Fattorini H.O., Kerber A. The Cauchy Problem

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Preface

xxi

self-contained with the exception of numerous results on Sobolev spaces and

two on singular integrals, which are stated without proof in Chapter 4. Some

distribution theory is used through the text; the few facts on vector-valued

distributions needed in Chapter 8 are presented there with complete proofs.

The contents of this book can be described as follows. Chapter 1 consists

almost exclusively of examples drawn from problems of mathematical

physics; the resulting equations are treated by ad hoc methods (Fourier

series and transforms) and the results provide motivation for the definition

of well-posed Cauchy problem. The resulting theory is examined in Chapter

2. Chapter 3 discusses the particular case corresponding to dissipative

operators and some related facts (such as semigroups in Banach lattices)

with applications to second order ordinary differential operators and sym-

metric hyperbolic equations. Chapter 4 is on abstract parabolic equations

(chiefly the analytic case) and on applications to second order parabolic

equations. Chapter 5 deals with perturbation theory; the applications

include the neutron transport equation and the Schrodinger and Dirac

equations with potentials. Other topics include continuous and discrete

approximations to abstract differential equations, among them finite dif-

ference methods, which are illustrated with a parabolic initial-boundary

value problem. Further considerations on the idea of well posed problem

are found in Chapter 6, where formulations different from that of Cauchy

problem are introduced in several examples. The theory of the equation (2)

with A depending on t is the subject of Chapter 7. Finally, we present in

Chapter 8 a brief account of the theory of the Cauchy problem in the sense

of distributions. Here the restriction to purely differential equations of the

type of (2) is unreasonable and we consider instead hereditary equations;

relations with system theory are pointed out. In the case of the equation (2)

this formulation is seen to be equivalent to the so-called mildly well-posed

Cauchy problem, brought into existence as a tool for the treatment of

hyperbolic equations with multiple characteristics.

Some shortcuts through the text will be evident to the reader. For

instance, Chapter 1 may be passed over except for the definition of

well-posed Cauchy problem and for some results on self-adjointness of the

unperturbed Schrodinger and Dirac operators to be used in Chapter 5.

Chapters 2, 3, and the first two sections of Chapter 4 are basic for the

understanding of most of the subsequent material, as are Sections 1 and 3 of

Chapter 5; the rest of these chapters and the remaining ones are fairly

independent of each other.

Numerous paragraphs labelled "Example" can be found throughout the

text. Some of these are worked out in detail; others are not and should be

understood as exercises. The most difficult ones are starred and references

are given.

Sections and paragraphs in small type can be omitted without detriment

to the comprehension of subsequent material.

xxii

Preface

Each chapter ends with "Miscellaneous" comments; these give historical

information, discuss parts of the theory not treated in detail, and provide

bibliographical indications.

The selection of topics reflects of course the tastes and limitations of the

author. Without a doubt, the most important subject we have not covered is

that of nonlinear equations (although some references to quasilinear equa-

tions are found in Chapter 2). Even within the field of linear equations

much has been omitted or is barely mentioned in passing. Some of these

topics are: applications of semigroup theory to probability, random evolu-

tions and differential operators of order greater than two, as well as the

theory of the equation (2) not directly associated with the well posed

Cauchy problem.

I am happy to acknowledge my indebtedness to numerous individuals

and institutions in relation to the writing of this book. A set of lecture notes

that became part of Chapters 2, 3, 4, and 5 was written during 1972 and

1973 at the Universidad de Buenos Aires, in preparation for a projected

course on applications of functional analysis. My thanks go to my col-

leagues there for making possible a fruitful stay, which would not have been

viable without the support of the Consejo Nacional de Investigaciones

Cientificas y Tecnicas of Argentina. Likewise, I am grateful to my col-

leagues at the University di Firenze and to the Consiglio Nazionale delle

Richerche of Italy for their arrangements for my visit during the spring of

1975, at which time the actual writing began.

Finally, I wish to point out that the completion of this book would have

been impossible without the effective and understanding support of the

National Science Foundation given during the entire period of its prepara-

tion.

To my wife Natalia go my sincere thanks for her patience and under-

standing, as well as for her constant encouragement.

H. 0. FATTORINI

The Cauchy Problem

Chapter 0

Elements of Functional Analysis

1. BANACH AND HILBERT SPACES. BOUNDED OPERATORS.

INTEGRATION OF VECTOR-VALUED FUNCTIONS

We shall denote by E, F, X, Y... etc. real or complex Banach spaces; a

Hilbert space will be named H. Unless otherwise stated all spaces will be

assumed complex. We denote by I I - I I

the norm in a Banach space E, by

(., -) y the scalar product in a Hilbert space H, and by (E; F) the space of

all linear bounded operators from E into F endowed with the norm

IIAII = sup{ II AuII F; II uII E < l) (the "uniform operator norm"). Clearly we

have

II AuII F _< IIAII(E; F)IIuIIE;

also, if A E (F; X) and BE (E; F), then

ABC= (E; X) and IIABII(E;

x) :! IIAII(F; X)IIBII(E; F). When

confusion is not

likely (that is, nearly always), we shall drop the subindices from norms and

scalar products; also (E; E) sometimes will be abbreviated to (E).

The following two results are basic.

1.1 Banach - Steinhaus Theorem.

Let (A,,) be a sequence in (E; F)

such that (a) I I A

I I <_ C for all n. (b) lim An u exists for u in a dense subset of E.

Then Au = lim Anu exists for all u E E, A E (E; F), and IIAII _< liminflIAnii

1.2 Uniform Boundedness Theorem.

Let

be a subset of (E; F)

such that (Bu; B (=- i) is bounded in F for each u E E. Then

I I B I I _< C

(B E 61); that is, % is bounded in (E; F).

Elements of Functional Analysis

For proofs see Hille-Phillips [1957: 1, pp. 41 and 26].

We recall that invertible operators are an open subset of (E; F). In

fact, if A E (E; F) possesses an inverse A -1 E (F; E), then so does every

operator B E (E; F) with IIB - All < IIA-'II-' and

B-'=A-1 E ((A-B)A-I)ne(F;E), (1.1)

n=0

the series convergent in the norm of (F; E) in the range of B indicated

above.

Given A E (E), the resolvent set of A (written p or p(A)) is the set of

all complex A such that XI - A (I the identity operator in E) has an inverse

R (X) = (XI - A) - I E (E), which is called the resolvent or resolvent operator

of A. The complement a = a(A) is the spectrum of A. As a consequence of

our previous observations on inverses, we deduce that p(A) is always open;

precisely, if A E p(A), then every it with lµ - Al < IIR(A)II-' belongs to p(A)

and

R(IL) = E (X

-µ)nR(X)n+I

(1.2)

n=0

the series convergent in the indicated range; moreover, p(A) contains the set

(A; IAI > IIAII) and

R(A) =E X-(n+I)An.

n=0

(1.3)

In particular, p(A) is never empty and a(A) is compact. It can be shown

(Hille-Phillips [1957: 1, p. 125]) that a(A) is nonempty as well.

If E is a real Banach space, all the previous statements have an

immediate counterpart, except that a(A) may be empty (this can be

exemplified by a real matrix without real eigenvalues).

Back in the general case, let f be a function defined in a compact

interval a< t < b and taking values in E. If f is continuous (or only

piecewise continuous), the Riemann integral off can be defined and shown

to exist in the same way as in the scalar case as a limit of Riemann sums,

Ibf(t)dt=lim E (tk-tk-I)f( k),

k=1

the limit understood in the norm of E. Integrals over multidimensional sets

are similarly defined, and improper Riemann integrals (that is, integrals

over noncompact sets) are obtained in the usual way as limits of integrals on

a convenient sequence of compact subsets. All the properties of ordinary

Riemann integrals that make sense in the vector-valued setting can be

proved by rather trivial modifications of the classical proofs. An important

existence criterion for improper integrals (which, to fix ideas, we state only

1. Banach and Hilbert Spaces. 3

for the real line) is the following: if J is a (finite or infinite) interval in

= (- oo, oo) and f is an E-valued piecewise continuous function defined in

J and such that f( - )I I is integrable in J, then f is integrable over J and

f f(t)dt

<f IIf(t)Ildt.

(1.4)

This result holds as well, of course, in multidimensional regions. We note in

passing that the fundamental theorem of calculus holds: if f is an E-valued

continuous function defined in a < t < b and

g(t) = f

tf(s) ds,

(1.5)

then

g'(t)=f(t) (a<t<b), (1.6)

the derivative understood as the limit in E of the corresponding quotient of

increments. Conversely, if g is continuously differentiable in a< t:!- b with

derivative f, then g is given by (1.5) modulo a constant (that is, modulo an

element of E that does not depend on t).

Let A(-) be a function defined in an interval J of the real line with

values in (E; F). If A is continuous, the previously outlined integration

theory can be automatically applied to it. However, we shall have to

contend most of the time with functions that are merely strongly continuous,

that is, such that t -> A(t)u is continuous in F for each u E E. The integral

can then be defined "elementwise"; that is, although fA(t) dt may not make

sense, fA(t)udt does for every u E E. We state the following useful result,

restricted for definiteness to the real line.

1.3 Lemma. Let be strongly continuous in an interval J. As-

sume that I I A (- )I

is integrable in J. Then the operator defined by

Au= f A(t)udt (1.7)

belongs to (E; F); precisely,

IIAII < f IIA(t)II dt.

(1.8)

The proof is an immediate consequence of (1.4) and the preceding

comments.

We note in passing that the integrand in (1.8) is not necessarily

continuous under the present

assumptions.

However,

IIA(t)II =

sup(IIA(t)uII;IIuII <1), where each t- IIA(t)uII is continuous, so that

is lower semicontinuous. On the other hand, if J' is a compact subinterval of

J, is continuous, hence bounded in J' for each u E E. It follows

4 Elements of Functional Analysis

from the uniform boundedness theorem (1.2) that IIA(.)II is bounded in J'.

This gives sense to the integrability assumption in Lemma 1.3.

A rather complete exposition of the theory of Riemann (or, more

generally, Riemann-Stieltjes) integration of vector-valued functions can be

found in Hille-Phillips [1957: 1, Sec. 3.2] and in Hille [1972: 1, Sec. 7.4]. We

shall also use a few times the theory of Lebesgue integrals (as generalized by

Bochner) of vector-valued functions, which is in Hille-Phillips [1957: 1, Sec.

3.7] and Hille [1972: 1, Sec. 7.5]. We note that both integration theories can

be made to function equally well in a real or in a complex space.

2. LINEAR FUNCTIONALS: THE DUAL SPACE.

VECTOR-VALUED ANALYTIC FUNCTIONS

Using the notation in the previous section we define E*, the dual of E, as

E* = (E; C) (C the complex numbers). The elements of E* are the bounded

linear functionals in E. Application of a functional u* E E* to an element u

of E will be denoted (u*, u), u*(u), or (u, u*). It is clear from the

definition of the norm in E* that the "Cauchy-Schwarz inequality"

Ku*, u)I:5 Ilu*II Hull

(2.1)

holds for all u E E, u* (-= E*.

The existence of abundant linear functionals in E is assured by the

2.1 Hahn - Banach Theorem. Let N be a subspace of E, f : N -C a

linear functional such that I f(u)l < Cllull (u E N). Then there exists a u* E E*

such that (a) u*(u) = f(u) (u E N). (b) Ilu*II

Theorem 2.1 is actually a very particular case of the general

Hahn-Banach theorem whose proof can be found in Banach [1932: 1, p. 27]

or Hille-Phillips [1957: 1, p. 29]. We note that the Hahn-Banach theorem

holds also in real spaces (actually, this was the case originally proved by

Banach).

2.2

Corollary.

Let N be a closed subspace of E, u E. Then there

exists a functional u* E E* such that u*(N) = (0), u*(u) = 1,

llu*Il =1/d,

where d = dist(u, N) > 0.

The proof is immediately obtained by defining a functional f in the

subspace (w = v + Au; v EN,XEC)generated by Nand uby f(v+Au)=A

and applying Theorem 2.1 (see Hille-Phillips [1957: 1, p. 30]). Taking N

itself equal to (0), we obtain

2.3 Corollary. Let u E E. Then there exists a u* E E* with

(u*, u)

= IIu112 = Ilu*1l2.

(2.2)

2. Linear Functionals: The Dual Space. Vector-Valued Analytic Functions

By virtue of the obvious bilinearity of the function (u*, u) -* (u*, u),

an element u E E gives rise to a u** E E** = (E*)* through the formula

(u**, u*) = (u, u*). (2.3)

It is clear that the map u - u** from E into E** thus obtained is linear;

moreover, it follows from (2.1) that Ilu**IIE** <_ IIUIIE, and from (2.2) that we

actually have IlU**IIE** = Ilull We thus see that E can be identified (linearly

and metrically) with a subspace of E**; in a less precise fashion we may say

that "E is a closed subspace of E**." If E = E**-that is,

if every

continuous linear functional in E* can be expressed by the formula (2.3) for

some u E E -we say that the space E is reflexive.

Let 0 a domain (open connected set) in the complex plane, and f()

a function defined in I with values in E. We say that f is analytic, or

holomorphic, in 9 if the derivative

exists (as the limit in the norm of E

of the corresponding quotient of increments) for all E Q. We can also

define analyticity by means of linear functionals as follows: f is analytic in a

if and only if the scalar function (u*, f(.)) is analytic in 9 for all u* E E*.

Clearly the first definition implies the second. Remarkably enough, the two

are actually equivalent. If A(.) is an (E; F)-valued function defined in 0,

the first definition of analyticity (existence of the limit of the quotient of

increments in the norm of (E; F)) is likewise equivalent to the statement

that (u*, A(.)u) is an ordinary analytic function for any u E E and any

u* E F* (for a proof see Hille-Phillips [1957:

1, p. 93]). Vector-valued

analytic functions share most properties of ordinary analytic functions.

Roughly speaking, the proofs are carried out by imitating the proof for the

scalar case (replacing when appropriate modulus by norm) or by applying

functionals and then using the scalar theory. We state a few facts that will

be of use later.

2.4

Lemma.

Let f be a E-valued analytic function in Q. Then (a) if F

is a simple closed contour such that F and its interior are contained in 9, then

f ("l (z) 2

! f(( f

)n +

(2.4)

7Ti

if z belongs to the interior of r and IF is oriented counterclockwise. (b) If z E S2

f can be developed in power series about z,

fM = E

(2.5)

converges absolutely and uniformly

in l - zl < d' if d' < d = dist(z, boundary of 0) (that is, EllanlO- zl"

converges uniformly in l - zI < d').

Elements of Functional Analysis

2.5 Lemma. An arbitrary power series Ean( - z)" with coefficients

in E converges absolutely and uniformly in I - zI < d' if d' < r,

1/r = limsupllanll1/"

(2.6)

and diverges if I - zj > r. The function

Ea"( - z)" is analytic in

I - zI < r and the successive derivatives are obtained by differentiation of (2.5)

term by term.

For the proofs of these and other results the reader may consult

Hille-Phillips [1957: 1, Sec. 3.11]. We note the following interesting conse-

quence of Lemma 2.5. Let A E (E). Define r = r(A), the spectral radius of

A, by r = sup(IXI; A E o(A)). It follows from (1.2) that R(A) is analytic in

p (A ). Applying then Lemma 2.4 to R(1/A), we deduce that (1.3) must

converge for IAI > r; thus it results from Lemma 2.5 that

r(A) = limsupllAnlll/"

(2.7)

We note finally that the limsup on the right-hand side of (2.7) is actually a

limit; this follows from Pblya-Szego [1954: 1, p. 17, Nr. 98] and from the

obvious inequality IIAm+nll _< IIAIImIIAII"

A thorough treatment of the theory of vector-valued analytic func-

tions can be found in Hille-Phillips [1957: 1, Ch. 3] and also in Hille [1972:

1, Ch. 8]. We note in particular that the facts concerning Laurent series and

development in power series about singular points extend to the present

case.

3. UNBOUNDED OPERATORS;

THE RESOLVENT. CLOSED OPERATORS

We shall deal most of the time with operators (e.g., differential operators)

that are neither bounded nor everywhere defined. We extend then our

previous definition of linear operator by allowing A to be defined in a

subspace D(A) of a Banach space E and to take values in another Banach

space F. Only the case E = F will be used. The subspace D(A) is called the

domain of A; in most instances D(A) will be dense in E (we say then that A

is densely defined) but not always. The sum of two operators is defined by

(A + B)u = Au + Bu with D(A+ B) = D(A)f1 D(B) while the product or

composition is (AB)u = A(Bu), where D(AB) consists of all u e D(B) with

Bu E D(A). (Note that, even if A and B are densely defined, A + B and AB

may not be; in fact, both D(A + B) and D(AB) may reduce to (0).) If A is

bounded (IIAull -< CIIuII, u E D(A)) and densely defined, we can extend A by

continuity to all of E; we shall then usually assume that all bounded

operators are everywhere defined.

An intermediate notion between that of bounded operator and the

general definition introduced in this section is that of closed operator. An