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

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3.8. Miscellaneous Comments

167

before for partial differential operators by Friedrichs [1954: 1], [1958: 1],

Phillips himself [1957: 11, and others; we note that Friedrichs coined the

word accretive to indicate the negative of a dissipative operator. The bulk of

the theory of dissipative operators in Hilbert space as presented in this

chapter can be found in Phillips [1959: 1]. Dissipative operators in Banach

spaces were introduced by Lumer and Phillips [1961: 1] in a slightly

different form using the "semi-inner products" introduced by Lumer [1961:

1] instead of duality maps; the definition was cast in the present form by

Nelson [1964: 1]. The term "duality map" is taken from Goldstein [1970: 5];

the present treatment of dissipativity leans heavily on these lecture notes.

We note a definition of dissipative operators in Banach space due to Maz'ja

and Sobolevskii [1962: 1] under certain smoothness conditions on the norm.

The results in Section 3.1 are due to Lumer-Phillips [1961: 1] or are

adaptations of earlier results of the Hilbert space theory. Theorem 3.6.7 is

the theorem of Stone mentioned in Section 2.5, whereas Theorem 3.6.6 was

discovered by J.

L. B. Cooper [1947:

1]; both results were originally

formulated in the semigroup-generator language. The version of Cooper's

theorem for Banach spaces presented here (Theorem 3.6.1) seems to be a

folk result. A related result can be found in Papini [1971: 1].

The present theory of semigroups of positive operators in Banach

lattices is due to Phillips [1962: 1]; dispersive operators were introduced by

W. J. Firey in a particular instance (see Phillips [1962: 1, p. 295]). Some of

the ideas in Section 3.7 have interesting finite-dimensional ancestors; we

refer the reader to Bellman [1970: 1, Ch. 14].

We comment finally on the results on ordinary and partial differen-

tial operators in this chapter. The theory of symmetric hyperbolic operators

in Section 3.5 is due to Friedrichs [1944: 11. His result on identity of Coo (the

strong extension of Coo) and (d')* (the weak extension of do) has been

extensively generalized (for instance, to situations involving boundary con-

ditions as in Lax-Phillips [1960: 1]). We note that assumption (3.5.12) on the

function p is nothing but Wintner's condition in [1945: 1] for global

existence of solutions of the ordinary differential system x'(t) = a(x(t)),

where is a m-dimensional vector function, maps IJ m into itself, and

I a (x) I < p (I x 1). This is not surprising since it has been shown by Povzner

[1964:

that, under the assumption that EDkak=0, global existence

is equivalent to the identity of the weak and strong extensions of

610 = Eak(x)Dk; many other related results are proved there. For connected

material see Lelong-Ferrand [1958: 1].

One of the results in the investigations of Weyl [1909: 1], [1910: 1],

[1910: 2] on the spectral theory of ordinary differential operators in infinite

intervals can be stated in modern terminology as follows: the operator

Aou= (p(x)u')'+q(x)u (3.8.3)

in x >, 0 (with domain and smoothness assumptions as in Section 3.3) has a

168 Dissipative Operators and Applications

self-adjoint extension in L2(0, cc) obtained by imposition of a boundary

condition at zero if

p(x)>0,q(x)<w (-oo<x<oo).

(3.8.4)

(This corresponds to what Weyl calls the limit point case; in the limit circle

case, to obtain a self-adjoint extension we must impose in addition a

boundary condition at infinity of the form

lim

(wo(x)u(x)+w,(x)u'(x)) = 0

x-+oo

for suitable functions wo, w1.) Under conditions (3.8.4), the extension is

semibounded above; weaker conditions on q, even if pertaining to the limit

point case, produce self-adjoint extensions not necessarily semibounded

above (for a complete account of the theory see Dunford-Schwartz [1963: 1,

Ch. 13]). Roughly along the same lines, Yosida [1949: 2] and Hille [1954: 2]

considered a general non-self-adjoint A0 in relation to applications to

unidimensional Markov processes. The problem is that of obtaining exten-

sions of the operator A0 in Lt and in spaces of continuous functions by

means of boundary conditions at finite points and at infinity; these exten-

sions are required to be infinitesimal generators of positive semigroups. The

works of Feller [ 1954: 11, [1954: 2], [1955: 1], [1955: 2], [1956: 11, and others

proceed in the same direction but use extremely weak assumptions on

the operators (see Mandl [1968: 1] or Ito-McKean [1974: 1] for modern

treatments). The results in Sections 3.2 and 3.3 concentrate only on the

simpler problem of obtaining m-dissipative extensions of A0 imposing no

boundary conditions at infinity in the case - oo < x < oo or only one

boundary condition at 0 in the case x >, 0. The generation properties of

ordinary differential operators in a finite interval were investigated by

Mullikin [1959: 1] in LP and C spaces; his results are more general than

those in Section 3.4 since dissipativity conditions like (3.4.8) on the boundary

conditions are absent, and will be met again (this time in almost full

generality) in Section 4.3.

Many papers on the theory and applications of dissipative operators

will be found later under other headings. Here we refer the reader to

Hasegawa [1966: 1], [1969: 1], [1971: 1], Lj ubic [1963: 11, [1965: 1], Mlak

[1960: 1], [1960/61: 11, [1966: 11, Moore [1971: 1], [1971: 2], Nagumo [1956:

1], Olobummo [1963: 11, [1967:

1], Olobummo-Phillips [1965:

1], Papini

[1969: 1], [1969: 2], [1971: 1], Phillips [1957: 1], [1959: 1], [1959: 3], [1961:

1], [1961: 2], and Crandall-Phillips [1968: 1].

(a) Special Semigroups. Partly along the lines of Stone's theorem,

a large number of results dealing with semigroups

where each

S(t) belongs to certain classes of operators in Hilbert or Banach spaces

have been investigated. For S(t) self-adjoint we have already pointed

out in Section 2.5 the works of Hille [1938: 1] and Sz.-Nagy [1938: 1]; an

3.8. Miscellaneous Comments

169

extension to unbounded self-adjoint

operators has been given by

Devinatz [1954: 1]. Normal operators are treated in Sova [1968: 4],

and spectral operators in McCarthy-Stampfli [1964: 1], Berkson [1966: 11,

Panchapagesan [1969: 1], and Sourur [1974: 1], [1974: 2]. For generalized

spectral operators in the sense of Colojoara-Foias [1968:

1], see Lutz

[1977: 1].

Other works deal with, for instance, semigroups or groups of isom-

etries in special spaces like LP or H. See Berkson-Porta [1974: 1], [1976: 1],

[1977: 1], Berkson-Kaufman-Porta [1974: 1],

Berkson-Fleming-Jamison

[ 1976: 1 ], and Goldstein [ 1973: 1 ]. Stone's theorem has also been generalized

in the direction of allowing the parameter t to roam over groups more

general than the real line. See Salehi [ 1972: 1 ].

(b) Contraction Semigroups and Dissipative Operators in Hilbert

Spaces.

Let H+ be a Hilbert space, H a closed subspace of H+, P the

orthogonal projection of H+ into H. If B+ is an operator in H+ we define

prB+, the projection of B+ into H by

Bu=(prB+)u=PB+u (uEH).

It was proved by Sz.-Nagy [1953: 1], [1954: 11 that if S(.) is a strongly

continuous semigroup in a Hilbert space such that

IIS(t)II < 1

(3.8.5)

for t > 0, then there exists a second Hilbert space H+ having H as a closed

subspace and a strongly continuous group U(.) of unitary operators such

that

S(t) = prU(t) (t > 0). (3.8.6)

Moreover, the space H+ is minimal (in the sense of being generated by all

elements of the form U(t) u, u E H) and uniquely determined (up to isomor-

phisms). For additional details, see Sz.-Nagy-Foias [1966: 1]. We note that

the result above was proved by Cooper [1947: 1] in the particular case where

each S(t) is an isometry; in this setting, the theorem is a consequence of the

well known fact that every symmetric operator A in H can always be

extended to a self-adjoint operator in a larger space H.

Sz.-Nagy's result combined with Stone's theorem shows that (modulo

the projection P) the study of strongly continuous contraction semigroups

in a Hilbert space can be reduced to that of operator-valued functions of the

form exp(itA), A self-adjoint, with a corresponding relation between their

generators (maximal dissipative operators) and self-adjoint operators. We

refer the reader to Sz.-Nagy and Foias [1969: 1] for a thorough exploitation

of this line of approach and note the following discrete version of (3.8.6)

also due to Sz.-Nagy [1953: 1]: if S is an operator in H with IISII 51, then

there exists H+ containing H as a closed subspace and a unitary operator U

in H+ such that

S"=pr U" (n> 0). (3.8.7)

170

Dissipative Operators and Applications

The space H+ and the operator U enjoy the same properties of minimality

and uniqueness up to isomorphism as in the continuous case.

Hilbert Space. In view of the results in the previous subsection, the

following question is of considerable interest. Let H be a Hilbert space, S(.)

a semigroup (group) of linear bounded operators in H such that

IIS(t)II <C

(3.8.8)

in t >, 0 (in - oo < t < oo). Can H be given an equivalent Hilbert norm in

which (3.8.5) holds in t >, 0 (in - oo < t < oo)?

For a group, the answer is in the affirmative, as shown by Sz.-Nagy

[1947: 1]. We note that, since S(.) satisfies (3.8.5), each S(t) is invertible

and isometric, hence unitary in the new norm. Sz.-Nagy also proves the

discrete version of the result above: if S is an operator such that

11S"11 < C (n =...,-1,0,1,...), (3.8.9)

then a similar renorming of the space makes S unitary. The corresponding

theorems for uniformly bounded semigroups in t > 0 are false: in the

discrete case, a counterexample was given by Foguel [1964: 1] and in the

continuous case by Packel [1969: 11. See Halmos [1964: 1] for additional

information. An analogue of Sz.-Nagy's result for cosine functions was

proved by the author [1970: 4]: under the hypothesis that

is uniformly

bounded in - oo < t < oo (or, equivalently, in t e 0), an equivalent Hilbert

norm can be found in which each C(t) is self-adjoint. A discrete analogue

holds for uniformly bounded families of operators (Cn) satisfying Co = I,

C. + Cn_m = 2CnCm (m, n =...,-1,0,1.... ). Sz.-Nagy's theorem has been

generalized by Dixmier [1950: 1] to uniformly bounded representation of

dertain groups into the algebra of linear bounded operators in a Hilbert

space. A similar generalization of the author's result was given by Kurepa

[1972: 1].

We note that the problem of renorming a Banach space with an

equivalent norm, which improves (3.8.8) into (3.8.5), is rather trivial; see

Example 3.1.10.

(d) Truly Nonlinear Equations and Semigroups.

The theory of con-

traction semigroups, and of dissipative operators as their infinitesimal

generators, has been generalized in the last two decades to nonlinear

operators. In the nonlinear realm, a contraction semigroup is a family

(S(t); t > 0) of nonlinear maps defined in a Banach space E or a subset X

thereof and satisfying the semigroup equations (2.1.2). The nonlinear coun-

terpart of (3.8.5) is

IIS(t)u-S(t)vll<Ilu-V11

(u,v(=-X),

while the dissipativity condition (3.1.5) reads

Re(w*,Au-Av)<0 (w*EO(u-v)).

3.8. Miscellaneous Comments

171

The relation between semigroup and generator roughly parallels the linear

case, but many interesting new phenomena appear (for instance, the infini-

tesimal generator may be multiply valued). For an account of the theory

and the applications we refer the reader to Barbu [ 1974: 1 ], Brezis [ 1973: 11,

Cioranescu [1973: 1], and Pazy [1970: 1]. A short exposition of some of the

central points of the theory can be found on Yosida [1978: 1].

Chapter 4

Abstract Parabolic Equations:

Applications to Second Order

Parabolic Equations

An operator A E C+ is abstract elliptic if the solutions of the differen-

tial equation u'(t) = Au(t) are infinitely differentiable in t > 0. Partic-

ularly important examples of abstract elliptic operators are those in

the class 6, where solutions can be analytically extended to a sector

about the positive real axis. The class d is characterized in the first

two sections of this chapter; the rest of the material is on applications

to differential operators. Section 4.3 deals with the ordinary differen-

tial operators in compact intervals last seen in Section 3.4; in Sections

4.4 to 4.10 we study second order partial differential operators in

smooth domains of Euclidean space, beginning with dissipativity and

assignation of boundary conditions. Extensions in 0Y, of differential

operators in various spaces are then constructed. We treat Lz spaces

in Sections 4.6 and 4.7, and LP spaces and spaces of continuous

functions in Section 4.8. Section 4.9 is devoted to the proof that the

LP extensions belong in fact to the class a (with attendant smoothness

results for solutions of the equation u'(t) = Au(t)) and to the removal

of certain dissipativity assumptions on the boundary conditions; these

assumptions are only significant in L' and in spaces of continuous

functions and play a strictly auxiliary role in other spaces. Finally, we

study in Section 4.10 compactness of families of solutions and pre-

servation of positivity of initial data.

For the ordinary differential operators in Section 4.3, the

treatment is essentially the same in all spaces; in contrast, the Lz case

172

4.1. Abstract Parabolic Equations

173

for partial differential operators is more elementary and is thus

presented first. The reader interested only in the LZ theory may

proceed directly to Section 4.6 from Section 4.2, skipping most of the

intermediate material.

4.1. ABSTRACT PARABOLIC EQUATIONS

Let A be an operator in e and let u(t) = S(t)u be a generalized solution of

u'(t) = Au(t)

(4.1.1)

in (- oo, oo). We know (Section 2.2) that if the initial value u(0) belongs to

D(A), then

will be a genuine solution of (4.1.1); in particular, u(.) is

continuously differentiable and belongs to D(A) for all t. On the other

hand, let u(.) be differentiable at to. Then, as h-'(S(to+h)-S(to))u=

h-'(S(h)- I)S(to)u, it

is immediate that S(to)u E D(A); thus u(0) =

S(- to)S(to)u e D(A) as well. In other words, only solutions that originate

in D(A) may be smooth. This property is based, roughly speaking, on the

fact that the solution operator of (4.1.1) is defined for all t. We study in this

section operators A having properties in a sense opposite to those of

elements of the class e.

The operator A E C'+ is said to be abstract elliptic (in symbols,

A E 13°°) and the equation (4.1.1) abstract parabolic if and only if every

generalized solution of (4.1.1) is continuously differentiable in t > 0 (even if

u(0) does not belong to D(A)). The following result shows that the smooth-

ness properties implied by this definition are stronger than apparent.

4.1.1 Lemma. The following four statements are equivalent:

(a) S( - ) u is continuously differentiable in t > 0 for all u E E.

(b) S( ) u is infinitely differentiable in t > 0 for all u E E.

(t > 0).

(d)

(t > 0).

Proof. It is plain that (b) implies (a) and (d) implies (c). We show

that (c) implies (d). If t > 0, we have S(t)u = S(t/2)S(t/2)u; since A and S

commute, AS(t)u = S(t/2)AS(t/2)u E D(A) and thus S(t)u E D(A2). The

general statement is obtained on the basis of the identity S(t) = S(t/ n)".

We show next the implication (a) - (c). Let 0 < a < $ < oo and

consider the operator

u- (S(t)u)'='Su

from E into the space C = C(a, /3; E) of all continuous E-valued functions

ft-) defined in a < t < /3, endowed with the supremum norm. Assume

174

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

u -* u in E and S u = u,,)' - ft-) in C. Then

ftf(s)ds (a<t<f3),

so that letting n - oo, we obtain Su = (S(t) u)'= f(t). This shows that 5 is

closed, hence bounded because of the closed graph theorem. But if u E D(A),

(S(t) u )' = AS(t) u; thus AS(t) is bounded and it results from the closedness

of A that S(t) c D(A) for a < t < ft. Since a and $ are arbitrary, (c) follows.

We prove finally that (b) is a consequence of (d). To this end we

observe that if u E D(A°°), then (S(t)u)'= AS(t)u = S(t)Au, which is

again continuously differentiable; then (S(t) u )" = (S(t)Au)' = S(t)A2 ,

and so on. It follows that S(.) u is infinitely differentiable; making use of (d)

and writing S(t) = S(t - to)S(to)u for t > to, we see that (b) holds. This

ends the proof.

We have proved as well the fact that AS(t) = S'(t) is a bounded

operator and that t - AS(t) is strongly continuous for t > 0. Clearly the

same is true of AS(t) = S(t - to)(AS(to/m))'. We define

y(t)

= IIAS(t)II = IIS'(t)II

(t > 0). (4.1.2)

Because of the strong continuity of AS(-) and of the uniform boundedness

theorem, y is bounded on compacts of t > 0. The same is true of

Ym(t)=IIAmS(t)II=IIS(m)(t)II

(t>o)

foranym>,l.If0<a<t<t'<fi<oo,

IIs(m)(t')u-s(m)(t)oll < f

ills(-+')(s)ullds<C(t'-t)IIuII,

where C only depends on a, $ so that t - S(m)(t) is continuous in t > 0 as a

(E)-valued function. Since, on the other hand, we have

S(m)(t')-s(m)(t) = tt's(m+')(s) ds

(as can be seen applying the operators on both sides to an arbitrary element

u (=- E), the following result is immediate.

4.1.2 Corollary.

Let A E (2°0. Then t - S(t) E (E) is infinitely dif-

ferentiable in t > 0 and S(-)(t) = AmS(t) (t > 0).

We examine more closely the growth properties of the function y. In

view of the inequality IIAS(t)II < IIAS(to)II II S(t - to)II, we see that

Y(t)<Cte "t

(t%to) (4.1.3)

if A E C'+(w). In contrast, y may grow arbitrarily fast when t 0.

The following result provides a complete characterization of the class

e°°.

4. 1. Abstract Parabolic Equations 175

4.1.3 Theorem. Let A E (+. Assume that for every a> 0 there exists

/3 = /3(a) > 0 such that p(A) contains the region

R(a,$)=(A; ReX fl-alogIXI)

(4.1.4)

and

IIR(X)II <CIAI"'

(X E SI(a,/3)) (4.1.5)

where m is an integer independent of a. Then A E e°°. Conversely, let A E (200.

Then for every a there exists /3 such that Sl(a, /3) c p(A) and

IIR(X)II <CIXI

(X E Sl(a, 8)). (4.1.6)

Theorem 4.1.3 is a particular case of the much more general Theorem

8.5.1, thus we omit the proof (see also Figure 8.5.1 for a region S2 (a, /3)).

Several subclasses of E °° characterized by the growth of the function

y at the origin have been identified.

4.1.4

Example. Assume y is integrable near zero. Then A is bounded.

In fact, we have

III- XR(X)II <

f °°e-ary(t) dt

for A large enough; for A - oo the right-hand side tends to zero and the

result follows in the same way as Lemma 2.3.5.

An extremely important subclass of E °° (whose characterization will

be completed in the next section) is the class 9 of all A E e°° such that

y(t)=0(t-t)ast-+0. (4.1.7)

It is a remarkable consequence of (4.1.7) that S, which is only assumed to be

differentiable, admits in fact an extension to part of the complex plane as an

analytic function. To simplify the statement of this result (which will be

proved later in this section), we introduce some notations.

Let 0 < qq < 77/2. An operator A E (2+ is said to belong to 6T(99) if S

can be extended to a (E)-valued function defined and analytic in

(here we take arg in (- 77, 77]) and strongly continuous in

Iarg I <9))=7- (()

We note that the strong continuity assumption simply reduces to

lint U.

larglI-< ro,l'-O

It is easy to see by means of standard analytic continuation arguments that

the semigroup equation (2.1.2) holds in 2(q'), that is,

S(z+0=S(z)S(0

(4.1.8)

176

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

In fact, if (2.1.2) holds for s, t >, 0, it will hold for s >, 0,

E I+(qq) (both

sides of (4.1.8) are an analytic continuation to Y.+(T) oft - S(s)S(t)) and

a similar argument takes care of the variable s.

It is a consequence of the uniform boundedness theorem that

I < a. A reasoning based in this and on

(4.1.8) and entirely similar to the one leading to (2.1.3) shows that

Ce"11"1

( (-= 7-(T))

(4.1.9)

for some constants C, w (not necessarily the same in (2.1.3)).

For 0 < q, < 'ir/2 we define

(ro-)= U

d((P').

0,9"<q

A real variable characterization of the classes d ((p -) is made explicit in the

following result.

4.1.5 Theorem. Let C >- 1/e. Assume A E (°° and

II AS(t )II < Ct-1 (4.1.10)

for t near zero. Then

AEQ(p-),

where q = aresin(Ce)-1. Conversely, let A E Q(T -). Then A E (2°° and

(4.1.10) holds near zero.

Let d be the class of all A E C2°° satisfying (4.1.10) near zero. Then

Theorem 4.1.5 can be restated in the single equality

l(q-)=

l(ro)

0,op ,1,/2

0,q,<i/2

have

Proof.

Assume A E d so that (4.1.10) holds for 0 < t < 8 > 0. We

IIS`"'(t)II = IIA"S(t)II

=II(AS(t/n))"II <(Cn/t)"

(n>, t/8,0<t <oo).

(4.1.11)

Accordingly, given t > 0, the coefficients of the power series

S("'(t)

G -

t)"

n=0

(4.1.12)

(except perhaps for a finite number), are bounded in the norm of (E) by

ri1(Cn) "=

(Ce)"(1+o(1))

(n - oo)

(4.1.13)

by virtue of Stirling's formula. Then the series (4.1.12) is actually convergent

in the disk I - t I < t/Ce and defines there a (E )-valued holomorphic

extension of S. Carrying out this construction for all t > 0 and showing in