= 2IIua(t)II•

3.17)

Fr om (3 .17), it follows that

-(Axua(t), ua(t))

is non-incre asing with respec t to

t. Integra ting (3.16) betwee n 0 a nd t > 0, it follows that

-

t(Aaua(t),ua(t))

< - f A),ua(s),ua(s))ds <

2 I x 1 1 2 .

3.18)

0

Integra ting (3.17), we obtain

2tIIua(t)II


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Two important special cases 37

addition (by (3.14)), for a ll t > 0, IIAAJ au a(t)I ) is bounde d a s .1 j 0. The refore

u(t) E D(A), for a ll t > 0, with

A u(t) = 11i

a

A,\J.,ua(t),

in X weak. (3.10), (3.11), and (3.12) now follow from Theorem 3.1.1, and

(3.13), (3.14), and (3.15) ar e obta ined by pa ssing to the limit in (3.20), (3.18),

and (3.19).

It remains to show the uniqueness of

u. To do this, take t

> 0 and 0 0,

which completes the proof.

Rem ark 3.2.2. Theorem 3.2.1 m ea ns that T(t ) has a smoothing ef fect on the

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