Analytic Semigroups and Semilinear Initial Boundary Value Problems (London Mathematical Society Lecture Note Series)

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London Mathematical Society Lecture Note Series. 223

Analytic Semigroups and Semilinear Initial Boundary Value Problems

Kazuaki Taira Hiroshima University

AMBRIDGE

UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1995 First published 1995 Library of Congress cataloging in publication data available British Library cataloguing in publication data available

ISBN 0 521 55603 1 paperback Transferred to digital printing 2004

TABLE OF CONTENTS Preface Introduction and Results Chapter I. Theory of Analytic Semigroups

ix 1

8

Generation Theorem for Analytic Semigroups 1.2 Fractional Powers 1.3 The Linear Cauchy Problem 1.4 The Semilinear Cauchy Problem 1.1

Chapter II. Sobolev Imbedding Theorems

8 19 31 38

46

2.1 Holder Spaces and Sobolev Spaces

46 48 74 86

2.2 Interpolation Theorems 2.3 Imbeddings of the Spaces Hm'P(R') 2.4 Imbeddings of the Spaces Hm'P(Sl)

Chapter III. LP Theory of Pseudo-Differential Operators 3.1 Generalized Sobolev Spaces and Besov Spaces

3.2 Fourier Integral Operators 3.2A Symbol Classes 3.2B Phase Functions 3.2C Oscillatory Integrals 3.2D Fourier Integral Operators 3.3 Pseudo-DifferentiaJ Operators

93 93 97 97 99 100 102 102

Chapter IV. LP Approach to Elliptic Boundary Value Problems

109

4.1 The Dirichlet Problem 4.2 Formulation of a Boundary Value Problem 4.3 Reduction to the Boundary 4.4 Operator 17

109

Chapter V. Proof of Theorem 1 5.1 Regularity Theorem for Problem (*)

5.2 Uniqueness Theorem for Problem (*) 5.3 Existence Theorem for Problem (*) 5.3A Proof of Theorem 5.7 5.3B Proof of Proposition 5.10

Chapter VI. Proof of Theorem 2

111

115 120

122 122 126 127 128 134

137 Typeset by ASS-TjX

TABLE OF CONTENTS

vi

6.1 A Priori Estimates 6.2 Generation of Analytic Semigroups

Chapter VII. Proof of Theorems 3 and 4 7.1 Fractional Powers and Imbedding Theorems

7.2 Semilinear Initial-Boundary Value Problems 7.2A Proof of Theorem 3 7.2B Proof of Theorem 4

137 142

148 148 153 153 153

Appendix: The Maximum Principle References

157 159

Index

161

TO MY MOTHER

PREFACE

This monograph is devoted to the functional analytic approach to initial boundary value problems for semilinear parabolic differential equations. First we study non-homogeneous boundary value problems for second-order elliptic differential operators, in the framework of Sobolev spaces of LP style, which include as particular cases the Dirichlet and Neumann problems. We prove that these boundary value problems provide an example of analytic semigroups in the LP topology. The essential point in the proof is to define a function space which is a tool well suited to investigating our boundary conditions. By virtue of the theory of analytic semigroups, one can apply this result to the study of the initial boundary value problems for semilinear parabolic differential equations in the framework of LP spaces. This monograph grew out of a set of lecture notes "On initial boundary value problems for semilinear parabolic differential equations" for graduate courses given at the University of Tsukuba in winter 1994/95. In order to make this monograph accessible to a broad readership, I have tried to start from scratch. In the preparatory chapters, we even prove fundamental results like a generation theorem for analytic semigroups in functional analysis and Sobolev imbeddings theorems in partial differential equations. Furthermore, we summarize the basic definitions and results about the LP theory of pseudo-differential operators which is considered as a modern theory of potentials. The LP theory of pseudo-differential operators forms a most convenient tool in the study of elliptic boundary value problems in the framework of Sobolev spaces of LP style. The material in these preparatory chapters is given for completeness, to minimize the necessity of consulting too many outside references. This makes the monograph fairly self-contained.

This work was begun at the University of Turin and the University of Bologna in May 1988 under the sponsorship of the Italian "Consiglio Nazionale delle Ricerche" and a major part of the work was done at the University of the Philippines in the course of the JSPS-DOST exchange program from January 1989 to March 1989 while I was on leave from the University of Tsukuba. I take this opportunity to express my gratitude to all these

institutions for their hospitality and support. Thanks are also due to the editorial staff of Cambridge University Press for their unfailing helpfulness and cooperation during the production of the book. I hope that this monograph will lead to a better insight into the study Typeset by AMS-TEX

x

PREFACE

of initial boundary value problems for semilinear parabolic differential equations. For probabilistic information on the topics discussed here, I would like to call attention to my previous book Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics, No. 1499, Springer-Verlag, 1991. Higashi-Hiroshima, June 1995 Kazuaki Taira

INTRODUCTION AND RESULTS

Let ci be a bounded domain of Euclidean space Rn, with C°° boundary r; its closure Si = S2 U r is an n-dimensional, compact C°° manifold with boundary. We let

L axix

A=

Eazj(x) axi j=1

=1

+b'(x) axia +c(x) i=1

be a second-order elliptic differential operator with real C°° coefficients on S2 such that:

(1) a"(x) = a3'(x), x E 7',j < n. (2) There exists a constant ao > 0 such that n

2,

az-1

xESZ, ERn.

(3) c(x) < 0 in S2, and c does not vanish identically in 9. We consider the following boundary value problem: Given functions f and cp defined in S2 and on r, respectively, find a function u in S2 such that

(*)

Au = .f

in 52,

Bu := at + bulr =

on r.

Here:

(1) a and b are real-valued, C°° functions on r. (2) a/av is the conormal derivative associated with the operator A: a

av n = (n1, n2,

,

= i

1

a zj

n.i-, axi

nn) being the unit exterior normal to the boundary r (see Typeset by AMS-T 1

INTRODUCTION AND RESULTS

Figure 1).

Figure 1

We remark that if a- 1 and b=O on r(resp.a=0 and b- 1onI'), then the boundary condition B is the so-called Neumann (resp. Dirichlet) condition. It is easy to see that problem (*) is non-degenerate (or coercive)

if and only if either a#0 on For a=0 and b#0 on I'. The first purpose of this book is to prove an existence and uniqueness theorem for problem (*) under the condition a > 0 on r in the framework of Sobolev spaces of LP style. The essential point is to define a function space which is a suitable tool to investigate the degenerate boundary condition B. If 1 < p < oo, we let LP(Q) =the space of (equivalence classes of) Lebesgue measurable functions u on S2 such that lulp is integrable on Q.

The space LP(Q) is a Banach space with the norm \ 1/p

l1ullp

( u(x)lpdx 1 = \L l

Ifsis a positive integer, we define the Sobolev space H'°p(Q) =the space of (equivalence classes of) functions u E L°(1) whose derivatives D«u, jal < s, in the sense of distributions are in Lp(S2).

The space H''p(92) is a Banach space with the norm 1/p

f (D«u(x)lpdx

I1uIl9,P = I«I<9

INTRODUCTION AND RESULTS

3

Furthermore we let

B'-,'P'P(r) = the space of the boundary values cp of functions u E H5'P(1 ).

In the space B'-1/P'P(r), we introduce a norm inf Ilulls,p,

where the infimum is taken over all functions u E H3>P(S2) which equal co on

the boundary. The space B'-1/P'P(r) is a Banach space with respect to this norm I Is-1/P,P; more precisely, it is a Besov space (cf. [BL], [Tr]). We introduce a subspace of Bs-1-1/P,P(r) which is associated with the

boundary condition

Bu = a

au/

=0 on r.

+ bu F

We let

B*-1-11P,P(r)

_

a(p1 +

E

Bs-1-11P,P(r), 02 E B3-11P,P(r)

and define

p = apl + bIP2 }

k0I9-1-1/P,P = inf {

.

B:-1-1/p'P(F) is a Banach space with Then it is easy to verify that the space respect to the norm I We remark that the space B*-1-11"(r) is I9-i-i/p,p' an "interpolation space" between the spaces Bs-1/P,P(r) and B8-1-1'P'P(r). More precisely, we have B*-1-11p,p(r) = Bs-1/p,P(r) if a = 0 on r, B*-1-1IP,P(r) = Bs-1-1/p,p(r) if a > 0 on r,

and, for general a, the continuous injections Bs-1/P,P(r) C

B;-1-1/P,P(r)

C

Bs-1-1/P,p(r).

Now we can state our existence and uniqueness theorem for problem (*) (cf. [Um, Theorem 1]):

4

INTRODUCTION AND RESULTS

Theorem 1. Let 1 < p < oo and s > 1 + 1/p. Assume that the following two conditions (H.1) and (H.2) are satisfied:

(H.1)a(x')>_0andb(x')>_0onI'.

(H.2)b(x')>0onro={x'EF:a(x')=0}. Then the mapping (A, B) : Hs,P(1l) -p

Hs-2,P(1l)

Co

B:-1-1iP,P(r)

is an algebraic and topological isomorphism. In particular, for any f E Hs-2'P(I) and any cp E B*-1-1/P'P(F), there exists a unique solution u E H3'1'(1) of problem (*). The second purpose of this book is to study problem (*) from the point of view of analytic semigroup theory in functional analysis. The generation theorem for analytic semigroups is well established in the non-degenerate case in the LP topology (cf. [Fr]). We shall generalize this generation theorem for analytic semigroups to the degenerate case. First we state a generation theorem for analytic semigroups in the LP topology. We associate with problem (*) an unbounded linear operator 2l from LP(Il) into itself as follows: (a) The domain of definition D(2[) of 2l is the set

D(2l)_{uEH2'1'(1):Bu=0}. (b) 2lu = Au, U E D(2l). The next theorem is an LP version of [Tal, Theorem 1]:

Theorem 2. Let 1 < p < oo. If conditions (H.1) and (H.2) are satisfied, then we have the following:

(i) For every 0 < e < 7r/2, there exists a constant r(E) > 0 such that the resolvent set of 2l contains the set

E(E)={A=r2eie:r>r(e),-7r+E<0<7r-E}, and that the resolvent (2l - Al)-' satisfies the estimate II(%-

A1)-1 11 <

A E E(E),

where c(e) > 0 is a constant depending on E. (ii) The operator 2l generates a semigroup U(z) on the space LP (Q) which is analytic in the sector Ae = {z = t + is : z # 0, I arg zl < 7r/2 - e} for any 0 < e < 7r/2 (see Figure 2).

INTRODUCTION AND RESULTS

5

Figure 2 Next, as an application of Theorem 2, we consider the following semilinear

initial boundary value problem: Given functions f and uo defined in 1 x [0, T) x R x RN and in St, respectively, find a function u in SZ x [0, T) such

that (**)

(at - A)u(.x, t) = f (x, t, u, grad u)

in SZ x (0, T),

Bu(x',t):= a(x')au(x',t)+b(x')u(x',t)Irx1oT) =0 on r x [0,T), in Q.

u(x,0) = u0(X)

By using the operator 21, one can formulate problem (**) in terms of the abstract Cauchy problem in the space LP(5l) as follows:

(**')

f dt = 2tu(t) + F(t, u(t)), 0 < t < T, t

u1t=o = uo

Here u(t) = t) and F(t, u(t)) = f t, u(t), grad u(t)) are functions defined on the interval [0, T), taking values in the space LP(1). First we consider the case p > n:

Theorem 3. Assume that conditions (H.1) and (H.2) are satisfied. Let n < p < oo and let f (x, t, u, 6) be a locally Lipschitz continuous function of all its variables with the possible exception of the x variable. Then, for every function uo of D(2l), problem (**') has a unique local solution u E C([0,T']; LP(S2)) fl C' ((0,T'); LP(S2)) where T' = T'(p,uo) > 0. Here C ([0, T']; LP(t )) denotes the space of continuous functions on [0, T'] taking values in LP(1l), and C' ((0, T'); LP(S2)) denotes the space of continuously differentiable functions on (0, T') taking values in LP(S2), respectively.

In the case p < n, the domain D(21) is large compared with the case n < p < oo. Hence we must impose some growth conditions on the function f:

INTRODUCTION AND RESULTS

6

Theorem 4. Assume that conditions (H.1) and (H.2) are satisfied. Let n/2 < p < n and let f (x, t, u, 6) be a locally Lipschitz continuous function of all its variables with the possible exception of the x variable. Further assume that there exist a non-negative continuous function p(t, r) on R x R and a

constant 1 < y < On - p) such that: (a) If(x,t,u,e)I <- P(t, lul)(1 + ICI-) (b) I f(x, t, u, 6) - f(x, s, u, e)I

(c)If(x,t,u, )-f(x,t,u,01

p(t, Jul) (1 + I61-1) It - sl. P(t,Jul)(1+161-1-1+I7IIry-1)

16 -tnl.

(d) If (x, t, u, 6) - f (x, t, v, 6)1 <_P(t,Jul+lvl)(1+161-1)lu-vl.

Then, for every function uo of D(2t), problem (**') has a unique local solution u E C ([0, T']; LP(SZ))f C1 ((0, T'); LP(1)) where T' = T'(p, uo) > 0. Theorems 3 and 4 are a generalization of Pazy [Pa, Section 8.4, Theorems 4.4 and 4.5] to the degenerate case. The rest of this book is organized as follows. Chapter 1 is devoted to a review of standard topics from the theory of analytic semigroups which forms a functional analytic background for the proof of Theorems 2, 3 and 4. First we prove a generation theorem of analytic semigroups (Theorem 1.2). Next we consider the abstract linear and semilinear Cauchy problems, and give a complete proof of local existence and uniqueness theorems (Theorems 1.16 and 1.18). In Chapter 2, we study the imbedding characteristics of Sobolev spaces of LP style that render these spaces so useful in the study of partial differential equations. We give a complete proof of the most important of the imbedding properties of the spaces Hm'P(R") (Theorems 2.15 and 2.18 and Corollary 2.22).

In Chapter 3, we present a brief description of the basic concepts and results of the LP theory of pseudo-differential operators which may be considered as a generalization of the classical potential theory. In particular, we state a Besov-space boundedness theorem (Theorem 3.17) due to Bourdaud [Bo] and a criterion for hypoellipticity for pseudo-differential operators (Theorem 3.19) due to Hormander [Ho]. In Chapter 4, we study the boundary value problem (*) in the framework of Sobolev spaces of LP style, by using the LP theory of pseudo-differential

operators. We prove that problem (*) can be reduced to the study of a pseudo-differential operator T on the boundary r (Theorems 4.10-4.12). In Chapter 5, we study the pseudo-differential operator T in question, and prove Theorem 1. In Section 5.1 we prove a regularity theorem for problem (*) (Theorem 5.1). More precisely, we prove that if conditions (H.1) and (H.2) are satisfied, one can construct a parametrix S for T in the Hormander class L° 112(F) (Lemma 5.2), and then apply a Besov-space boundedness theorem to the parametrix S to obtain the regularity theorem for problem (*). Section 5.2 is devoted to a uniqueness theorem for problem (*) (Theorem 5.5). In the proof we make good use of the maximum principle stated in the

INTRODUCTION AND RESULTS

7

Appendix (Theorems A.1 and A.2). Section 5.3 is devoted to an existence theorem for problem (*) (Theorem 5.7) which is an essential step in the proof of Theorem 1. To prove this, we make use of a method essentially due to Agmon [Ag] (Proposition 5.10). This is a technique of treating a spectral parameter as a second-order differential operator of an extra variable and relating the old problem to a new one with the additional variable. Chapter 6 is devoted to the proof of Theorem 2 (Theorems 6.6 and 6.8). Once again the method of Agmon plays an important role in the proof of the surjectivity of 2t - AI (Proposition 6.7). In Section 6.1 we study the operator 2t, and prove fundamental a priori estimates for 2t - Al (Theorem 6.3) which play an important role in the proof of Theorem 2 in Section 6.2. In the final chapter, Chapter 7, we prove Theorems 3 and 4. In Section 7.1 we study the imbedding properties of the domains of the fractional powers (-2t)a, 0 < a < 1, into Sobolev spaces of LP style (Theorem 7.1). This allows us to solve by successive approximations the semilinear initial boundary value problem (**) in Section 7.2, proving Theorems 3 and 4.

CHAPTER I

THEORY OF ANALYTIC SEMIGROUPS This chapter is devoted to the theory of analytic semigroups which forms a functional analytic background for the proof of Theorems 2, 3 and 4. First we prove a generation theorem for analytic semigroups (Theorem 1.2). Next we consider the abstract linear and semilinear Cauchy problems, and give a complete proof of local existence and uniqueness theorems (Theorems 1.16 and 1.18). For more leisurely treatments of analytic semigroups, the reader is referred to Friedman [Fr], Pazy [Pa] and also Yosida [Yo].

1.1 Generation Theorem for Analytic Semigroups Let E be a Banach space over the real number field R or the complex number field C. Let A : E -i E be a densely defined, closed linear operator, that is, the domain of definition D(A) of A is dense in the space E. Assume that the operator A satisfies the following two conditions: (1) The resolvent set of A contains the region (see Figure 1.1)

E,={AEC:a#0,Iarg\I 0, there exists a constant MM > 0 such that the resolvent

R(A) = (A - Al)-' satisfies the estimate (1.1)

IIR(A)II-MI, AEEW={AEC:A

0,IargAl <7r/2+w-e}.

Figure 1.1 Typeset by A,MS-TEX 8

1.1 GENERATION THEOREM FOR ANALYTIC SEMIGROUPS

9

We let (1.2)

U(t) _

-2 7ri- Jr dtR(\) d\,

t > 0,

where IF is a path in the set Eu, consisting of the following three curves (see Figure 1.2): r(1) = {re-i(7r/2+w-E) : 1 < r < oc}

,

r(2) = {eie : -(ir/2 +w - e) < 0 < it/2 +w - e) r(3) = re i(7rl2+w-e) : 1 < r < oo}

.

Figure 1.2

Then, by estimate (1.1), it is easy to see that the integral 3

1

U(t)

27ri

E Jp(k) eR() d

converges in the Banach space L(E, E) of all bounded linear operators on E into itself, for every t > 0, and thus it defines a bounded linear operator on E. Furthermore, the operators U(t) form a semigroup on E; more precisely, we have the following:

Proposition 1.1. The operators U(t), defined by formula (1.2), enjoy the semigroup property, that is,

U(t + s) = U(t) U(s), t, s > 0. Proof. By Cauchy's theorem, one may assume that U(s)

27ri

J eµ8R(p)dp, ,

s > 0.

THEORY OF ANALYTIC SEMIGROUPS

I.

10

Here r' is a path obtained from the path IF by translating each point of F to the right by a fixed small positive distance (see Figure 1.3).

Figure 1.3

Then we have by Fubini's theorem U(t) - U(s) = (21ri)2 __

1

(2iri)2 1

=

2,7ri

fir' teµsR(a) R(µ) da dµ

JJ

Ie

_

eaceµ9 R

A-µ

'

µs

at At

f

- µd) dµ

27ri

r , Ae µdµ dA at

Ita

dA dµ. 2ri , e R(µ) [ 2 - Jr Ae µ We calculate the two terms in the last part. (a) We let A"3µ f(µ) _

Then, applying the residue theorem, we obtain that (see Figure 1.4)

J '(1)n{Iµl
+ -(ir12+W-E)

f (reie)rie`B dO

= 2iriRes [f (µ)]µ=A

= -2irieAs. But we have, as r

oo,

r1(')n


f (i) dii

)

j

(l) f (y) dN,

1.1 GENERATION THEOREM FOR ANALYTIC SEMIGROUPS

J

11

) J'(3) f(µ)dµ,

'(3)n{IµI
Pf(µ)dµ

and

f

/2+w-e

W/2+w-e

f(reie)rie'B d6

d8 0.

(7r/2+w-e)

Figure 1.4

Therefore, we find that eµ9

1 _2-7ri

A-µdµ=-e

r,

ae .

(b) Similarly, since the path r lies to the left of the path F', we find that eat

1

27ri

rA-µ

dA=0.

Summing up, we obtain that

U(t) . U(s) = - 11 ev(t+9)R(A) dA = U(t + s). 27ri

r

The proof of Proposition 1.1 is complete.

The next theorem states that the semigroup U(t) can be extended to an analytic semigroup in some sector containing the positive real axis.

I. THEORY OF ANALYTIC SEMIGROUPS

12

Theorem 1.2. The semigroup U(t) can be extended to a semigroup U(z) which is analytic in the sector 4, = {z = t + is : z 54 0, I arg z I < w}, and enjoys the following properties: (a) The operators AU(z) and d4 (z) are bounded operators on E for each

z E A, and satisfy the relation

U(z)=AU(z), zE&. (b) For each 0 < e < w/2, there exist constants Mo(e) > 0 and Mi(e) > 0 such that IIU(z)II < Mo(e), IIAU(z)II C

(Ie),

z E ,Ye zE

where

QWE={zEC: z54 0,Iargzl <w-2e}. (c) For each x E E, we have as z --+ 0, z E L ,E,

U(z)x -f x in E.

4W

Figure 1.5

Proof. (i) The analyticity of U(z): If A E V-3) and z E

A= I\Ieie z = Izle",

8=it/2+w-e, I(pI C w - 2e,

then we have

Az = Al Izle`(B+w)

that is, if

1.1 GENERATION THEOREM FOR ANALYTIC SEMIGROUPS

13

with it/2 + e < 0 + co < 7r/2 + 2w - 3e < 37r/2 - 3e.

Note that cos(0 +

cp)

< cos(ir/2 + e) _ - sine.

Hence it follows that Ie)zI < e-IXI IzIsine,

Ieaz1

< .-JAI Izi sine,

AEri3> zE

AEr(1), zEL

.

For each small e > 0, we let

K = Wen{zcc:IzI>e}= {z cc:IzI>e,Iargzl<w-2e}. Then, combining estimates (1.1), (1.6) and (1.7), we obtain that (1.8)

lieAzR(A)II <

Me-esine.IAI

A E r(1) Ur(3', z E K.

On the other hand, we have (1.9)

11e'\=R(A)11 < Meelzl

A E r(2), z E K.

Therefore, we find that the integral (1.2')

U(z) _ -

12i

3

e- zR(A) dA = -

12i E

Jr(k)

converges in the Banach space L(E, E), uniformly in z E Kw, for every e > 0. This proves that the operator U(z) is analytic in the domain A,,, = Ue>0 Kr,. By the analyticity of U(z), it follows that the operators U(z) also enjoy the semigroup property

U(z + w) = U(z) U(w),

z, w E 4,.,

(ii) We prove that the operators U(z) enjoy properties (a), (b) and (c). (b) First, using Cauchy's theorem, we obtain that U(z)

21ri

l r

eazR(A) dA 21ri

j

eazR(A) dA, 1 ,1

I.

14

THEORY OF ANALYTIC SEMIGROUPS

-z

where rizl is a path consisting of the following three curves (see Figure 1.6) r(1 ) _ Iz l IZI

r(2=

re-:(n/2+w-e)

{lio

: :

Izl

< r < oo

,

-(/2+w-e) < 0
)))

r(3) I=I

-

re:(7r/2+w-e)

:

1zl < r < oo

Figure 1.6

But, by estimates (1.1), (1.6) and (1.7), it follows that

M

zE E r(1) (3), I=I V r(3) IZI

e-Ia11zlsine

IIeAzR(A)ll C

Hence we have for k = 1, 3 IleazR(A)II

ME

I dA

e-PlzlsinP-1 dP

IT 1

=

= Me

ds. 1

We have also for k = 2 n/2+w-e IleazR(A)ll I dAl

Jr(2) 1.1

< Me

J

e dB

(n/2+w-e)

= 2eMf(7r/2 + w - e) < 2ireMe.

Summing up, we obtain the following estimate: 3

11U(z)II <

27r

k=1

f

(k) rl=1

IleazR(A)II dAl

2e w

1.1 GENERATION THEOREM FOR ANALYTIC SEMIGROUPS

<1

(2Me

s-1 e- gin "' ds + 27reMe I

/

1

Me

=

7r

15

s-1 e- sin

Ul

ds + ire I

.

Th is proves estimate (1.4), with Mo(e)

=

Me (Js_1e_s8ds + Ire

To prove estimate (1.5), note that AR(A) = (A - AI + AI)R(A) = I + AR(A),

so that IIAR(A)II C 1 + Me7

A E EL.

Hence, arguing as in the proof of estimate (1.4), we obtain that (1.10)

Jr

e-'zAR(.A) d.A


<2

lT

+

I

7r/2+w-e

(ir/2+w-e)

1

(1+Me)e-d9 IzI

2(1 + Me) (1100 e-

I1

This proves that the integral fr eAzAR(.A) dA is convergent for every z E LXwe. By the closedness of A, it follows that, for all z E QWe,

U(z) E D(A), and

AU(z)=-2l7ril e)zAR(A) dA.

(1.11)

Therefore, estimate (1.5) follows from estimate (1.10), with

Mi(e) =

i+

m"

(Je'cis+ire).

We remark that formula (1.11) remains valid for all z E Qw, since Ow = Ue>0

Q2e w

I. THEORY OF ANALYTIC SEMIGROUPS

16

(a) By estimates (1.8) and (1.9), one can differentiate formula (1.2') under

the integral sign to obtain that (1.12)

dU(z) =

f e'\-AR(A)dv, -2' -7r i r

z E A,.

On the other hand, it follows from formula (1.11) that (1.13)

AU(z) = -2jri J eAZAR(A)dA

eaz(I + AR(A))da _ -1 2-7ri r _

--L j reaz\R(\)d,A, z E&, 2-7ri

since we have by Cauchy's theorem

Jr

eAXd) = 0.

Therefore, formula (1.3) follows immediately from formulas (1.12) and (1.13).

(c) Now let x° be an arbitrary element of D(A). By the residue theorem, it follows that eaz

x0

2iri

lr A

x°

Hence we have U(z)xo - x°

27ri 27ri

f eaz (R)+ c I x° d A r

,

eAz

R(A)Ax° dA.

r

Here we remark that

AEF, leazl < 2e-lal Izlsin e +elzl,

z E Awe, x E r.

Thus it follows from an application of the Lebesgue dominated convergence theorem that, as z -* 0, z E Ate,

U(z)x° - x° -> -21

7ri

jr

R(,A)Ax° d).

1.1 GENERATION THEOREM FOR ANALYTIC SEMIGROUPS

But we have

17

11R(A)Axod,v=0.

r Indeed, by Cauchy's theorem, it follows that

r

f1

1

R(A)Axo d,\


lim r-. Jr

AR(A)Axo dA

= 0,

where Cr is a closed path shown in Figure 1.7.

Figure 1.7

Summing up, we have proved that

U(z)xo -> xo as z --+ 0, z E

AW,,

for each xo E D(A). Since the domain D(A) is dense in E and it follows that, for each x E E,

Mo(e) for all z E AWE,

U(z)x -mix asz-*0,zEAWf The proof of Theorem 1.2 is now complete.

Remark 1.3. Assume that the operator A satisfies a stronger condition than condition (1.1) (1.1')

MM

IIR(A)II C

lal + 1'

A E Eu,.

18

THEORY OF ANALYTIC SEMIGROUPS

I.

Then we have the estimates

zE

Mo(e)e-6-R,!.,

(1.4')

IIU(z)II

(1.5')

II AU(z) II <

M'

-6.Rez,

(e)

zE

IzI

with some constant b > 0. Proof. Take a real number b such that 1

0
E .

Then we have by estimate (1.1') 6 II (A -

AI)-1 II <

ME1 < 6ME < 1,

A E w.

I

Hence it follows that the operator (A + 61) - AI has the inverse

((A + 61) - AI)-1 = (I + b(A - AI)-1)-'(A - AI)-1, and

II((A + 61) -

11(1 + b(A - AI)-')-' II - II(A -

AI)-1 II

ME

AI)-1 II

1

JAI+11- IIS(A-AI)-'II <

ME

1

JAI +11-SME ME

1

1-6ME

ICI

This proves that the operator A+bI satisfies condition (1.1), so that estimates (1.4) and (1.5) remain valid for the operator A + ff. Mo(e),

(1.14)

IIV(z)II

(1.15)

II(A + SI)v(z)II <-

z E AWE,

M1 00

z E a2E,

IzI

where

V(Z)=-je'\-(A+b1--A1)-1dA. 2l 7rz

But we have by Cauchy's theorem (1.16)

V(z)

AI)-1 dA

-217ri1 re'1T(A+611

27ri

r+6

1

29ri

r

eaz(A+ 61

eµze"(A

-

-

AI)-1 dA

µI)-1 dp = ebzU(z)

In view of formula (1.16), the desired estimates (1.4') and (1.5') follow from estimates (1.14) and (1.15). 0

1.2 FRACTIONAL POWERS

19

1.2 Fractional Powers Assume that the operator A satisfies a stronger condition than condition (1.1).

(1) The resolvent set of A contains the region E shown in Figure 1.8. (2) There exists a constant M > 0 such that the resolvent R(A) = (A AI)-1 satisfies the estimate M (1.17)

A E Z.

IIR(A)II <- 1 + IAI,

E

0

Figure 1.8

If a > 0, we can define the fractional power (-A)-" of -A by the following formula: (1.18)

(-A)- - -27ri

(-A)-"R(A)dA.

Here the path r runs in the set E from ooe-2W to ooe2U, avoiding the positive

real axis and the origin (see Figure 1.9), and for the function (-A)-" = e-"'g(-A), we choose the branch whose argument lies between -air and air; it is analytic in the region obtained by omitting the positive real axis.

Figure 1.9

20

I.

THEORY OF ANALYTIC SEMIGROUPS

The integral (1.18) converges in the uniform operator topology for a > 0, and thus defines a bounded linear operator on E. Indeed, it suffices to note the following:

I(-))-al = le-alog(-X) I = e-«logiAI = JAI-a, IIR(A)II

- 1 Ma

Some basic properties of the fractional power (-A)-a are summarized in the following:

Proposition 1.4. (i) We have for all a, 0 > 0

(-A)-a(-A)-p = (-A)-(a+a). (ii) If a is a positive integer n, then we have

(-A)-c' = ((-A)-1)n (iii) The fractional power (-A)-a is invertible for all a > 0. Proof. (i) The proof of part (i) is similar to that of Proposition 1.1. By Cauchy's theorem, one may assume that

(-A)-Q =

- 211ri

r

(-µ)-0R(I1) dµ,

where F' is a path obtained from the path r by translating each point of r to the right by a fixed small positive distance (see Figure 1.10).

Figure 1.10

Then we have by Fubini's theorem

(_A)-0'(_A)-1' = (27ri)2 1

1 Jr

(-A)-a(-,u)-OR(A)R(w)dAdp

1.2 FRACTIONAL POWERS

21

R(A (2,7ri)2

27ri 27ri

j(-A)-"R(A) [27ri A

2

-

R(µ)

dA dµ

dµ1 d\

, (-µ)-QR(µ) [;J ( A - µ" d\] dµ.

Just as in the proof of Proposition 1.1, we can calculate each term in the last part as follows:

dµ = -(-.X)-1; 2iri r, 1

27ri

A-µ

fr (-A) A - d)=0.

Therefore, we obtain that

(-A)-"(-A)-a =

- 21

1(-A)-("+13)R(A) dA

=

(-A)-("+Q).

(ii) Since we have by estimate (1.17) lim roo

J

(-reza)-nR(reie)ire`a dO = 0 for all integer n > 1, w

it follows that

(-A) -n

lim = 27ri where Cr is a closed path shown in Figure 1.11. 217ri

1(-A)-"R(A) da

Figure 1.11

fcr(-A)-nR(A) da,

22

I.

THEORY OF ANALYTIC SEMIGROUPS

Thus, by the residue theorem, we obtain that (-A)-n

= Res [(-A) -n R(A)]),=o (_1)n

do-1

(n - 1)!

_

d\n-1

((A - I) 1 )

V

A=0

(-1)n(A-1)n

_ ((_A)-1)n

(iii) Since the operator (-A)-' is injective, it follows that (-A)-n ((-A)-' )n

is injective for all integer n > 1. Assume that

(-A)-ax = 0,

a > 0.

Then, taking an integer n > a, we obtain that

(-A)-nx = (-A)-(n-a)(-A)-ax = 0, so that

x=0.

This proves part (iii). The proof of Proposition 1.4 is complete.

If 0 < a < 1, we have the following useful formula for the fractional power (-A)-a: Theorem 1.5. We have for 0 < a < 1 -'in oor

(1.19)

100 s-aR(s) ds.

7r

Proof. By Cauchy's theorem, one may deform the path r in formula (1.18) into the upper and lower sides of the positive real axis. But we have AI -aeon:

(-A)-a =

if ImA > 0, if ImA < 0.

e-alog(-X)

JAI-ae-ani

Hence it follows that

(-A)-a =-

f00 1

dS -

ri

- - 1 ea7ri - eaai 2i sin air f .) 7r

J

0

J°0

1J

Ir0

27ri

sR(s) ds

s-aR(s) ds.

o0

s-ae-aR(S) ds

1.2 FRACTIONAL POWERS

23

Corollary 1.6. We have for 0 < a < 1

II(-A)-"II < M.

(1.20)

Proof. By formula (1.19), it follows that sin air

II(-A)-"II

it

°°

s-"IIR(s)jI

o

sin a7r / O°

i

o

But we have s-"(1+s)-1ds=

Jl 1(:)

v)

0

1000

=

do,

(1- 0)2

0)a-1 d0 fo

=B(1-a,a) = r(1 - a)r(a), and also the well-known formula sin air

i

1

= r(a)r(1- a).

Summing up, we obtain that

II(-A)-"II < M. O By Remark 1.3, one may assume that there exist positive constants M0, M1 and a such that Moe-at,

(1.21)

IIU(t)II

(1.22)

IIAU(t)II < Mle-at

t > 0, t > 0.

Then we can prove still another useful formula for the fractional power

(-A)-",0< a < 1. To this end, we need a representation formula for the resolvent R(s) _

(A - sI)-1, s > 0.

I.

24

THEORY OF ANALYTIC SEMIGROUPS

Lemma 1.7. We have for all s > 0

R(s) = - f 00 e-`U(t) dt.

(1.23)

0

Proof. Let s > 0. For any T > 0, we have by Fubini's theorem T

e-9tU(t)

(1.24)

e-9t (

dt

10

217rz

*

_

fo

eµtR(p) dµdt

(JT) \r

eµt-9tdt

1

27ri

_

-1 2,7ri

_

r µ -S

R(p)dµ 1) R(µ)dµ

(e

Jr

R(It) dµ e (µ-s)T /.t - s 27n Jr µ - s R(µ)dµ - 2iri Jr 1

1

1

-R(s) -

1

27ri

j

(µ-s)T R([i)s dµ.

µ-

e

But we find that the second term in the last line tends to zero as T -> oo:

Jr

e(µ-s)T R(µ) dy /L - s

<e-ST

J 1MµIIlIdl1I -40.

Thus formula (1.23) for s > 0 follows by letting T -> oo in formula (1.24). In view of estimate (1.21), it follows that the integral f0,30 e-stU(t)dt converges uniformly in s E [0, oo) in the space L(E, E). Hence, applying the Lebesgue dominated convergence theorem, we obtain that formula (1.23) remains valid for s = 0.

Theorem 1.8. We have for 0 < a < 1 1

100"t-1U(t)dt.

I(a )

Proof. Substituting formula (1.23) into formula (1.19), we obtain that

(-A)-c' =

sin a7r

1

7r

=

'

\ Jo

Jo

sin air

A U(t) (fo 0

ds

esds dt r

sin

/

-,tU(t)dt)

U(t)

0

\ t' -a d r

dt

1.2 FRACTIONAL POWERS

=

sin a7r

°O

J

7r

F(a)

e-rT-« d7-

°°

J 't«-1U(t)dt. 0

25

t«-'U(t) dt

In view of part (iii) of Proposition 1.4, we can define the fractional power

(-A)« for a > 0 as follows:

(-A)« = the inverse of (-A)-«,

a > 0.

The next theorem states that the domain D((-A)«) of (-A)« is bigger than the domain D(A) of A when 0 < a < 1.

Theorem 1.9. We have for any 0 < a < 1 D(A) C D ((-A)"). Proof. Let x be an arbitrary element of D(A). Then there exists a unique element y E E such that x= (-A)-1y.

If 0 < a < 1, one can define the fractional powers and write (-A)-1 as follows:

(-A)-« and

Hence we have

x=

(-A)-ly = (-A)-«

((-A)-(1-«)y)

This proves that

x E D((-A)«). The proof of Theorem 1.9 is complete. 0 We can give an explicit formula for the fractional power (-A)&, 0 < a < 1, on the domain D(A).

Theorem 1.10. Let 0 < a < 1. We have for any x E D(A) (1.25)

(-A)«x =

sin air it

°°

sa-1R(s)Axds.

o

Proof. First we remark that

(_A)a = (-A)(-A)-(1-«)

I.

26

THEORY OF ANALYTIC SEMIGROUPS

By formula (1.19) with 1 - a for a, it follows that sin(1 - a)7r /'°° sa-1R(s)ds it o sinaIT O° sa-1R(s) ds. 7r

0

But we have for x c D(A) s«-1(1

(1.26)

Is«-1R(s)AxI <

{

+ M)IIxII Sa-2MIIAxII

near s = 0, near s = oo,

since R(s)A = I + sR(s) and I R(s) I < Ml s, s > 0. This implies that < oo.

J Hence, by the closedness of A, it follows that

(-A)-(1-a)x E D(A) and

(-A)ax =

(-A)(-A)-(1-")x

sin air

J"'o

sa-1R(s)Ax ds.

0

Corollary 1.11 (The moment inequality). Let 0 < a < 1. Then we have (1.27)

II(-A)axll < 2M(1 +M)Ilxll1-aIIAxIIa,

x E D(A)-

Proof. By formula (1.25) and estimate (1.26), it follows that for every

p>0 sin air

II(-A)axll =

it

sin air a7f

+

°O

s«-1R(s)Ax ds

J

o

f

11

P

asa- IIR(s)Axll ds

0

sin(1 - a)7r

(1 - a)7r

J -

a)s«-1IIR(s)AxII ds

P

p sin air asa-1(1 +M)ds IIx1I air fo

1.2 FRACTIONAL POWERS

+ sin(1 - a)7r

(1 - a)ir

27

f(i - a)s'-2M ds <-(1+M)IlxIIP"+MIIAxIIp

IIAxII

1.

Therefore, the moment inequality (1.27) follows by taking P

M IIAxil 1+M IIxi)

The proof of Corollary 1.11 is complete.

Now we study some useful relationships between the fractional powers

(-A)', 0 < a < 1, and the semigroup U(t). Theorem 1.12. Let 0 < a < 1. For all t > 0, we have the following:

(a) U(t) : E--) D((-A)"). (b) U(t)(-A)°x = (-A)"U(t)x, x E D((-A)"). 2M(1+M)MJ-«Mi t-"e-°.t.

(c) II(-A)'U(t)II C

(d) II U(t)x - xll < 2M(a+M) Mo M1-"tall (-A)'xll, x E D((-A)`I).

Proof. (a) This is obvious, since U(t) : E -> D(A) for every t > 0 and D(A) C D((-A)"). (b) If x E D((-A)«), then we have by formula (1.19')

U(t)x = U(t)((-A)-°`)(-A)x _ 1 °° 8 r (a)

_ _

1

'

°-1 U(t)(U(s)(-A) x) ds

10

r (0

°°

s

«-' U(s)(U(t)(-A)« x) ds

= (-A)-'U(t)(-A)ax. This proves part (b).

(c) Since (-A)" = ((-A)-")-' is closed and U(t) is bounded, it follows that the operator (-A)aU(t) is closed. But part (a) tells us that it is everywhere defined on E. Hence, by the closed graph theorem, it follows that (-A)aU(t) is a bounded operator on E. More precisely, applying the moment inequality (1.27) to U(t)x, we obtain from estimates (1.21) and (1.22)

that II(-A)«U(t)xii

2M(1 + M)II U(t)xii'-"II AU(t)xil

< 2M(1 + M) (Moe-atll xll

)1-a

2M(1 + M)Ma-"Mi t-'c-°tlixli This proves part (c).

(Mle-°tt-1Il xl i)a

28

I.

THEORY OF ANALYTIC SEMIGROUPS

(d) By part (b), we have for all x E D((-A)") U (t)x

-x=

f U'(s)x ds

= JAU(s)xds

= ft

ds.

Hence it follows from part (c) with 1 - a for a that IIUOt

x-xII

t

II(-A)1-"u(s)II

<

- II(-A)"xII ds

_ JO

t

< 2M(1 + M)MM MM -" 1 s"-i ds ll(-A)"xll 0

< 2M(1+M)M00'Mi-0't0'II(-A)0xlj a

The proof of Theorem 1.12 is complete.

The next lemma is useful in applications to concrete initial boundary value problems for semilinear parabolic differential equations (cf. [He, Section 1.4, Exercise 11]).

Lemma 1.13. Let A : E -+ E be a densely defined, closed linear operator which satisfies condition (1.17), and let B be a closed linear operator from D(B) C E into a Banach space F. Assume that D(B) D D(A), and that there exist constants 0 < ry < 1 and 6o > 0 such that, for all 0 < S < bo, (1.28)

II BxII F < C (s-lllxll + al-'IlAxll)

,

x E D(A).

Then we have for all y
IIBzliF < K.11(-A)"zll,

z E D((-A)"),

with a constant K" > 0.

Proof. (1) We show that: If (-A)-"x E D(B), then we have (1.29)

B((-A)-'x)= 1,(a) f s"-'BU(s)x ds.

We use the representation formula (1.19') for the fractional power (-A)-"

r(a) j"o

s"-1U(s) ds.

1.2 FRACTIONAL POWERS

29

Since B is closed, it suffices to show that the integral f °° sa-1BU(s)x ds is convergent in F. First we have 00

sa-1BU(s)x ds F

10

00

bo

<

sa-1 11 BU(s)x11F ds +lb fo

S«-1 JjBU(s)xiiF ds. 0

We estimate each term in the last line. (1) Applying inequality (1.28) with U(s)x for x and s for 6, we obtain from estimates (1.21) and (1.22) that s1-7iiAU(s)xiI)

II BU(s)xl iF < C (s-'1iIU(s)xil + < C (Mos-1 + s1- rM1s-1) lixii +Mi)s-yiIxii.

= C(M0

Hence we have for y < a < 1 (1.30)

f

6o

sa-1IIBU(s)xll F ds < C(Mo +M1)

(f

6o

81--y-1

ds

114

= C(Mo +M1)s0-ry114 a - ry

(2) Similarly, applying inequality (1.28) with x = U(s)x and 6 = So, we obtain that (1.31)

1600" s°-1 JjBU(s)xjjF ds


J/bo

s°-1 (So 7jjU(s)xjj + S01-1jJAU(s)xjj) ds

00

CJ

(So'YM0

sc-1

bo


(Mo6_1

j

0

+ So-ryM1s-1)

e-as ds + M161--Y

< C (Mobo --r-1 a +

+

1(2-a) f a

M16o--r

sa-3ds

bo

(M0 + 2M1) bo -ry-1 11x 1j

<-

a

f e-as

ds . jlxii

s«-2e-as ds) jjxii

/

([1_as _ s « -21 l 1lxii

Jbo

I.

30

THEORY OF ANALYTIC SEMIGROUPS

Therefore, combining inequalities (1.30) and (1.31), we have 1

fO°

r(a)

s"-1II BU(s)xII F ds < K"IIx1I,

with

K.

F(a) C«

(M.+2M1)bo--r-1l.

7(Mo+M1)8o--y +a

1

This proves formula (1.29) and also (1.32) II B((-A)")xII F < K"IIxII, (2) We remark that

J x E D(B(-A)-").

(-A)1-" = (-A)(-A)-". Hence we have for all x E D((-A)1-")

(-A)-"x E D(A) C D(B). Thus we can apply inequality (1.32) to obtain that (1.32') II F < K"IIxII, x E D((-A)1-"). B((-A)-")xII

Now let x be an arbitrary element of E. Since the domain D(A) is dense

in E and since D(A) C D((-A)1-0'), we can choose a sequence {xj} in D((-A)1-") such that (1.33) xj--px in E. Then, by inequality (1.32'), it follows that {B((-A)-")xj} is a Cauchy sequence in F. Hence there exists an element y of F such that (1.34) B((-A)-")xj -> y in F. But, since B is closed and (-A)-" is bounded, it follows that B((-A)-") is a closed operator. Hence we have by (1.33) and (1.34)

x E D(B((-A)-")) B ((-A)-"x) = y This proves that

D (B((-A)-")) = E, so that

D((-A)") C D(B)Finally, we can pass to the limit in the inequality (1.32')

JIB ((-A)-") xjIIF

K"IIxjII

to obtain that

JIB ((-A)-") xIIF : Ii"II4II, or equivalently (letting z = (-A)-"x) II BzII F < K. II(-A)"zII, The proof of Lemma 1.13 is complete.

x E E,

z E D((-A)).

1.3 THE LINEAR CAUCHY PROBLEM

31

1.3 The Linear Cauchy Problem In this section, we consider the following Cauchy problem:

dt = Ax(t),

0 < t < T,

(P)

{ x(0) = xo.

A function x(t) : [0, T) --+ E is called a solution of problem (P) if it satisfies the following three conditions: (1) X(t) E C([0, T); E) n C'((O, T); E) and x(0) = xo. (2) x(t) E D(A) for all 0 < t < T.

(3) d' = Ax(t) for all 0 < t < T. Here C([0, T); E) denotes the space of continuous functions on [0, T) taking values in E, and Cl ((0, T); E) denotes the space of continuously differentiable functions on (0, T) taking values in E, respectively. Assume that the operator A satisfies condition (1.17): (1.17)

JIR(A)II <

A E E.

M , 1+JAI

The next theorem states that problem (P) has a unique solution x(t) for any initial condition xo E E.

Theorem 1.14. Assume that the operator A satisfies condition (1.17). Then, for every xo E E, the function

belongs to the space C([0, oo); E) n C°°((0, oo); E), and is a unique solution of problem (P).

Proof. By Theorem 1.2, it suffices to prove that the function U(t)xo is the only solution of problem (P), for every T > 0. Let y(t) be an arbitrary solution of problem (P), and let z(t,s) = U(t - s)y(s),

0 < s < t < T.

Then we obtain that the function

s -+ z(t, s) belongs to the space C([0, t]; E) n C' ((0, t); E), and satisfies s

a(z(t, s)) = U(t - s)y'(s) - AU(t - s)y(s)

= U(t - s)(y'(s) - Ay(s))

32

I.

THEORY OF ANALYTIC SEMIGROUPS

=0, 0<s
z(t, 0) = z(t, t), so that

y(t) = U(t)xo,

0 < t < T.

This proves Theorem 1.14.

Let f : [0, T) --+ E be a continuous function. Now we consider the following non-homogeneous Cauchy problem:

d =Ax(t)+f(t), 0
(NP)

x(0) = xo.

The next theorem gives an explicit formula for the solutions of problem (NP).

Theorem 1.15. Assume that the operator A satisfies condition (1.17). Then a solution x(t) of problem (NP), if it exists, is given by the following formula:

X(t) = U(t)xo + J t U(t - s) f(s) ds,

(1.36)

0 < t < T.

0

Proof. Applying the operators U(t - s), 0 < s < t, to the equation x'(s) = Ax(s) + f(s),

0 < s < T,

we obtain that

U(t - s)x'(s) = U(t - s)Ax(s) + U(t - s) f(s), 0 < s < t.

(1.37)

On the other hand, it follows that (1.38)

d ds

(U(t - s x s = lim

(U(t - s - Q)x(s + a) - U(t - s)x(s))

o-o

= lim { U(t - s - a) 0-0

-U(t - s - o)

Cx(s + a) - X ( S ) 01 U(O')

Q

-f

= U(t - s)x'(s) - U(t - s)Ax(s),

x(s)} 0 < s < t.

Hence it follows from formulas (1.37) and (1.38) that

d (U(t - s)x(s)) = U(t - s) f (s),

0< s < t < T.

1.3 THE LINEAR CAUCHY PROBLEM

33

Integrating this equation from 0 to t - h, h > 0, with respect to s, we obtain that

t-h U(t - s) f (s) ds = pt - s)x(s)]0 h 10

= U(h)x(t - h) - U(t)xo. But we have, as h 10, II U(h)x(t - h) - x(t)II <-IIU(h)II II x(t - h) - x(t)II + II (U(h) - I)x(t)II

->0. Indeed, it suffices to note the following: (1) supo
(3) x(t) E C([0,T);E). Therefore, we find that the improper integral fo U(t - s) f (s) ds exists, and satisfies

t

I';

U(t - s) f (s) ds = x(t) - U(t)xo.

This proves formula (1.36). The proof of Theorem 1.15 is complete.

The next theorem states that the function x(t), defined by formula (1.36), is a solution of problem (NP) (cf. [He, Theorem 3.2.2]; [Pa, Chapter 4, Corollary 3.3]).

Theorem 1.16. Assume that the operator A satisfies condition (1.17). Let f be a locally Holder continuous function on (0, T), with exponent 0 < -y < 1, which satisfies the condition (1.39)

1 11f (s) 11 ds < oo. T

Then, for every xo E E, the function (1.36)

X(t) = U(t)xo + f U(t - s) f (s) ds 0

belongs to the space Q0, T]; E) fl C'((0, T); E), and is a unique solution of problem (NP). Proof. Theorem 1.14 tells us that the function U(t)xo, given by formula (1.35), belongs to C([0, oo); E) fl C°0((0, oo); E), and is a (unique) solution of problem (P).

I. THEORY OF ANALYTIC SEMIGROUPS

34

Thus it suffices to consider the function y(t) = 10 U(t - s)f(s) ds,

(1.36')

0 < t < T.

0

First we have by estimate (1.21)

1104 < f IIt U(t - s)II

- 11f(s)II

ds < Mo f IIf(s)II ds,

0

0

t

so that

limy(t) y(t) = 0. (a) The continuity of y(t): For each small 6 > 0, we let

if0
0

y6(t) f0

-6U(t-s)f(s)ds

Then we have f°

11y6(t) - y(t)II <

if0
IIU(t - s) 11 - IIf(s)II ds

ft`

6IIU(t-s)II

- IIf(s)IIds

f IIU(t - s)II ' Iif(s)II ds +

IIU(t - s)II t-6 t

< Mo f

IIf(s)II ds

6

IIf(s)II ds +

f

t

IIf(s)II ds 6

Thus, by condition (1.39), it follows that (1.40)

The function y6(t) converges to the function y(t) as b 10, uniformly in t E [0, T].

But we have for0
t-6

y6(t + h) - y6(t) = f

U(t + h - s)f(s) ds - f

0

0

U(t - s)f(s) ds

t+h-6

=

Jt-6 +

f 0

U(t + h - s) f (s) ds t-6

[U(t + h - s) - U(t - s)] f (s) ds

1.3 THE LINEAR CAUCHY PROBLEM

35

-b

[U(h) - I] I

U(t - s) f (s) ds t

t+h-b

+I

t-b

U(t + h - s)f (s) ds,

so that as h 10 II yb(t + h) - y6(t)II t

[U(h) - I]

/0

t -b

U(t - S) f (s) ds + Mo

b+h

it -b

IIf(s)II ds

0.

This proves that Y6 (t) E C([O,T]; E).

Hence, in view of assertion (1.40), it follows that y(t) E C([0, T]; E).

(b) Next we show that

0 < t < T,

y(t) E D(A), (1.41)

1 Ay(t) = .fo AU(t - s)(f(s) - f(t)) ds + (U(t) - I).f(t). Since we have by estimate (1.22) and condition (1.39) t-b

t-6

IIAU(t-s)f(s)II ds <M, f 10

t

11f(s)II ds

0

M,

6

T

1 II.f(s)II ds

< oo,

in view of the closedness of A it follows that

f yb(t) = fo

-b

Ayb(t) = fo

U(t - s).f (s) E D(A)

-b

for S < t < T,

AU(t - .9)f(s) d$.

We remark that -b Ayb(t) =

=

f

t-b

AU(t - s) (f (s) - f (t)) ds + 1 t

AU(t - s) f (t) ds t-b

0t-b

AU(t - s)(f (s) - f (t)) ds -

ds (U(t - s)f (t)) ds

I.

36

THEORY OF ANALYTIC SEMIGROUPS

_ f'-'AU(t - s)(f (s) - f(t)) ds + (U(t) - U(S)) f (t). Let [a, b] be an arbitrary closed interval of (0, T), and let Lab be a Holder constant for the function f on the interval [a/2, b]: 11f(t) - f(s)JI < Lablt - 81-',

t, s E [a/2, b].

Then, by estimate (1.22), we have for 0 < 6' < 6 < a/2

-

b'

t

t

AU (t - s) (f (s) - f (t)) ds b

< M1Lab

=

M1

Jt-b

(t - s)^'-1 ds

Lab (S-" - S'-Y),

t E [a, b].

'Y

This proves that The improper integral (1.42)

10AU(t - s)(f (s) - f (t)) ds = lim / 610 0

AU(t - s)(f(s) - f(t)) ds

t

exists, and the convergence is uniform in t E [a, b] C (0, T). Hence we have, as b 10, (1.43) Ayb(t)

10 AU(t-s)(f(s)-f(t)) ds+(U(t)-I)f(t), 0 < t < T.

Therefore, assertion (1.41) follows from assertions (1.40) and (1.43). (c) Finally we show that

y(t) E C1((0,T);E), y'(t) = Ay(t) + f (t),

0 < t < T.

First we remark that the function y6(t) is continuously differentiable on the interval (6, T) and satisfies

tad ys(t) =

dt

10 t

_

(U(t - s) f (s)) ds + U(S) f (t - 6)

b

AU(t - s)f(s) ds + U(6)f(t - 6), 0

1.3 THE LINEAR CAUCHY PROBLEM

37

since the semigroup U(t) is real analytic for t > 0. Thus we have for 6 < t < T t

t=yeO

o

AU(t 0

+

I

- s)(.f (s) - .f (t)) ds + U(b)f(t - 6)

t-b AU(t - s)f(t) ds

AU(t - s)(f (s) - f (t)) ds + U(b) f (t - b)

= 0

-

jt- ds (U(t - s) f(t)) ds AU(t - s)(f(s) - f(t)) ds + U(b)(f(t - 6) - f(t)) + U(t)f(t). 0

But, by estimate (1.21), we have for any closed interval [a, b] of (0, T)

IIU(6)(f(t - b) - f(t))II < MoLabb-',

(1.44)

t E [a, b],

where 0 < 6 < a/2 and Lab > 0 is a Holder constant for the function f on the interval [a/2, b]. Therefore, combining assertions (1.42) and (1.44), we find that as b 10

f AU(t - s)(f(s) - f(t)) ds + U(t)f(t) = Ay(t) + .f (t),

Y,6(0

0t

uniformly in t over closed intervals of (0, T). Thus one can let 6 10 in the formula t

Y6(t) =

y6(r) dT + yb(e),

0 < b < e,

to obtain that y(t) =

t

(Ay(r) + f(T))d7 + y(E),

0 < e < t < T.

Since E is arbitrary, this proves that

y(t) E C'((O, T); E), and also

y'(t) = Ay(t) + f (t),

0 < t < T.

Summing up, we have proved that the function y(t), defined by formula (1.36'), belongs to C([0, T]; E) fl C' ((O, T); E), and is a (unique) solution of problem (NP) with initial condition y(O) = 0. The proof of Theorem 1.16 is now complete.

I.

38

THEORY OF ANALYTIC SEMIGROUPS

1.4 The Semilinear Cauchy Problem Assume that the operator A satisfies condition (1.17). Then we can define

the fractional power (-A)a for 0 < a < 1. The operator (-A)a is a closed linear, invertible operator with domain D((-A)a) 3 D(A). We let

Ea =the space D((-A)') endowed with the graph norm

of

(-A)a where

lixiJa = (IIxJJ' + II(-A)axII2)1/2 Then we have the following:

x E D((-A)a).

,

Proposition 1.17. (i) The space Ea is a Banach space. (ii) The graph norm IIxIIa is equivalent to the norm JJ(-A)axJJ. (iii) If 0 < a < Q < 1, then we have EQ C Ea with continuous injection.

Proof. (i) Assume that {xj} is a Cauchy sequence in Ea, that is, {xj} is a Cauchy sequence in E, { {(-A)axj} is a Cauchy sequence in E.

Then there exist elements x, y E E such that

f xj --+xinE, 1 (-A)axj -> y in E. Hence, by the closedness of (-A)a, we have

f x E D((-A)), 1 (-A)ax = y This proves that

JxEEa, xj --->x in E,,,.

(ii) Recall that (1.20)

II(-A)-all < M, 0
Hence we have for all x E ((-A)a) II(-A)axll2 <11x112 + II(-A)axll2 <

(M2

+ 1)II(-A)ax112,

so that (1.45)

Jl(-A)axJJ < JJXJJ. <- (M2 + 1)1/211(-A)axJJ,

x E Ea.

1.4 THE SEMILINEAR CAUCHY PROBLEM

39

This proves part (ii). (iii) We remark that

Hence we have D((-A)a) C D((-A)a'). Furthermore, in view of inequalities (1.20) and (1.45), it follows that, for all x E Ep = D((-A)Q), IIx$I« -< (1 + M2)1/2II(-A)'xII

< M(1 + < M(1 +

M2)1/21l(-A)axll

M2)1/2 1IxII

a

This proves part (iii). The proof of Proposition 1.17 is complete. Now we consider the following semilinear Cauchy problem: (SLP)

de u(to)

Au(t) + f (t, u(t)), = xo.

to < t < tl,

Here f is a function defined on an open subset U of [0, oo) x Ea, 0 < a < 1, taking values in E. We assume that f (t, x) is locally Holder continuous in t and locally Lipschitz continuous in x. That is, for each point (t, x) of U, there exist a neighborhood V C U, constants L = L(t, x, V) > 0 and 0 < -y < 1 such that (1.46)

IIf(S1,y1)-f(s2,y2)IICL(Is1-521'+IIy1-y2IIa), (Sl, yl ), (82, y2) E V

A function u(t) : [to, t1) -f E is called a solution of problem (SLP) if it satisfies the following three conditions:

(1) u(t) E C([to,t1);E) n C'((to,tl);E) and u(to) = xo. (2) u(t) E D(A) and (t, u(t)) E U for all to < t < t1. (3) ai = Au(t) + f (t, u(t)) for all to < t < ti. Our main result is the following local existence and uniqueness theorem (cf. [He, Theorem 3.3.3]; [Pa, Chapter 6, Theorem 3.1]):

Theorem 1.18. Let f be a function defined on an open subset U of [0, oo) x Ea, 0 < a < 1, taking values in E. Assume that f (t, x) is locally Holder continuous in t and locally Lipschitz continuous in x. Then, for every

(to,xo) E U, problem (CP) has a unique local solution u(t) in the space C([to,t1]; E) n Cl((to,t1); E) where t1 = t1(to, xo) > to. Proof. (1) We fix a point (to,xo) of U, and choose constants s > 0 and S > 0 such that V = {(t, x) E [0, oo) x Ea : to < t < to + e, II(-A)ax - (-A)axoll < s} C U,

I. THEORY OF ANALYTIC SEMIGROUPS

40

Il f(t, x) - f(s, y)II <_ L (It - sI1 + II(-A)a'x - (-A)ayII), (t, x), (s, y) E V.

Here L = L(to,xo,V) > 0 is a local Holder constant for the function f. By part (c) of Theorem 1.12, it follows that, for some constant Ma > 0, II(-A)aU(t)II < Mat-a,

(1.47)

t > 0.

Now choose a real number t1 such that (1.48)

0
11(1-a)

where

B=

(1.49)

max

to
II f(t,xo)II.

Further, one may assume that t1 - to is so small that (1.50)

II U(t - to)(-A)axo - (-A)axo11 < 2,

to < t < t1.

(2) We let

Y = the space C([to, t1]; E) of continuous functions on the interval [to, t1] taking values in E.

The space Y is a Banach space with the maximum norm I IIYI I I

= tomax IIy(t)II.

We define a mapping as follows: t

U(t - to)(-A)axo + j (-A)aU(t -

(-A)-ay(s)) ds.

to

We have to verify the well-definedness of the mapping 4. (a) First, by Theorem 1.16, it follows that U(t - to)(-A)axo E C([to, oo); E) fl C°°((to, oo); E).

1.4 THE SEMILINEAR CAUCHY PROBLEM

41

(b) Next we show that, for 0 < Q < 1 - a, (1.51)

j (_A)U(t - s)f (s, (-A)y(s)) ds E t

o

which proves that (1.52)

-by(t) E C([to, ti]; E) fl C'((to, ti]; E), 0<0<1-a.

We have for to
(-A)"U(t + h - s) f (s, (-A)-IXy(s)) ds

L -

t(-A)aU(t - s)f(s, (-A)-ay(s)) ds t

_ L(_A)u(t + h - s) - U(t - s)] f (s, (-A)y(s)) ds t+ h

+ ft

(-A)°U(t + h - s) f (s, (-A)-IXy(s)) ds.

We estimate each term on the right of the last equality.

(b-1) By the continuity of y(t) and inequality (1.46`), it follows that f (t, (-A)-°'y(t)) is continuous on [to, tl]. Hence there exists a constant N > 0 such that

jIf(t,(-A)-'y(t))II < N, t E [to,ti]

(1.53)

Further, by part (d) of Theorem 1.12, we have for each 0 < /3 < 1 - a (1.54)

11(U(h) - I)(-A)°U(t - s)II < Cphajj(-A)a+I U(t - s)II,

h > 0,

with a constant Co > 0. Thus, combining inequalities (1.54) and (1.47), we obtain that (1.55)

II (U(h) - I)(-A)°U(t - s)II < CghQM°+p(t -

h > 0.

Hence, it follows from inequalities (1.55) and (1.53) that t

(1.56)

to it.,

(-A)"[U(t + h - s) - U(t - s)] f (s, (-A)-°y(s)) ds t

< C#M°+pN J (t - s)-"-a ds by to

((t1 -

to)1-a-al

ha,

0 < a < 1 - a.

42

THEORY OF ANALYTIC SEMIGROUPS

I.

(b-2) Similarly, we have t+ h (1.57)

(-A)U(t + h - s) f (s, (-A)y(s))

II J rt+h


ll(-A)'U(t + h - s)JI ds

t

t+h

< NMQ.

It.

(t+h-s)-ads

(NM,,

1-c) h1

Therefore, assertion (1.51) follows immediately from estimates (1.56) and (1.57).

(3) We let

S = {y E Y : y(to) _ (-A)axo, t m x II y(t) - (-A)axojj< 6 We remark that S is a non-empty closed and bounded subset of Y, and so it is a complete metric space. We show the following: Claim I. 4D :S->S.

To show this, it suffices to verify that (1.58)

(-A)axoll < 1,

1I

to < t < ti.

By (1.50), (1.47), (1.46') and (1.49), we have for all t e [to,ti] 11,1y(t)

- (-A)axo 11

U(t -

to)(-A)axo

- (-A)axo

t

+

(-A)aU(t - s)f (s, (-A)-ay(-S)) ds to

< IIU(t - to)(-A)axo - (-A)axoJI

i + <

b

2

i

t

' t

(-A)a'U(t - s) f (s, xo) ds

+ MaL

f(t'- s)-a IJy(s) - (-A)axo Ip ds o

1.4 THE SEMILINEAR CAUCHY PROBLEM

+ M"B

J

43

(t - s)-" ds

itto

+M"(Lb+B)J (t-s)'"ds to it.

M,,, (L6 + B)

b

In view of condition (1.48), this proves inequality (1.58) and hence Claim 1.

Next we show the following: Claim H. III(Dyi - 41)y2lII <- I I I Iyi - y2I I I, yi, y2 E S.

By (1.47), (1.46') and (1.48), we have for all t E [to,t1]

II(yi(t) - 'Dy2(0II

ft

= 11

(-A)"U(t - s) [f (s, (-A)-"yi(s)) -f (s, (-A)"y2(s))] dsl

to o

<

o

t

< M"L

II(-A)"U(t-s)II

'

If (s,(-A)-"yl(s)) -f (s,(-A)"y2(s))IIds

t

J (t - s)-"IIyl(s) - y2(s)II ds to

<M"L (t1

-to)1-" 1

-a

IIJy1 -Y2111

I IIIY1- y2I II.

Therefore, by Claims I and II, one can apply the contraction mapping theorem to 4 to obtain that There exists a unique fixed point y E S of the mapping 4D, that is, 4Dy = y, or equivalently, (1.59)

Y(t) = U(t - to)(-A)"xo + Jt (-A)"U(t - s)f(s, (-A)-"y(s)) ds. to

Since y = it follows from assertion (1.52) that For each to > to, there exists a constant Lto > 0 such that

IIy(t) - y(s)II < Lt' It - sIn,

t, s E [to, tl].

By inequality (1.46'), this implies the local Holder continuity of the function f (t, (-A)-"y(t)). Indeed, we have for all ] t, s E [t'o, t1 ], to < t'0 < ti,

11f (t, (-A)-"y(t)) - f (s, (-A)-"y(s))

11

44

THEORY OF ANALYTIC SEMIGROUPS

I.



(NP)

de = Ax(t) + f (t, (-A)-"y(t)), 0 < t < T, x(0) = x0.

But we know that f (t, (-A)-"y(t)) belongs to C([to, tl]; E) fl CQ'((to, tl]; E)

where 0 < Q' < min(y,1 - a). Hence, applying Theorem 1.16, we obtain that the function

u(t) = U(t - to)xo + f U(t - s) f (s, (-A)-"y(s)) ds tot

belongs to C([to, ti]; E) fl C1((to, ti ); E), and is a unique solution of problem (NP). It remains to show that

u(t) = (-A)-"y(t), to < t < ti,

(1.60)

which proves that the function u is a solution of problem (SLP). Since we have for all to < t < t1

U(t - to)xo E D(A) C D ((-A)01), 1 ftt° U(t - s) f (s, (-A)-"y(s)) ds E D(A) C D ((-A)") , in view of part (b) of Theorem 1.12 and formula (1.59) it follows that

(-A)"u(t) = (-A)"U(t - to)xo +

J

t

(-A)"U(t - s).f(s, (-A)-"y(s)) ds

to

= U(t - to)(-A)"xo + j

A)"U(t - s)f (s, (-A)y(s)) ds t(

o

= y(t) This proves formula (1.60). (5) Finally, we prove the uniqueness of solutions of problem (SLP). Assume that u, v are two solutions of problem (SLP):

di = Au(t) + f (t, u(t)), to < t < t1, { u(to) = xo;

di = Av(t) + f (t, v(t)), to < t < t1i v(to) = xo.

1.4 THE SEMILINEAR CAUCHY PROBLEM

45

Then we have by Theorem 1.15

u(t) = U(t - to)xo +

J U(t - s) f(s, u(s)) ds, o

v(t) = U(t - to)xo +

f U(t - s) f(s, v(s)) ds, o

and hence

(-A)"u(t) = U(t - to)(-A)"xo + J t(-A)"U(t - s)f(s,u(s)) ds, (-A)"v(t) = U(t - to)(-A)"xo + J (-A)"U(t - s) f (s, v(s)) ds. t to

This implies that the functions (-A)"u(t) and (-A)"v(t) are both fixed points of the mapping '. (Here we remark that the interval [to,tl] should be replaced by a small interval [to, t1 - h], h > 0.) Thus, by the uniqueness of fixed points of ', it follows that

(-A)"u(t) = (-A)"v(t), to < t < ti, so that

u(t) = v(t), to < t < ti. The proof of Theorem 1.18 is now complete.

CHAPTER II

SOBOLEV IMBEDDING THEOREMS In this chapter we study the imbedding characteristics of Sobolev spaces of LP style that render these spaces so useful in the study of partial differential equations. We give a complete proof of the most important of the imbedding

properties of the spaces Hm'P(R') (Theorems 2.15 and 2.18 and Corollary 2.22). For more leisurely treatments of Sobolev spaces, the reader might refer to Adams [Ad] and also Friedman [Fr].

2.1 Holder Spaces and Sobolev Spaces Let S2 be an open subset of R. If m is a non-negative integer, we let C'(St) = the space of m times continuously differentiable functions u in Q, equipped with the topology of uniform convergence on every compact subset of 1 of the functions u and their derivatives D'u of order lal < m, and

Bm(f) =the subspace of Cm(S?) consisting of functions u having all their derivatives Dau, I aI < m, bounded in the whole of SZ, equipped with the topology of uniform convergence over 52 of the functions u and their derivatives D"u. On the space Bm(c), we introduce seminorms

lulj,,,,n = max sup ID"u(x)l,

0 < j < in,

frI_) xES2

and a norm I1ullm,-,t1 = om x lul,,,,,q. We remark that 1lu1Io,o0,il _ lulo,oo,n = sup lu(x)l, xE1

Typeset by .AMS-TjX 46

2.1 HOLDER SPACES AND SOBOLEV SPACES

47

lu m,oo,n = max D ul0,oo,fj. 1«I=m

If S2 = R', we simply write ' Im,oo = I ' Im,oo,R^,

I

' Ilm,oo = I ' IIm,oo,R'

II

Let 0 < 9 < 1. A function u defined on S2 is said to be uniformly Holder continuous with exponent 9 on Sl if the quantity [u]9Q

=

Iu(I )-yley)I

sup x#y

is finite.

If a = m + 9, where m is a non-negative integer and 0 < 9 < 1, we let Ba(st) =the space of functions in B"Z(Q) whose m-th order derivatives are uniformly Holder continuous with exponent 0 on R. We define a seminorm

IuIa,oo,SZ = max [D«u]e;jj = max sup

ID u(x) - D u(y) I

I«I=m m'yEn

I«I=m

=#y

IX - yI6

and a norm Ilulla,oo,Sl = max {Ilullm,oo,52, Iula,oo,12}

If SI = R'", we simply write I

II

la,oo = 1

Ia,oo,R' ,

Ia,oo = II Ila,oo,R^.

If m is a non-negative integer, we define the Sobolev space

H' P(Il) =the space of (equivalence classes of) functions u E LP(S2) whose distribution derivatives Dau of order IaI < m are all in LP(Sl). We introduce seminorms I ' Ik,P,fl, 0 < k < in, as follows:

lulk,p,Q =

(fn EI«I=k I D«u(x)IP dx)1/P

if 1 < p < oo,

maxi«I=k ess supxES I D«u(x)I

if p = oo.

II. SOBOLEV IMBEDDING THEOREMS

48

If fI = R', we simply write I

' Ik,p = I ' Ik,P,R^.

Furthermore, we introduce a norm 1/p

if 1 < p < oo,

I D°u(x)IP dx)

IIuIIm,P,S1=

maxi«j<m ess supxEc JD"u(x)I 1/P

_

if p = 00

if 1 < p < 00,

(I:-k=,, lulk,p,fZ)

maxo
if p = 00.

We remark that Ho,P(1) = LP(Sl),

'

11

1Io,p,S1 = I '

Io,p,iz.

If fI = Rn, we simply write 11

'

11-'P =

II

'

IIm,p,R'.

2.2 Interpolation Theorems In this section, we consider the problem of determining upper bounds for L''-norms of derivatives DQu, 101 < m, of functions u E H',P(Rn), in terms of the L9-norms of u and its derivatives D"u of order Ial = m. Following Friedman [Fr], we define

luIP,S1 =

(f s11 u(x)IP dx)1/P

if 0 < p < oo,

1u1-n/P,oc),Q

if -00 < p < 0,

lulo,.,s1

if p = ±00.

If Q = Rn, we simply write I

' Ip = I ' IP,Rn.

The purpose of this section is to prove the following:

Theorem 2.1. Let 1 < p < oo, 1 < q < oo and let j, m be integers such that 0 < j < m. If m - j - n/p is not a non-negative integer and if 1

r

+a

_ n

1

m +(1-a)1

p

n

q

then we have, for all functions u E Co (Rn), (2.1)

1D'ulr < CI uI ',plula,9".

Here C = C(m, n, p, q, j, a) is a positive constant.

m

< a < 1,

2.2 INTERPOLATION THEOREMS

49

Remark 2.2. Since a > j /m, it follows that 1

m

j

r

n

m

+a1+(1-a)1
a

Hence we have r > 1 or r < 0. The proof of Theorem 2.1 is carried out in a chain of auxiliary lemmas. The proof given here is essentially due to Gagliardo [Gal. Although it is rather lengthy, the techniques involved are quite elementary and based on little more than simple calculus combined with astute applications of Holder's inequality.

Lemma 2.3. Let 0 < a < b < c < oo. Then there exists a constant -y = 7(a, b, c) > 0 such that (2.2)

(lulc,.)(b-a)/(,-a)

lulb,0 < y

uE

oo)(c-b)/(c-a)

(lula

B' (Rn).

Proof. (i) The case 0 < a < b < c < 1: We remark that (2.3)

1U(X) u(y)I

(lu(s) - u(y)I

Ix ylIx-yl

(lu(s) - u(y)I

-_

lx -yl

Hence, if 0 < a < b < c < 1, then we have lulb.

.)(b-a)/(c-a) (hula

< (I ul c

oo)(c-b)/(c-a)

Thus we have only to consider the following two cases:

a=0 and c=1. (i-1) The case c = 1: We have for u E B1(R)

lu(x) - u(y)I =

1

dt (u(y + t(x - y))) dt

J0

au

1

o

i=1

8x(y + t(x - y))(xi - yi) dt i

n

JDiulo,.lxi - y i=1

lull,.v/nlx - yl. But we remark that 8u 01xi

(x)I

- lim h-0

u(x + hei) - u(x) I

h

< sup XOY

lu(x) - u(y)l lx

- yl

(c-b)/(c-a)

II. SOBOLEV IMBEDDING THEOREMS

50

so that < sup Iu(x) - u(y)I

lull,,,. = max sup

I
lx

x0y

- yI

Summing up, we have proved that

lull, C sup lu(I)

y)I

< Mull,,,,,,

- yI

u E BI(Rn).

In view of formula (2.3) with c = 1, this proves that Iulb,oo C

(b-a)/(1-a)

(v'

(Iula,oo)

(1-b)/(1-a)

u E B1(Rn).

(i-2) The case a = 0: Since we have I u(x) - u(y)I C 21ulo,00,

it follows that (2lulo,oo)(c-')/c,

Iulb,. < (lulc,oo)b/c

u E Bc(Rn).

(ii) The case 0 < a < b = 1 < c < 2: We have forallp>0 49U

au

1

p

a xi

au

JZ,+P (Ox (x)

ayi

(y)) dyi + n (u(x +Pei) - u(x))

But it follows that Iula,o0pa

l u(x + Pei) - u(x)I < {

21ulo,00

if a > 0, if a = 0,

and that

N

(x) C7xi

lulc,oolyi - xiIc-1

au

yi

(y)

Vnlulc,oolyi - xilc-I

if 1 < c < 2, if c = 2.

Thus, if a > 0, we have xi+p

1

C

=

xilc-1 dyi +

P

xi

1/-n- luIc,ooPc-I

c

+

IuIa,ooPa-1,

and hence (2.5)

lull,.

`Llulc,o0Po-1

+

lula,.Pa-I

for all p > 0.

Iula,ooPa-1

2.2 INTERPOLATION THEOREMS

51

Therefore, inequality (2.2) for a > 0 follows by minimizing the right-hand side of (2.5) with respect to p: lull,.

y

oo)(c-1)/(c-a)

(lulc,oo)(lula

u E Bc(Rn).

Similarly, if a = 0, we have (2.6)

lull,. <

NFn

lul,,,.pc-1

for all p > 0.

+ 2lulo,,,,,P-1

C

Hence inequality (2.2) for a = 0 follows by minimizing the right-hand side of (2.6) with respect to p: 7(lulc,.)l/c(lulo,.)(c-1)/c,

lull,.

u E Bc(Rn).

The proof of Lemma 2.3 is complete.

Lemma 2.4. Let p < -n, q > 0. Then we have (2.7)

lulo,.

(lulq)gl(q-p),

<7(lulp)p1(p-q)

u E co (Rn).

Proof. (i) The case p < -n: We remark that l u(x) - u(y)i

Iul-n/p,,,.Ix - yl

-n/P = I uI Pl x - yl -n/p,

since 0 < -n/p < 1. Hence it follows that (2.8)

lu(x)l

Julplx - yl-n/p + I u(y)I

(i-a) If q > 1, we integrate both sides of inequality (2.8) over the ball Bp(x) = {y E Rn : l Y - xI < p) to obtain that P

Iu(x)IVnPn < Iuln

(10

r-nlprn-1dr wn+ n-n/p

+

< lul pwn np-

f

f

o(x)

lu(1J)I dy 1/q

lu(y) I' dy

Bo(x) (n/p) n-n/p P < lulp(nVn)n (n/p) +(jlnpn)1-1/, lulq

f

1-1/q 1 dy

Bo(x)

Here Vn is the volume of the unit ball and wn = nVn is the surface area of the unit ball. Hence we have (2.9)

lulo,oo C lulp 1

P

-n/p (1/p) - (Vnpn)-l/,lulq for all p > 0.

II. SOBOLEV IMBEDDING THEOREMS

52

Therefore, inequality (2.7) for q > 1 follows by minimizing the right-hand side of (2.9) with respect to p. (i-b) If 0 < q < 1, we find from inequality (2.8) that Iu(x)I C IuIPIx-yl-n/P+(Iulo,oo)1-glu(y)lq.

Hence, integrating both sides over the ball BP(x), we obtain that

nIu(x)IVnpn (-nn//pp) +(lu1o,oo)1-gl.ulq,

C

so that -nlP (2.10)

lulo,.

Iul p

<

(1/P) +

1

(Vnpn)-1(IulO

oo)1-glulq

Therefore, inequality (2.7) for 0 < q < 1 follows by minimizing the right-hand side of (2.10) with respect to p.

(ii) The case p = -n: By inequality (2.4), it follows that lu(x) - u(y)I + Iu(y)I

lu(x)l

< vIuI1,.Ix - yI + Iu(y)I = v nlulplx - yI + I u(1J)I

Therefore, inequality (2.7) can be obtained just as in case (i). The proof of Lemma 2.4 is complete. 0

Lemma 2.5. Let r < p < -n, q > 0. Then we have (2.11)

<'Y(lUlP)P(r-q)/r(P-q)(Iulq)q(r-P)/r(q-P)

lulr

u E Co (R n).

Proof. We recall that lull,,,,, < sup Iu(x) ix

X:AY

lu(x) (x) -

I

-

p/ r

- (Iu(x) l

< \nlul1,oo.

U(y)I

- U(/P l

/

l u(x) - u(y) I 00)(r-P)lr

Iuln/r (2lu10

V/n,UlP/r

(r-P)/r

if p < -n, if p = -n, (2Iulo,00)(r-P)lr

2.2 INTERPOLATION THEOREMS

53

and hence (2.12)

(lul0,oo)(r-P)lr.

P)Plr

lulr < y (l ul

.

But we have by Lemma 2.4 lul0,oo

(2.7)

1 (l ul

P)Pl(P-q)

(l ul

q)gl(q-P)

Therefore, combining inequalities (2.12) and (2.7), we obtain that P)P/(P-q)

l ul r < 7 (lulP )P/r ((l ul P)P(r-q)/r(P-q)

<1(lul

(l u l

(l ul q)ql

(q-P)) (r-P)/r

q)q(,-P)l r(q-P)

This proves Lemma 2.5.

Lemma 2.6. Let p < -n, 0 < q < r. Then we have (2.13)

Jul,_

1(lulP)P(r-q)/r(P-q) (lulq)q(r-P)lr(q-P)

C

u E Co (R").

Proof. By Lemma 2.4, it follows that lul0,oo

:5

(lulq)gl(q-P)

1(lulP)p1(p-q)

Hence we have 1/r

lu(x)lr dx)

l ul r =

(JR-

=

Un (l

)1/r

lu(x)l r-glu(x)lq dx 1/r ul0,oo)(r-q)lr

u

(x)dx

(J"

<

< 1 (l ul

(7(lulP)P/(P-q)

P)P(r-q)l r(P-q)

(lulq)gl(q-P))(r-q)/r(lulq)g/r (l ul

q)q(r-P)l r(q-P)

The proof of Lemma 2.6 is complete.

Proposition 2.7. If A < ,u < v, then there exists a constant 'y(A,,u, v) > 0 such that (2.14)

lull/µ < y

(lull/),)(v-µ)/(-a)

(lull/,,)u E Co (R").

II. SOBOLEV IMBEDDING THEOREMS

54

Proof. (i) The case 0 < A <,a < v: We let y(v - A) P = A(v - /1)

Then we have

1
P=

= µ(v - A) A).

P

v(p -

p - 1

We remark that

p
A>0

A=0

If A > 0, by Holder's inequality, it follows that It

lull/I' = Un l u(x)I '/I' dx I /

I'

= (JR \lu (x)IlI u(x)I

dx J

1

n

, l'/Y

lu(x)ll /A dx)

<

(lul1/,X)(v-µ)/(v-A)

(JR n lull/v)

u(x)ll/v dx ) (µ-A)/(-A)

This proves inequality (2.14) for y = 1. If A = 0, we have

lull/µ =

Un

lu(x)l(--I')/I'v!u(x)ll/vdx 1

(Iulo,oo)(°-µ)/v l

I'

u(x)ll/v dx )

I'

Un

(lul1/,\)(v-I')/(v-A) (lull/,)(``-A)/(v-A)

This proves inequality (2.14) for y = 1. (ii) The case A < p < v < 0: Since we have

0 < -nv < -nµ < -nA, it follows from an application of Lemma 2.3 with a = -nv, b = -nµ and c = -nA that I uI

-nI',O

y

(lul-nv,.)(µ-A)/(v-A)

.

2.2 INTERPOLATION THEOREMS

55

This implies that

(lull/a)(lull/v)(µ-a)/(v-a)

lull/N. c

(iii) The case -1/n
Then we have

.

< 0, we let

r=µ, q=v 1

1

r0.

Hence it follows from an application of Lemma 2.5 that (lull/µ)(µ-a)/(v-a)

(lull/a)(v-µ)/(v-a)

lull/µ < 7

.

If y = 0, applying Lemma 2.4, we obtain that

lull/µ =

lulo,c

oull/v)

-Y (lull/,\)

(iv) The case A < -1/n < µ < 0 < v: If we apply part (iii) with A = -1/n, we obtain that lull/µ <_ -r(lul-n)

n(v-µ)/(nv+1)

(lull/v)(nµ+l)/(nv+l)

But we have by Lemma 2.3

lul-n =

lull,.

-r(lul-na,oo)(nµ+l)/n(µ-A) (lull/a)(nµ+l)/n(µ-A)

(lul-nµ,.)(nA+l)/n(A-0)

(lull/µ)(nA+l)/n(A-µ)

_ `Y

Hence it follows that lull/µ <_ 7

(lull/v)(nµ+l)/(nv+l) ((lull!

a)(nµ+l)/n(µ-A)

(lull/")(na+l)/n(a-µ) X

=

(lull/v)

(nµ+l)/(nv+l)

"ate =,, -+1 A-4

np+l -4

(lull/a)

-+1 4-X

so that lull/µ _< y

(lull/A)(v-µ)/(v-A)

(lull/v)(µ-\)/(v-a)

.

II. SOBOLEV IMBEDDING THEOREMS

56

(v) The case A < µ = -1/n < 0 < v: Since -nA > 1, applying Lemma 2.3 with a = 0, b = 1 and c = -nA, we obtain that (lulo,oo)(-nA-1)/(-na).

(lul-na,oo)-1/(nA)

y

IuI-n = lull,oo

But we have by Lemma 2.4 with p = -n and q = 1/v y(lul-n)nv/(nv+1)

Iulo,oo C

(lull/v)1/(nv+l).

Hence it follows that

lul-n < - y (lull/a)

1/(na)

)nv/(nv+1)

(lull/v)1/(nv+1)

1+1/na

((IuI-n

so that (lull/A)(nv+l)/n(v-A)

y

Iu111µ = lul-n

(lull/v)

-(nA+l)/n(v-A)

(lull/v)(µ-a)/(U-a)

=y

(vi) The case A < p < -1/n < 0 < v: If we apply part (v) with A = p, we obtain that lu

lull/µ)(nv+1)/n(v-µ)

-n C y

)(nµ+1)/n(v-µ) (,u,1/

But, since 1 < -nµ < -nA, it follows from an application of Lemma 2.3 with a = 1, b = -nµ and c = -nA that lull/µ = lul-nµ,00

y(lul-na,oo)(nµ+1)/(nA+l)

- y (lull/a)

(nµ+1)/(nA+1)

(lull,o0)-n(µ-A)/(nA+1)

(IuI-n)n(µ-A)/(nA+1)

Hence we have (lull/a)(nµ+l)/(na+1)

lull/µ C y x

(lull/,,)

((lull/µ)(nv+l)/n(v-µ)

(nµ+1)/n(v-µ))-n(µ-a)/(nA+1),

J

so that lull/µ

y

(vii) The case -1/n < A < 0 < p < v: Since we have 1

0
1

2.2 INTERPOLATION THEOREMS

57

it follows from an application of Lemma 2.6 with p = 1/A, q = 1/v and

r = 1/p that y (lull/,\)

lull/p

(v- 1i)/(v-A)

lull/v)

(viii) The case A < -1/n < 0 < p < v: If we apply part (v) with v = p, we obtain that (lull/a)(np+l)/n(p-A)

l ul -n C 7

(lull/p)-(na+l)/n(p-A)

But we have by part (vii) with A = -1/n (lull/v)(np+1)/(nv+1)

y(luI-n)n(v-p)/(nv+1)

lull/, C Hence it follows that

(lull/v)(np+l)/(nv+l)

lull/, X

(lull/A)(np+l)/n(p-A) )-n(v-p)/(nv+l),

(lul1/p)(nA+1)/n(p-A)

so that

lull/p S y

(lull/a)(v-p)/(v-a)

(lull/v)(p-a)/(v-a)

The proof of Proposition 2.7 is now complete.

First we consider inequality (2.1) when m = 1 and a = 1:

Lemma 2.8. If n < p < oc, then there exists a constant C = C(p, n) > 0 such that

lull-n/,,. < Clull,r, u E Co(Rn).

(2.15)

Proof. It suffices to show that (2.15')

lu(x) - u(y)l < Clull,nl x -

yll-n/P,

X, Y E Rn.

We let

d=lx - yl, and

G={zERn:lz-xl
lu(x) - u(y)l dz < IG lu(x) - u(z)l dz + f lu(z) - u(y)l dz, IG

II. SOBOLEV IMBEDDING THEOREMS

58

so that

cdnlu(x) - u(y)I < f

(2.16)

for some constant c > 0. Indeed, it suffices to note that the set G contains a ball Bd/2 of radius d/2 and that the volume of Bd/2 is equal to ,Irn/2

2nf(n/2 + 1)

dn.

We estimate each term on the right of inequality (2.16). (a) We make the change of variables

z = x+ p0,

p> 0, 0' E S1(0),

where S1(0) is the unit sphere about the origin. Then we have by Holder's inequality

IG lu(x) - u(z)Idz

I

d

<

f

I u(x + po,) - u(x))Pn-1 dp dQ

S1(o)

j j JO d

1,(0)

dl

(u(x+to,))dtpn-Idpdjd

o

, (o)

pE

On

f0

(x+ta)dt

pn-1dpda

j-1

JPjgradu(x±ta) d tp1 dp d0 1(0)

jd

P

{JSl(0)

pd / p

Igrad u(x + ta) I

t('-I)/P

.

t(I-n) 'P dt d,

pn-1dp

o

pp

a) 1P t"-' dt do, (P-I>lP

p

x

(jSj(0)j0

t p=i dtdcr d

(

(p - 1)(p_1)/p wnJ

_ p-n p-1 C

(p-nWn)

Pn-I dP 1/P

P.-n/P

0

dp

Igradu(z)IP dz

(11--xl
(P-1)/P do+1-n/p

n+1-n/p

\IlP (R^ Igradu(z)IPdz)

,

2.2 INTERPOLATION THEOREMS

59

where con is the surface area of the unit sphere S1(0): 7rn/2

wn =

2n1'(n/2 + 1)

This proves that

J lu(x) - u(z)ldz <

(2.17)

C'Iu1l,pdn-n/P+1

(b) Similarly, we have

f Iu(y) - u(z)l dz <

(2.18)

C'Iul1,pdn-n1P+1

G

Therefore, carrying inequalities (2.17) and (2.18) into inequality (2.16). we obtain that clu(x) - u(y)l C C'l,ull,pol-n/p so that I u(x) - u(y)I C Clull,plx - yl1-n/P x, y E R'. The proof of Lemma 2.8 is complete. Lemma 2.9. Let 1 < p < n. We have, for all functions u E Co (Rn ),

p(n - 1) Iulo,np/(n-p) C 2(n - p)

(2.19)

n

/n

JD=ul1o,p

=]1 i1

Proof. (i) The case p = 1: (2.20)

lulo,n/(n-1)

11/n

1 `2

IDiulo 1 i=1

Since we have for 1 < j < n x;

U(X) = f Dju(x1 ... x.j ...

jDju(xi... ,

xj ...

xn)dx.j xn)dx.j

;

it follows that

2lu(x)l <

f

00

ID;u(x)ldx; + f j ID;u(x)ldxj

x' 0o 00

x

I D.ju(x)l dxj.

= 00

This gives that (2lu(x)I)n/(n-1)

_ ((

((2lu(x)I)n)1/(n-1)

n

j=j Here we need the following:

(f: lD'u(x)ldx'I

H. SOBOLEV IMBEDDING THEOREMS

60

Lemma 2.10 (Gagliardo). Let F ; be a function defined o n Rn-1 de, x;, , xn (1 < j < n) and belonging to the pending o n the variables x 1 , space Ln-1(Rn-1). Then the function F(x) on Rn, defined by the formula

xn)...Fj(xl .... .xj ... xn)...Fn(xl x = xl x2, ... ,xn E R n,

F(x)

=Fl(x2,...

xn-1),

belongs to the space L1(Rn) and satisfies the inequality

J IF(x)I dx < 11 n

IIF;IILn-1(Rn-1).

j=1

Granting Lemma 2.10 for the moment, we shall prove inequality (2.20). Now, applying Lemma 2.10 with Fj(xi,...

xj ..,xn)

00

Dju(x1i... xj... xn)I dx7 (f_oo

I

we obtain that n

/

0(2Iu(x)I)n/(n-1)dx

dx;)

< R^

Rn

dx

0

j=1

1/(n-1)

n

< )1, (fitn

IDju(x)I dx

This proves inequality (2.20). Proof of Lemma ,2.10. We prove Lemma 2.10 by induction on the dimension n.

(a) The case n = 2: Assume that F1(x2), F2(x1) E L1(R) Then we have by Fubini's theorem

ffrt2 I F(xi, x2)I dxldx2 =

=

ffIt2

f

R

IF1( x2)F2(x1)I dxldx2

IF1(x2)Idx2

f

R

IF2(x1)Idx1.

This proves the lemma for n = 2. (b) Now we assume that the lemma has been established for n - 1.

2.2 INTERPOLATION THEOREMS

61

By Holder's inequality, it follows that

J

F(x)Idxl I

IFi(xi)I (JR

IF2(2)IIF3(3)IF(x)Idx1)///

2(x2)in-1dx1ff

IFl(xl)I U IF

(n-2)/(n-1)

/ X

IFi(xi)I

I

F2(x2)In-idxl1/(n-2)

(JR

F3(3)I(n-1)/(n-2)(n-2)dxi)

X [(J

(n-3)/(n-2) (n-2)/(n-1) X

JR

(IF4(x4)I ... IFn(xn)I n-z n-3 dxl/ 1/(n-1)

// C IF1(xi)I 1

\ R

X

IF2(-i2

-idx1)

(JR (IF4(x4)I ...

n

1/(n-1) IF3(x3)In-idx1)

(JR

(n -3)/(n-1)

IFn(xn)I)(n-1)/(n-3) dxl)

IFj(xj)In-1dxl)

IF1(xl)I (JYR'

H ence

we have again by Holder's inequality

(2.21) I F(x)Idxi dxdxn

JR 1/(n-1) f.n

<

-1

(J-

IFi(i1)I

,j=2 R

j)I

n-1dxi

dx2 . .. dxn

IFi(xl)In-1 dx2 ... dxn 1

n

X

rj f IFF

[JH(JIF()I'_1dxI) J-2

(n-2)/(n-1)

1/(n-2)

dx2 ... dxn

II. SOBOLEV IMBEDDING THEOREMS

62

But we remark that the function

= (JR

l

1/ (n-2)

()I'-1dx1/ I

, xn (2 < j < n) and belongs to depends on the variables x2i , x7, Ln-2(Rn-2). Hence, applying the induction hypothesis to the last part of inequality (2.21), we obtain that

fj

(2.22) JRl

_ 7-2

R

I)I1dx1 )

1/(n-2) dx2...dxn

n

dxn1/(n-2)

H [f.n- (J

IF,(x7)In-1dx1l dx2 ... dxj ...

/

j=2

l

n

fj

(IIFiIILn-'(Rn-))(n-1)/(n-2)

j=2

Therefore, combining inequalities (2.21) and (2.22), we find that

Jn IF(x)I dx < f j IIFj j=1

This completes the induction and hence the proof of Lemma 2.10. The proof of inequality (2.20) (inequality (2.19) with p = 1) is complete. (ii) The case 1 < p < n: If we let v = IuIP(n-1)/(n-P) e

then we have

IUIO,np/(n-P)

= (JRn

U;. (I v O,n I

and also vE

\ (n-P)/np nP/(n-P) dx) I u(x)I Iv(x)I

(n-p)/np nl(n-1) dx

l(n-1))(n-P)/P(n-1)

H1,1(Rn).

Indeed, it suffices to note that the function Jul is absolutely continuous on Rn and so

Dw = (n - 1)pIuln(P-1)/(n-P)H3(I uI) a.e. in Rn.

n-p

2.2 INTERPOLATION THEOREMS

63

Further we remark that by the triangle inequality ID;IuI I < ID

so that

IDjvl < (n - 1)pluln(p-1)l(n-p)IDjuj

n-p

a.e. in Rn.

Since the space CO(Rn) is dense in H1'1(Rn), we can apply inequality (2.20) to the function v to obtain that (2.23)

Ivlo,n/(n-1) < 2 X11

CJR^

I Djv(x)I dx

1n

But we have by Holder's inequality (2.24)

I

IDjv(x)I dx

Rn

<

Iu(x)In(p-1)l(n-p)IDjuI dx

(n -pp f n _

<

(n-pp

Iu(x)Inp/(n-p)dx

(J

)

(p-1)/p

(jR^

IDju(x)Ipdx )

1/p

Therefore, combining inequalities (2.23) and (2.24), we find that (n-1)p/(n-p)

(

= lVlo,np/(n-p)

<

1 2(nn

-- 1pp

1 (n - 1)p

2 n- p

(p-1)lp n

(1

Iu(x)Inpl(n-p)dxl

j=

l1/n n(p-1)/(n-p) 11 Djuo,p (Iulo,np/(n-p)) j=1

I

so that IUIO,np/(n-p) < 2p(n

- p) fj I Djul o/p j=1

This completes the proof of inequality (2.19) and hence that of Lemma 2.9.

Now we consider inequality (2.1) when a = j/m. It is convenient to begin with a straightforward one-dimensional interpolation inequality which typifies and provides a basis for the proof of the more general case (Lemma 2.14):

II. SOBOLEV IMBEDDING THEOREMS

64

Lemma 2.11. Let 1 < p < oo, 1 < q < oo, 2/r = 1/p + 1/q and let I = [a, b] be a bounded, closed interval. Then we have, for all functions u E C2(I), (2.25)

(jIul(xlrdx)

1/r

1/p

/

1111+1/r-1/p

I

1/q

+8111-1 -1/r+l/p

Iu(x)I q dx1 CJfI

Here III = b - a, the length of the interval I. Proof. We let

b-a

a

4

and

( xl E [a,a+a], X2 E [a + 3a, b].

Sl

Then, by virtue of the mean value theorem, one can find a point x12 E (x1, x2)

such that u (x12) =

u(x1) - u(x2)

I

xl -x2

Hence we have for any x E [a, b] x

u'(x) = u'(x12) +

u"(y) dy

I12 x = u(x1) - u(x2) + J12 x1 - x2

u"(y) dy,

X12

so that lu,(x)I < Iu(x1)I + Iu(x2)I +

b

2a

I0(y)I

dy

a

Integrating both sides from a to a + a with respect to x1 and then from a + 3a to b with respect to x2i we obtain that <

1

Ja+a Iu(x1)I dx1 + 1 b lu(x2)I dx2

a2lu,(x)I

f

+ a2

2

Iu"(y)I dy

a

1

2

a+3a

b

6

lu(y)I dy + a2

jb lu"(y)I dy.

2.2 INTERPOLATION THEOREMS

65

Hence, by Holder's inequality, it follows that 1/r

b

f (a2lu,(x)I)r dx a 1

b

b

-2

a

I u(y)I dy + a2 a

_ (b - a)1/r

a

2

< (b -2 a)

f

f

(b

1/r

r

b

dx

I u"/ (y)I dy

b

Iu(y)I dy + a2

jb Iu"(y)I

dy

a

- a) 1-11q f b Iu(y)I9 dy a

1/p

b

+ a 2(b - a)1/r

(b - a)1-1/p f I u(y)I pdy a

2(b

- a)1/r+l-1/q

Iu(y)I' dy b

+ a 2(b -

a)1/r+1-1/p

(Ja

\1

l/p

I u"(y)Ip

dy/ Therefore, dividing both sides by a2 = (b - a)2/16, we obtain that 1/r

fb a

I u,(x)Ir dx I

b

< 8(b - a)-1-1/r+l/P

u(y)) q dy 1/p

b

+ (b -

a)1+1/r-1/p

f I u"(y)I p dy a

since 1/q = 2/r - 1/p. The proof of Lemma 2.11 is complete.

Lemma 2.12. Let 1 < p< oo, 1 < q < oo and 2/r = 1/p + 1/q. Then we have, for all functions u E C(O) ([0, oo)),

f

1/r

(00

0

Iu'(x)Irdx

< 4v

(f

1/2p

00

Iu"(x)Ip dx

0

l/2q

(oo

f

Iu(x)Iq dx

0

Here oo)) is the subspace of C2([0, oo)) consisting of functions u with compact support in the interval [0, oo).

Proof. (i) The case 1 < p, q < oo: For each function u E can choose a positive integer .£ = £(u) such that

u(x)=0 forallx>/.

([0, oo)), we

II. SOBOLEV IMBEDDING THEOREMS

66

If I is a bounded, closed interval in [0, oo), we introduce two functions K1(I) and K2(I) by the following formulas (cf. the right of inequality (2.25)): 1

K1(I) = III1+1/r-1/p

Iu"(x)I r

rl

dx)

(1,

1/q

K2(I) = 8111-(1+1/r-1/p) (f I u(x)I q dx) I

Let k be an arbitrary positive integer. Then we define a bounded, closed interval I1 = [0,a1],

where the positive constant al is chosen in such a way that

f al = t

if K1([0,

k]) > K2([0, 1]),

1 al > k, Kl([0,a1]) = K2([0,a,]) if K1([0,

<_ K2([0,

f]).

k]) Here we remark that K1([0, x]) and K2([0, x]) are continuous functions of x which enjoy the following properties: Kl ([0, x]) T oo

K2([0,x])

8

as x -4 oo,

(f

Iu(x)Ipdx)1/p

--' 0

as x

oo.

0

Similarly, we define a bounded, closed interval 12 = [al, a2],

where the positive constant a2 is chosen in such a way that

ifKi([a1,k])>K2([al,f]),

a2=a1+k 1 a2 > al + k, Kl([al,a2]) = K2([al,a2])

if Ki([a,,

C K2([al, f]) Continuing this process, we have after j steps (at most k steps)

[0, f] c U ii, i=1

since the length of each interval Ii = [ai_1, ai] is greater than £/k. But we have by Lemma 2.11 ) 1/r

Uli Iu'(x)I' dx < K1 ([ai-1, ai]) + K2 ([ai-1, ai])

f])

2.2 INTERPOLATION THEOREMS

if ai = ai-1 + 1,

2K1 ([ai-1, ai-1 + k])

G

67

2./K1 ([ai-1, ai]) K2 ([ai-1, ai]) if ai > ai_1 + k

2 (k)

1+1

/

r-l p /

lu"(x)IP dx)

(fl.

1

p

if ai = ai_1 + 112q

4 v " (i, I u"(x)jp dx 1/2p fll l7All (x)l9 dx)

ai > ai_1 + T.

if

Since r/2p+r/2q = 1, it follows from an application of Holder's inequality that lu'(x)lr dx

fo 0C

Iu'(x)Irdx

i

f r+l-r/p

< 12r \

l

r/Uli

(\k/l

(foo u"(x)lpdx) k

f

+ (4Y ,)r z=1

Iu(x)lq dx)

(x)IP dx)

(J Ilull r+1-r/p

r/2q

2p

1

+ (4V

< 2-

r/p

u"(x)Ip dx)

r/p

l

2p

E f I u(x)l9 dx

r/2q

7

Iu"(x)Ip dx

i=1

I

= 2r Qr+1-r/p kr/p-r

lull(x) Ip dx

I

r/ P I

(100c, r /2p

+ (4V 2-)r

(iu"(x)IP

dx)

/

Jo Therefore, letting k oo, we obtain that

r 2q

f

l

u(x)l9 dx)

l

(2.26) r/2p

00

f iu'(x)lr dx < (4V2)r (f

OF

r/2q

lu"(x)IP dx)

u(x)Iq (fo"o

dx)

I

(ii) The case p = oo or q = oo can be obtained by letting p --+ oc or q-+ oo in inequality (2.26), respectively. The proof of Lemma 2.12 is complete.

Corollary 2.13. Let 1 < p < oo, 1 < q < oo and 2/r = 1/p+ 11 q. Then we have, for all functions u E C0(-00,00), 1/r

00

CJ-

.

lu'(x)Ir dx)

1/2q

1/2p

< 4V2 (F. lu"(x)Ip dx)

u(x)Ig dx) U70.

II. SOBOLEV IMBEDDING THEOREMS

68

Proof. We let u+ = UI(0,001,

U_ =

UI(-00,0].

Then, applying Lemma 2.12 to the functions u+ and u_, we obtain that

f.

to

u'(x)I r dx

IoL (x)I r dx + 00

lu+(x)Ir dx o

r/2p

0

(4V2-)r

(100

r/2q

0

u"(x)IP dx)

(100 Iu-(x)Iq dx)

I

r 2p

) r/2q

(4,12)r (foo" I u+(x)I p dx)

+

((L

U000

Iu(x)Ipdx+fo

I u+(x)I q dx r/2p

00

Iu+(x)Ipdx)

00

x

Iu-(x)Iq dx + f 00

I u+(x)Iq

dx)

o

(1°

0o

r/2q

r/2p

Iu"(x)Ip

dx)

(F. I u(x)I q dx)

.

The proof of Corollary 2.13 is complete.

Lemma 2.14. Let 1 that 0 < j < m, and let r

n

p

,1

q

cc and let j, m be integers such

m

-mj

1

If m - j - n/p is not a non-negative integer, then we have, for all functions u E Co (R"), (2.27)

/"`

Iul,,r < CI uI !"p IUIo

Here C = C(m, n, p, q, j) is a positive constant.

Proof. Inequality (2.27) is obvious if j = 0. (I) The case j = 1: By induction on m, we show that inequality (2.27) holds for all m > 1. (I-1) The case m = 2: Applying Corollary 2.12 to the function Dlu(xl, ) (u E C0 2(R')), we obtain that

(f

jID1U(xi,xl)ITdxi < (4V2)r (J

ID iu(xl,x')Ipdxl) r

/2p

2.2 INTERPOLATION THEOREMS

69

r/24

x CC00 Iu(xl,x')I9dxl) Hence, by Holder's inequality, it follows that

J

fIDiu(xi,x1)Irdxi dx' R^-1

\/1

< (4,/2-)r j

R"-1

x

(JR I Diu(xl, x')I1 dx1

(fJu(xi,x1)Icixi)

r/2p

/

r/24

dx' r /2q

r/2p

UI ^Di

u(x)IP

dx)

(JR-

lu(x)I9 dx

< (4V2)rIuI2,p IUIo,q

so that

\ 1/r Dlu(x)Ir dx

(2.28)

< 4yGlul2/p IUloo/q

Un

1/r

(2.28')

Un

IDju(x)Irdx

< 4V2Iu12(p Iulo/q

Therefore, inequality (2.27) for j = 1 and m = 2 follows from inequalities (2.28) and (2.28'): (2.27)1,2

I UIl,r <- CIUll/p lulu/q ,

u E C2(Rn).

(1-2) Now we assume that inequality (2.27) has been established for m -1

(and j=1): (2.27)1,m-i

u 11,r

< C u 1/(m-1) Im-1,p I

u I

1-1/(m-1) Io,9

UE

Cm-1(Rn) 0

Then, applying inequality (2.27)1,,,,_l to the functions Dju, 1 < j < n (u E Co (R")), we obtain that ID7ull,p1 <_ CID,UIl

so that /(m-1)IUi1-rl/(m-1).

IUI2,p1 <_ CIuI

II. SOBOLEV IMBEDDING THEOREMS

70

But we have

Iull,r <

CIUII12

IUIO,q ,

u E Co (Rn).

-12

Hence we have

lull,r < C (lul /pm 1)lul1,r

1/(m-1)11/2

IuIO/q

/apm-1)lul(m-2)/2(m-1)Iul1/q

Clul

This proves inequality (2.27) for all m > 1 (and j = 1): (2.27)1,m

lulls <- CIuI!,P Iul0,g1/m,

u E Co (Rn).

(II) By induction on j, we show that inequality (2.27) holds for 0 < j < M.

(II-1) Step (I) establishes the case j = 1. (11-2) We assume that inequality (2.27) has been established for j - 1: (2.27)-l,e

lul;-l,r'

< Clul'' l)RIul0

g(j-1)1'

e > j - 1.

Let m be an arbitrary integer such that

0<j<m. Since m - 1 > j - 1, one can apply inequality (2.27) j-1,,,,,-l to the functions

Dju, 1 < j < n (u E Co (R')), we obtain that ID;uI;-l,r

<- CIDjul((

so that Clul(,y,P1)/(m-l)lUll,gi

ILIJ,r <

But we have

lull,q c Clul /,P

l ul i,r

Clul(,

1)/(m-1) (lull/ n, Iulo-1/m P

Clulm/,P lulu

P

,q

/"'.

This completes the induction and hence the proof of Lemma 2.14.

2.2 INTERPOLATION THEOREMS

71

Proof of Theorem 2.1. (i) The case a = j/m: Iulj,r < CIuI!;IuIo,9J/m. This is nothing but inequality (2.27) in Lemma 2.14. (ii) The case a = 1: (2.29)

ID'uIr < Clul,n,p,

1

j

1

m

r

n

p

n

Here we remark that

1
lulr < Clulm,P,

(2.30)

r

=

1

-m

p

n

(ii-1-a) The case m = 1:

lulr < Clull,p,

(2.30)1

1

r

=

1

-

p

1

n

Since 1 - n/p is not a non-negative integer, it follows that

p4n. If n < p < oo, then we have np r= n-p <0.

Hence it follows from Lemma 2.8 that

Iulr = IUI-./r,. = Ikl1-nIp,c < Clull p. If 1 < p < n, then we have

r>0.

Hence it follows from Lemma 2.9 that I uI

r=

p(n -1)

I

tIO,npl(n-p) <-

2(n - p) Iuli,p

Therefore, we have proved inequality (2.30)1.

II. SOBOLEV IMBEDDING THEOREMS

72

(ii-1-b) We assume that inequality (2.30) has been established for m - 1: 1

1

m-1

r'

p

n

lulr' < C lulm.-l,p,

(2.30)m-1

We let

m

1

1

r

p

n

1

1

1

1

m-1

r1

r

n

p

n

Then, since m -1- n/p is not a non-negative integer, we can apply inequality

(2.30),n_l with r' = r1 to the functions Diu, 1 < i < n, to obtain that (2.31)

IDiulr, < CIDi411m-1,p C CJUJm,p

If r < 0, then we have

1
Hence, by inequality (2.30)1 with p = rl, it follows that

lulr C Clull,rl.

(2.32)

Thus, combining inequalities (2.32) and (2.31), we obtain that Jul, < CJUJ1,ri < CIUJm,p.

If r < 0, then we have

rl > 0

r1 > n,

rl <0 = -

Ti

Julr =

=

[--] -1+h, h= -r r

JUJ-n/r,oo = 14111-n/rl,oo

[-nr

]

C1u11,r1.

Combining this inequality with inequality (2.31), we obtain that Julr < Clulm,p.

If rl < 0, then we have JUJr =

JUJ-n/r,oo

= sup IDLu(x) =0v

- Dtu(y)l

Jx - yJh

.

2.2 INTERPOLATION THEOREMS

73

IDe-1Diu(x) - De-1Diu(y)I

sup

Ix - yIh

=#v


max IDiulri.

1
Therefore, combining this inequality with inequality (2.31), we find that
Iulr

(ii-2) We assume that inequality (2.29) has been established for j - 1: I

Di_1

ulr,

1

< Cl u b ,,,,,,

r, =

j-1 + - m n n P 1

1

j

1

m

r

n

p

n

We let

.

Then we remark that

j-1 +-- m-1

1

1

r

n

p

n

and that

(m - 1) - (j - 1) - n/p is not a non-negative integer. Hence we have by the induction hypothesis IDi-1vlr < CIVIm,_1,p,

v E Co -1(R'n)

Therefore, applying this inequality to the functions Diu, 1 < i < n, we obtain

that ID'uIr = ID1-1(Du)Ir < C Dulm_1,p < Cluim,p. This completes the proof of inequality (2.29) (inequality (2.1) with a = 1).

(iii) The case J /m < a < 1: We let

11

q

m

1

r2

n

p

n

0=

1-a 1 - j/m*

and

II. SOBOLEV IMBEDDING THEOREMS

74

Then we have by parts (i) and (ii)

ID'ul,
lm,

ID1 ulr2 <- CIuIm,P.

But we remark that 1

=

9

+

1-9

rl

r

r2

0<9<1.

Therefore, applying Proposition 2.7 to our situation, we find that

ID'ulr, <- CID'ule1ID,ul


=

m,P

(IUlil-Iul'-3/m)(1-a)/(i-i/m) 9

(ILI

,P

m)(a-i/m)/(l-ilm)

Clulm,PIulq-a.

This completes the proof of Theorem 2.1.

2.3 Imbeddings of the Spaces Hm,P(R') It is the imbedding characteristics of Sobolev spaces that render these spaces so useful in the study of partial differential equations. The most important of the imbedding properties of the spaces Hm,P(Rn) are usually lumped together in a single theorem referred to as the Sobolev imbedding theorem.

Theorem 2.15. Let 1 < p < oo, 1 < q < oo and let j, m be integers such that 0 < j < m. If m - j - n/p is not a non-negative integer and if

i
IDjuIr <

C = C(m, n, p, q, j, a) is a positive constant.

Proof. (1) First we show the following:

Lemma 2.16. I f U E Hm,P(Rf) n Lg(Rn), then one can find a sequence C Co (Rn) such that (2.34) (2.35)

U, -i u

u-u

in Hm,P(Rf), in Lg(Rn).

2.3 IMBEDDINGS OF THE SPACES Hr,P(R')

Proof. Take a function p E Co (Rn) such that

f supppC{xERn: jxj <1}, fRn p(x) dx = 1,

and define pe(x)

x

Enp(E)'

e>0.

Then we have

= u * pe(x) = J n pe(x - y)u(y) dy c C°°(Rn), R

and

ue - u

in H'n'P(Rn) as E 10,

ue ru inL9(Rn)asEJ,0. Furthermore, take a function cp E Co (Rn) such that 1

if1xI <1,

0

if Jxj > 2,

and define

4'R(x)=`p(R)' R>0. Then it follows that (2.34')

'pRUe --

(2.35')

VRUe

in H'n°P(Rn)as R-+ooand E10, u inL9(Rn)asR--+ ooande1O. u

This proves the claim, since {SpRUe} C Co (R n). Indeed, we have for la I < m (2.36) IRn

ue) - DauIP dx < 2P-1

ue) - Da('R u)IP dx

(JRn

+I

ID-(PR U) - DaulP dx) . n

But, since we have YR(x) = 1,

Dl(s'R(x)) =

lxi C R, x

(R)

-rl (R1

)

,

75

II. SOBOLEV IMBEDDING THEOREMS

76

it follows that, as R -> oo, (2.37)

f

n

1

U) -

DauIP

dx = JxpR

U) - DauIP dx

1

I a I D&-R Du - D&u

=

I

C

dx

Q

LI>R

xJ>R

0.

ID"u1Pdx

Further, we have, as e 1 0., dx

JRn 1 Da(PR u)

(2.38)

=fR

n

IDa

u))IP dx P

_

J.n

a ) D--#1PR Dfl(u, -

- u)

dx

/3
C

IRJIP d x

0.

I$I<m

Therefore, assertion (2.34') follows from inequalities (2.36), (2.37) and (2.38). Assertion (2.35') is proved similarly.

The proof of Lemma 2.16 is complete. (2) By Lemma 2.16, one can find a sequence

u

u

u --> u

C Co (R') such that

in Hm,P(Rn), in Lq(Rn).

Hence, applying inequality (2.1) to the functions u and u - uµ, we obtain that (2.39)

ID'uvlr G

(2.40)

ID'uv - Diu,,1,

C1uvlmPluvlO,ga

Cluv -

uµlm,Pluv - 21µ10,q

(2-a) If r > 1, then it follows that

Diu,, -* Diu in Lr(R'). Thus, passing to the limit in inequality (2.39), we find that I D'uI r < Clulm PIUIo qa.

2.3 IMBEDDINGS OF THE SPACES Hm,P(Rn)

77

(2-b) If r < 0, then we let L

so that

j+f<

rJ n p

n

r

j--=am-a--(1-a)-n

q

<m.

One may assume that for J al < m

D'u(x) a.e. x E R'.

(2.41)

(2-b-i) The case -n/r E N: Then we have by inequality (2.39) (2.42)

ID'+'uvIo,,,, = ID2uvIC,,, = IDjuvIr

CluvIm,pIuvIO,ga.

This implies that the sequence is uniformly bounded on R'. Thus, by the mean value theorem, it follows that the sequence {Dauv} is equicontinuous on every bounded subset of R", for I a I = j + f - 1. Here we need the following elementary lemma:

Lemma 2.17. Let {uv} 1 be a sequence in the space Cm(R"). Assume that: (i) There is a point xo E R" such that the set

{D'uv(xo):v=1,2,..., IaI<m} is uniformly bounded, that is, for some constant M > 0 sup sup I

M.

1-1<m v

(ii) The set {Dauv : v = 1, 2, . , IaI = m} of functions is equicontinuous on every bounded subset of R". Then it follows that the set {Dauv} is uniformly bounded and equicontinuous on every bounded subset of R", for IaI < M.

Proof. First, conditions (i) and (ii) imply that (a) The set {Dauv} is uniformly bounded on every bounded subset of R", for Ial = in. Hence, by the mean value theorem, it follows that (b) The set is equicontinuous on every bounded subset of R",

for 101 =m-1. Therefore, this together with condition (i) implies that (c) The set {Dauv} is uniformly bounded on every bounded subset of R", for 101 = m - 1.

II. SOBOLEV IMBEDDING THEOREMS

78

Lemma 2.17 follows by repeating this process. is uniformly Now we find from Lemma 2.17 that the sequence bounded and equicontinuous on every bounded subset of R', for lal < j + 2 - 1. By the Ascoli-Arzela theorem, it follows that uE

C'+'-1(R,")

uv -p u

in

Cj+'-1(R")

But we have by inequality (2.40)

lDj+'uv - D' u lo,. = ID'uv - Diuµle,,, = ID'uv - D'u,,IT < CI2Lv -

uµla

m,PI uv -

uµl1-a

0,q

Therefore, we obtain that

u E C'+t(R"), IDj+'uv - Dj+tulo,,, -> 0 as v --, oo. Further, by passing to the limit in inequality (2.42), we have

ID'uIr = lDj+eulo,, < Clulmplula qa (2-b-ii) The case -n/r ' N: We let

-nr =t+h, 0
lDj+'u (x) - Dj+tu (y)I <_ Cluvlmpluvl0,galx - ylh. In view of (2.41), it follows from an application of Lemma 2.17 that the (2.43)

sequence {D"uv} is uniformly bounded and equicontinuous on every bounded

subset of R", for Ial < J+ t. Hence we find from the Ascoli-Arzela theorem that

u E Ci+,(R"),

uv -r u

in C'+e(R")

By passing to the limit in inequality (2.43), we obtain that IDj+'u(x) D'+'u(y)I <- CIulm,plul0,gals ylh,

-

-

so that ID'uIr = ID3ul-nir,c = lDj+'ulh,,, < CIul'M,plulo,ga. The proof of Theorem 2.15 is now complete.

The next imbedding theorem asserts the existence of imbeddings of the spaces Hm,p(R") into the spaces Ba(R"):

2.3 IMBEDDINGS OF THE SPACES Hm'P(R-)

79

Theorem 2.18. Let 1 < p < oo and let m be a positive integer such that m > n/p. If n/p is not an integer, then we have Hm,P(Rn) C Bm-n/P(R'n)

with continuous injection. Moreover, we have (2.44)

IUIm-n/P,oo C ^YIuIm,Pi

(2.45)

IIUIIm-[n/P]-1,oo C CIIuIIm,P-

Remark 2.19. Since elements of Hm,P(Rn) are not functions defined everywhere on Rn but equivalence classes of such functions defined and equal up to sets of measure zero, we must clarify what is meant by the imbedding Hm,P(Rn) Hm,P(Rn) into Bm-n/P(Rn). What is intended is that each u E can be redefined on a set of measure zero in Rn in such a way that the modified function u (which equals u in Hm,P(Rn)) belongs to Bm-n/P(Rn).

The proof of Theorem 2.18 is based on the following lemma:

Lemma 2.20. Let 1 < p < oo and let j, m be integers such that 0 < j < m - n/p. Then we have, for all functions u E Co (Rn), (2.46)

Iul

j, <

Imp

CIuI(n+)P)/rnp1U11-(n+7P)/-P

D,P

Here C = C(m, p, n) is a positive constant. Proof. (1) We let

f=m -j,

and so

n p

Then, to prove inequality (2.46), it suffices to show the following: (2.47)

Iul.o,oo C CIul7PPPIuIOPn/QP,

u E CO(R').

Indeed, applying inequality (2.47) to the functions

D'u, IaI = 7e u E C' (R'), we obtain that ID"ulo,,,, :5 CID"ul1'P JD"ul

so that (2.48)

Iulj'-

-

CIUIem,PI'U'I j,Pnl eP

o,Pn/1

P,

II. SOBOLEV IMBEDDING THEOREMS

80

But we have by Lemma 2.14 lUlj,P C Clul/m lull Therefore, combining this inequality with inequality (2.48), we find that P7/m.

11-nt/P

IuIJ,0
P(lul!PluluP'/P

< Clul(PAP)/-P lull -P(n+iP)/-P

(2) Proof of inequality (2.47). It suffices to show that There exists a constant C = C(e, p, n) > 0 such that Iu(x)I <-Clule/1plul1-nle1, 0,P ,P

(2.47')

x E Rn.

Without loss of generality, we may assume that

x=0. If u E CO(Rn), we let

f (t) = u(ta), t > 0, Q E E, where E is the unit sphere. Then, applying Taylor's formula to the function f, we obtain that (e 1) f(t1)=.f(t2)+.f'(t2)(t1-t2)+...+f(e-it)!)(t1-t2)e

I

(t1 - T)e 1 (Q - 1)!

el

f

1

dr.

If we take tl = 0, t2 = t,

it follows that U(O)

=k=0k (k !)k

(dt

k

(u(ta)) -

It v 7)1)1

(dt

(u(TV)) dr.

Hence, integrating both sides over the ball of radius h, we obtain that

1/J

nnWn/ 1/P

(2.49)

lu(0)I C (jh

u(0) lt1 dt dQ E e_1

<

k=O

k

h

k! 1

Jo

- 1)!

Jtkp

(d )

(fL i

/P

P

(u(tc)) to-1 dt do

) P

to-1 dt do,

2.3 IMBEDDINGS OF THE SPACES Hm'r(R')

81

Here n7rn/2

F(n/2 + 1)'

wn

the surface area of the unit ball Z.

We estimate each term on the right of inequality (2.49). Q-1 1

h

I

(a)

to-1 dt do,I

J0

k=O

1 I /P

P

tkP

1/P

P

to-1 dt do,

(b) If p > 1, we have by Holder's inequality t

f

d

71-1 ( t

o

P

e

dT) (u(TU)) dT

r(1-1-(n-1)lP)4 dr) P-1

1

P -n £p

t

t1p-n

Pl9

P

e

d 7e-1-(n-1)/P7 (n-1)/P ( dT

(U(7-o,)) dr

P

t

1Tn-1 (ddT)t d

rn-1

d)(u(,O,))

(u(To)) P

(

Jo

dT

dT.

If p = 1, we have t

e

Jt

((U(r)) dr)dT

T1-1

t1-n

d \1

(dT )

0

so that the number

p-1

(u(Tor))

Tn-1

P-1

fp-n should be replaced by 1 if p = 1. Hence it follows that t 7P-1 (-d- )' (u(7-o,)) dr J0

P

1/P

to-1dtdo.

dT,

II. SOBOLEV IMBEDDING THEOREMS

82

1 )P-1

h

LL L-)n x

C

ep-n

t

J 0Tn-1

1

\

h I tip-1

1/P

P

d

dT) to-1dtdo

dT) (u(Ta)) It

1/P

P

()t

d

n-1

r)

(u(TQ))

d-

.

But we have by integration by parts

htip-1 (f 1

0

P

tTn-1

(dT)e(u(TU))

0

ttP EP

f

t

d

rn-1

ptn-1

J

hep fh

P

e

t ep

I It

(dT)e(u(TU))

( )

(u(ta)) P

(d)F

Tn-1

dt

u(T0'))

dT.

(

0

Thus it follows that t

A h

,rn-1

1z 10

d) d

1/P

P

P

Te-1

(u(TQ)) dT

to-1 dt dQ 1/P

P

e

dTdo,

(Cdd) (u(Ta))


Chklulk,P,

Iu(0)I

k=0

that is, Iu(0)Ihn/p < C 1: hk I

Ik,p-

k=0

Therefore, combining this inequality with inequality (2.27) with j = k and m = .£, we find that e Iu(0)Ihn/p

<

C

ulo,pk/e

hkI uI 1 PeI k=0

2.3 IMBEDDINGS OF THE SPACES H'm,P(R') 2

(h

=C

Q

kit

83

1-k/t

Iulo,p

k=o

C e

<

('h1juj,,p+ (1

Iulo,p

k=0

< C (htiule,p + Iulo,p)

so that

Iu(0)I < C (h'-n/plult,p + h-n/plulo,p)

(2.50)

for all h > 0.

,

Now we choose a number ho so that the function he-n/plul$,p + h-nlp)

uI o,p

attains its minimum at h = ho, that is,

f - p\ ho =

Then it follows from inequality (2.50) that Iu(0)I C C'IuIP,p$plulo,pn/fp

This proves inequality (2.47').

Proof of Theorem 2.18. By Lemma 2.20, we find that Hm,p(Rn) C Bm-[nlp]-1(Rn), and (2.46)

luj(m iv)lmplul0,p(n+)p)lmp

uI7,. <

u E Hm,p(R )

Hence we have, for 0 < k < j, (2.51) IUIk ,oo

<

=C < li

CIuI(n+kp)lmpl,ull-(n+kp)lmp

m,p

o,p

(n+jp)lmp

(I uI m,p

1-(n+ip)/mp (n+kp)/(n+ip)

Iulo,p

)

1-(n+kp)/(n+jp)

Iulo,p

(Iul(npjp)lmplul0,p(n+jp)lmp + IuIO,p)

Therefore, combining inequalities (2.46) and (2.51), we obtain that

IIuIIj,.

C

(iui(P7p)lmplul1-(n+jp)/mp + Iulo,p)

II. SOBOLEV IMBEDDING THEOREMS

84


0 < j <m-[n/p]-1.

This proves inequality (2.45). On the other hand, inequality (2.44) follows from Theorem 2.15 with

q=p, j=0,

a=1.

Indeed, we have lulm-n/P,oo =

IUI-n/r,oo =

CIUIm,p

Iulr

The proof of Theorem 2.18 is now complete. 0 The next imbedding theorem asserts the existence of imbeddings of the spaces Hm,P(Rn) into the spaces H3'r(Rn), 0 < j < m:

Theorem 2.21. Let 1 < p < oo and let j, m be integers such that 0 < j < m. If m - j - n/p is not a non-negative integer and if

r

j

1

n

p

m n

m


then we have

Hm,P(Rn) C H7,r(Rn),

with continuous injection. Moreover, we have (2.52)

IIUIIj,r < C (lulm,pIuIO,pa + lulo,P),

u E Hm,P(Rn)

Here C = C(m, n, p, q, j, a) is a positive constant.

Proof. We remark that 1 1 r=p+-(jam) > p,1 1

so that

r>p>1.

If we take

q=p in Theorem 2.15, then we have, for J /m < a < 1, (2.53)

lul,,r < Club. plulo pa,

u E Hm,P(R")

In particular, we have, for a = j/m, (2.54)

IuI;,P

ClullPIUIo,p'lm

2.3 IMBEDDINGS OF THE SPACES H-,P(R-)

85

Now we take an arbitrary integer k such that

0
am-j+k m-j+k

k m-j+k
1

m-]'+k

1

r n p Hence, applying inequality (2.53) with

n

m=m-j+k, j=k,

a=b,

we obtain that (2.53')

I uI k,r < CI uI m-7+k,pI uI0 pb.

But, applying inequality (2.54) with

j:=m-j+k, we obtain that (2.54')

Iulm-l+k,p <

CIui(mP 7+k)/mlul1-(m-.+k)/m

Thus, combining inequalities (2.53') and (2.54'), we find that (2.55) (am-J+k)/('m-7+k)

Iulk,r

-

C (IulmmP j+k)/mI uI0p k)/m I C (Iula Iull_,)(.m-7+k)/am m,P

0,P

< C (Iulm,PIUII0'P-a + IuIo,P) ,

UIm(1-a)/(m-7+k)

IuI(j-k)/am

0,P

0< - k < j.

Therefore, inequality (2.52) follows from inequalities (2.55) and (2.53). Indeed, we have 7

Ilullj,r =

IUIk,r

< C (lulm,plulo,pa + IuIO,P)'

k=0

The proof of Theorem 2.21 is complete.

II. SOBOLEV IMBEDDING THEOREMS

86

Corollary 2.22. Let 1 < p < oo and let j, m be integers such that 0 < j < m and that m - j - n/p is not a non-negative integer. If p < r < oo and j/n + 1/p - m/n < 1/r, then we have Hm,P(Rn) C H',r(Rn), with continuous injection.

Proof. We let

(j p

Then we find that

j

n

1

nm mn

1

mn p r j nC1-11 m mp r

and that

--

j+1 n p

-am

n

j m'

---- -

= j+ -n -p1- nj

1

+p

1

1

r

r

Hence Corollary 2.22 is an immediate consequence of Theorem 2.21.

2.4 Imbeddings of the Spaces Hm'P(SZ) The purpose of this section is to prove Sobolev imbedding theorems for a bounded C°° domain Q.

The next theorem, due to Seeley [Sel], asserts that the functions in C°°(S2) are the restrictions to ci of functions in C°°(Rn):

Theorem 2.23. Let SZ be either the half space R+ or a C°° domain in Rn with bounded boundary r. Then there exists a continuous linear extension operator

E : C°°(N) - C°°(Rn). Furthermore, the restriction to the space C0 (St) of E is a continuous linear extension operator on Co (S2) into C0(Rn).

Proof. (i) First we consider the case 52 = R. The proof is based on the following:

Lemma 2.24. There exists a function w in the space S(R) such that supp w C [l, oo),

{ fl'°tnw(t)dt=(-1)n, Assuming Lemma 2.24 for the moment, we shall prove Theorem 2.23.

2.4 IMBEDDINGS OF THE SPACES Hm'p(1)

87

By using the function w in Lemma 2.24, we can define a linear operator

C'(R')

E : C' (R7) by the formula E(p(x , xn) =

1

p(x', xn)

if xn > 0,

/'O° w(s) 0(-xns) cp(x', -sxn) ds

if xn < 0,

where

x = (X', X,,) E R',

n-1 x' x = (XI i x2, ... , xn-1) E R

and

0 E Co (R), supp 0 C [-2, 2], 0(t) = 1 for Itl < 1. Then it is easy to verify the following: (1) E C°°(R' ). (2) The operator E maps C°°(UT) continuously into C°°(Rn).

(3) Ifsupp(pC{xER':jx'j 0and a>0,

then it follows that supp Eyo C {x E Rn : I x' j < r, I xn I < a}. This proves Theorem 2.23 for the half space R. (ii) Now assume that S2 is a C°° domain in Rn with bounded boundary r. Then we can choose a finite covering {Uj IN, of r by open subsets of Rn and C°° diffeomorphisms X j of Uj onto the unit ball B = {x E Rn : I x < 1 } such that the open sets Vj = X71

x E R" : Ix'I < 2,

Ixnl < 2 T

1

< j < N,

form an open covering of the tubular neighborhood 526 = {x E f2: dist (x, r) < b}

for some b > 0. Further we can choose an open set Vo in St, bounded away from r, such that (see Figure 2.1) N

QCVoU UVj j=1

.

II. SOBOLEV IMBEDDING THEOREMS

88

Figure 2.1

Let {wj} o be a partition of unity subordinate to the covering {V;} p E C°°(S2), we define

o. If

N Eyo

= w0Y +

j=1

X (E ((X

Then it is easy to verify that this operator E enjoys the desired properties. Theorem 2.23 is proved, apart from the proof of Lemma 2.24.

Proof of Lemma 2.24. (1) First take a function uo E Co (R) such that

uo>0onR, supp uo C [1, 2],

(2.56)

f°0 uo(t)dt=1. If k is a positive integer, we let (2.57)

uk(t) =

1

2k up C 2k/

.

Then we have the following:

I Uk>0onR, supp uk C [2k, 2k+1],

j

f °° uk(t) dt = 1.

Hence, for any sequence {ak}, the formal sum cc

w(t) =

ak Uk(t) k=0

2.4 IMBEDDINGS OF THE SPACES Hr,P(SZ)

89

make a sense. We shall choose a sequence {ak} in such a way that the function w has the desired properties. (2) Next fix an integer N > 0 and construct the N-th approximation to the function w: N

WN(t) = E aNk uk(t). k=0

The coefficients aNk will be picked to satisfy the conditions 00

(2.58)

J0

t" war(t) dt = (-1)",

n = 0,1,

,

N.

But, by conditions (2.56) and (2.57), this equation may be rewritten as N

n = 0,1, ... N

E 2k aNk = hn,

(2.58')

k=o

where

1n

"(

hn=

to uo(t) dt

fo

Now equation (2.58') is a linear system of N + 1 equations in N + 1 unknowns, and its determinant is the Vandermonde determinant: 1

1

1

2

... ...

1

2N

...

AN =

1

N

2N

f71 (2' -2`). i, j=0

i<j

2N2

Hence, using Cramer's rule, we find the solutions {aNk} of equation (2.58') as follows.

aNk = (-1)kONk

((_l)1h1s) 0-1 e-0

where 2j

-

2`)

and

SNk = the elementary symmetric polynomial of degree m in the elements {1, 2,

,

2-1 , 2 k41 ,

,

2}. N

II. SOBOLEV IMBEDDING THEOREMS

90

Since we have by condition (2.56) IhtI < 1

and

N

N

SNVk1=H(1+21), f=0 e=0 Pk it follows that N

k-1

fi (2k - 2e) II (21 - 2k)

Q=k+1

P=0

N

k-1

Ia NkI C J (1 + 2Q) 11 (1 + 2p)

I=k+1

P=0

Ak . BNk

where k-1

1 + 2e

Ak =1=0 fi

2(-2k,

N

BNk= II

e=k+1

1+21

21-2k

But we remark that k-1 21+1

(2.59)

jAkI

2k-1

=

2(3k-k2)/2

e=0

and also N

(2.60)

log BNk =

/

k

log I 1 + 2Q+ 2k ) 2=k+1

<

1+2k E (+22 f - k) N

P =k+1

N

1

< (1 + 2k) E 2Q-1-1 Q=k+1

< 4,

since we have the inequality log(1 + x) < x for all x > 0. Therefore, by combining estimates (2.59) and (2.60), we obtain that (2.61)

I aNk I <

e4 2(3k-k2)/2

k = 0, 1, ... , N.

2.4 IMBEDDINGS OF THE SPACES

91

To prove that a finite limN-0 aNk exists for every integer k > 0, it suffices to show that each sequence Since we have ,5N+1

kf

is a Cauchy sequence.

- 2N+1 SNk Q + SNk 1-1,

it follows that N

aN+1 k - aNk = (-1)k 1 E [(-1)' 2k h1 + (-1)e+1 h1+1] SNk f 1=o 1

X AN+lk'ON+17

so that I aN+1 k - aNk I

N Y

(1 + 2k)

SNk1

ON+1 k - ON+1

(1=0 k)-i

= IAkI . BNk (2N+1 -2

Hence we obtain from estimates (2.59) and (2.60) that for any integer M > 0 M

IaN+M k - aNk1

(1 + 2k) IAkI

e4

(2N+m

-2 k)-1

m[=1

< e4 (1+2 k) IAkI

This proves that the sequence {aNk }N=1 integer k > 0.

2-N

is a Cauchy sequence for every

(3) Finally we let

ak= lim aNk, N-,oo

k=0,1,

Then, letting N - oc in estimate (2.61), we have (2.62)

Iakl

e4 2(3k-k2)/2

In view of assertions (2.61), (2.62) and (2.58), it is easy to verify that wN --> w in the space S(R) and that the limit function w enjoys the desired properties. Now the proof of Lemma 2.24, and hence that of Theorem 2.23, is complete. By using Theorem 2.23, we can construct an extension operator

E : Hm,P(I) - Hm,P(R' ). Thus it is easy to see that H',P(Q) = the space of restrictions to 1 of functions in Hm P(R") with the norm IIuIIm,P,f1= inf IIUIIm,P,R_,

where the infimum is taken over all U E Hm,P(R") which equal u in n. Furthermore, we have for all functions u E Hm,P(cl) IIullj,r,Il < IIEuII;,r,R' < CI Eullm,P,Rr < CC'IlUhIm,P,ii. Therefore, by such extension arguments, one can prove the following:

92

II. SOBOLEV IMBEDDING THEOREMS

Theorem 2.25. Theorems 2.15, 2.18, 2.21 and Corollary 2.22 remain valid for a bounded C°° domain Q.

CHAPTER III

LP THEORY OF PSEUDO-DIFFERENTIAL OPERATORS In this chapter we present a brief description of the basic concepts and results of the LP theory of pseudo-differential operators - a modern theory of potentials - which will be used in subsequent chapters. For detailed studies of pseudo-differential operators, the reader is referred to Chazarain and Piriou [CP], Kumano-go [Ku] and Taylor [Ty].

3.1 Generalized Sobolev Spaces and Besov Spaces Let S2 be a bounded domain of Euclidean space R' with C°° boundary F. Its closure Sl = 92 U F is an n-dimensional compact C°° manifold with

boundary. One may assume that (see Figure 3.1): (a) The domain Sl is a relatively compact open subset of an n-dimensional compact C°° manifold M without boundary. (b) In a neighborhood W of F in M a normal coordinate t is chosen so

that the points of W are represented as (x', t), x' E F, -1 < t < 1; t > 0 in

fI,t<0inM\S2 andt=0 only on r. (c) The manifold M is equipped with a strictly positive density tc which, on W, is the product of a strictly positive density w on r and the Lebesgue measure dt on (-1, 1). This manifold M is called the double of Q.

Figure 3.1

The function spaces we shall treat are the following (cf. [BL], [Tr]): Typeset by A.IS-TjX 93

94

III. LP THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

(i) The generalized Sobolev spaces H''P(Q) and H''P(M), consisting of all potentials of order s of LP functions. When s is integral, these spaces coincide with the usual Sobolev spaces W'"P(Sl) and W',P(M). (ii) The Besov spaces B''P(r). These are function spaces defined in terms

of the LP modulus of continuity, and enter naturally in connection with boundary value problems. First we recall the basic definitions and facts about the Fourier transform. If f E L1(Rn), we define its (direct) Fourier transform Ff by the formula

.Ff( ) = Rn f e- x f f(x)dx,

S = (S1, 61- , bn),

where x = x1 1 + + xn n. We also denote Ff by f. Similarly, if g E L1(Rn ), we define .17*g(x) = (27r)n 1

I dx n

The function F*g is called the inverse Fourier transform of g. We introduce a subspace of L'(Rn) which is invariant under the Fourier transform. We let

S(Rn) =the space of C°° functions cP on Rn such that we have for any non-negative integer j

p.i(0 = sup {(1 + JxJ2)i12I o xERn

(x)I } < oo.

IaISi

The space S(Rn) is called the space of C°° functions on Rn rapidly decreasing

at infinity. We equip the space S(Rn) with the topology defined by the countable family {pj} of seminorms. The space S(Rn) is a Frechet space. The transforms F and F* map S(Rn) continuously into itself, and .F.F* _ .F*.F = I on S(Rn). Since the injection of Co (Rn) into S(Rn) is continuous, it follows that the dual space S'(Rn) of S(Rn) consists of those distributions T E D'(Rn) that have continuous extensions to S(Rn). The elements of S'(Rn) are called tempered distributions on Rn. The direct and inverse Fourier transforms can be extended to the space S'(Rn) by the following formulas: 1P E S(Rn), (.Fu, P) = (T* u, v) = (u,.F*w) , V E S(Rn)

Once again, the transforms 'F and F* map S'(Rn) continuously into itself,

and F.F* = F*T = I on S'(Rn).

3.1 GENERALIZED SOBOLEV SPACES AND BESOV SPACES

95

If s E R, we define a linear map

J':S'(Rn)-pS'(Rn) by the formula

J'u = -F* ((1 +

u E S'(Rn).

Then the map J' is an isomorphism of S'(Rn) onto itself, and its inverse is the map J-'. The function J'u is called the Bessel potential of order s of u. Now, if 8 E R and 1 < p < oc, we let

H',P(Rn) = the image of LP(Rn) under the mapping P. We equip H''P(Rn) with the norm llulls,p = IIJ-'ullp for u E H''P(Rn). The space H''P(Rn) is called the (generalized) Sobolev space of order s. We list some basic topological properties of H''P(Rn):

(1) The space S(R') is dense in H',P(Rn).

(2) The space H-''P'(Rn) is the dual space of H''P(Rn), where p' _ p/(p - 1) is the exponent conjugate to p. (3) If s > t, then we have the inclusions

S(Rn) c H9,P(R') c Ht,P(Rn) c S'(Rn), with continuous injections. (4) If s is a non-negative integer, then the space H''P(Rn) is isomorphic to the usual Sobolev space W''P(Rn), that is, the space H'"P(Rn) coincides with the space of functions u E LP(Rn) such that D'u E LP(Rn) for Ial < s, and the norm II I19,p is equivalent to the norm -

Rn

ID"u(x)IPdx

1a1<e

Next, if 1 < p < oo, we let

B",p(Rn-1) =the space of (equivalence classes of) functions yo E' LP(R"-1) for which

I

+ y) - 2(x) + (x n- , xRn-

dy dx

IyJn-1+P

The space B',P(Rn-1) is a Banach space with respect to the norm IVI1,p = (JRn-t Isv(x)IPdx

96

III. LP THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

+

Iso(x+y)-2
JfR^-1 xRn-1

1/p

dy dx

If s E R, we let

B.''1'(Rn-1) =the image of B1,P(Rn-1) under the mapping J'9-1, where J'4-1 is the Bessel potential of order s - 1 on

Rn-1

J'-9+1(P

We equip the space B',P(Rn-1) with the norm Icpls,p =

for I

cp E B',P(Rr-1). The space B''P(Rn-1) is called the Besov space of order s. We list some basic topological properties of B''P(Rn-1): (1) The space S(Rr-1) is dense in B8'P(Rn-1). (2) The space B-''P'(Rn-1) is the dual space of B'1P(Rn-1), where p' _

p/(p - 1).

(3) If s > t, then we have the inclusions

S(Rn-1) C B',P(Rn-1) C

Bt,P(Rn-1)

C

S'(Rn-1)

with continuous injections.

(4) If s = m + a where m is a non-negative integer and 0 < a < 1, then the Besov space B''P(Rn-1) coincides with the space of functions cp E

H',P(Rn-1) such that for Ial = m

- D(y)IP dx dy

JJn-1xRn-1

Ix - yln-l+Pa

Furthermore, the norm Icpl,g,, is equivalent to the norm

(im"R^-1

I D- p(x)l Pdx

I

Ia =m

R^-1 xR^-1

ix -

D",p(y)IP dx

dy)1/p

yln-1+Po

Now we define the generalized Sobolev spaces H''P(Il), H'"P(M) and the Besov spaces B',P(r) for arbitrary values of s. For each s E R, we define

H',P(Q) = the space of restrictions to SZ of functions in H'"P(Rn).

We equip the space H''P(1) with the norm IIuIIs,p = inf IIUII.,p,

3.2 FOURIER INTEGRAL OPERATORS

97

where the infimum is taken over all U E H'>P(R") which equal u in Q. The space H''P(1) is a Banach space with respect to the norm II II9,P We remark that Ho,P(1) = LP(st), II - IIo,P = II IIP. -

The spaces H''P(M) are defined to be locally the spaces H',P(Rn), upon using local coordinate systems flattening out M, together with a partition of unity. The spaces B'"P(r) are defined similarly, with H',P(Rn) replaced by B',P(R"-1). The norms of H',P(M) and B''P(r) will be denoted respectively by ll'II9,Pand I'I9,P. We state two important facts which will be used in the study of boundary value problems: (I) The restriction map

p : H',P(1) -f u1--)uIr

B9-1/P,P(F)

is well defined for all s > 1/p, and is surjective (cf. [St]). (II) (Rellich) If s > t, then the injections

H8,P(M) - Ht,P(M), Bt,P(r) B',P(r) are both compact (or completely continuous). Finally we introduce a space of distributions on SZ which behave locally

just like the distributions in H''P(R"): H;oP(n) = the space of distributions u E D'(5l) such that

WuEH'"P(R')forall
3.2 Fourier Integral Operators In this section, we present a brief description of the basic concepts and results of the theory of Fourier integral operators.

3.2A Symbol Classes. Let Q be an open subset of R'. If m E R and

0<6
SP b(fl x RN) = the set of all functions a E C°°(S2 x RN) with the property that, for any compact K C 52 and any

III. LP THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

98

multi-indices a, 0, there exists a constant CK,a,p > 0 such that we have for all x E K and 9 E RN 100 aRa(x, 0) 1 < CK,a,a(1 +

The elements of SP a(Q x RN) are called symbols of order m. We drop the 1Z x RN and use SP 6 when the context is clear.

Examples 3.1. (1) A polynomial p(x, 6)

aa(x)e of order m

with coefficients in C°°(fl) is in Sl o(SZ x R''). (2) If m E R, the function

52 x R'i 3 (x,6)'' (1 + 1612)m/2

is inS10(c x Rn). (3) A function a E C°°(1lx(RN\{0}) is said to be positively homogeneous of degree m in 9 if it satisfies a(x,t9) = tm a(x,9),

t > 0.

If a(x, 9) is positively homogeneous of degree m in 0 and if V(O) is a C°° function such that V(O) = 0 for 191 < 1/2 and V(O) = 1 for 101 > 1, then the function cp(9)a(x, 9) is in So (Q x RN). If K is a compact subset of 12 and j is a non-negative integer, we define a seminorm PK,j,m on Sp b(1 x RN) by Sp 6(1l x RN) 3 a

(a) =

jae aQa(x, 9)j su

XEp (1 + BERN IaI<7

We equip the space S, 7,,(Q x RN) with the topology defined by the family {PK,j,m} of seminorms where K ranges over all compact subsets of 12 and j = 0,1, . . . The space S( 52 x RN) is a Frechet space. We set

S-°°(12 x RN)

=mER n S' 6(Sl x RN).

The next theorem gives a meaning to a formal sum of symbols of decreasing order:

Theorem 3.2. Let aj E SP 6 (12 x RN), mj

-oo, j = 0,1,

Then there exists a symbol a E SP b (S2 x RN), unique modulo S-°°(12 x RN), such

that we have for all k > 0 k-1

(3.1)

a-

aj E Sp b (12 x RN).

i-o

. .

3.2 FOURIER INTEGRAL OPERATORS

99

If formula (3.1) holds, we write 00

a - E aj.

j=o

The formal sum >j aj is called an asymptotic expansion of a. A symbol a(x, 9) E Sl o(Sl x RN) is said to be classical if there exist C°° functions aj(x, 9), positively homogeneous of degree m - j in 9 for 101 > 1, such that 00 a - E aj. j=0

The homogeneous function ao of degree m is called the principal part of a. We let

S,', (St x RN) = the set of all classical symbols of order m.

For example, the symbols in Examples 3.1 are all classical, and they have respectively as principal part the following functions:

(1) pm(x,) _ =m a«(x)a (2)

(3) a(x, 0).

A symbol a(x, 9) in SP 6(9 x RN) is said to be elliptic of order in if, for any compact k C S2, there exists a constant CK > 0 such that I a(x, 0)I > Cx-(1 + I91)m,

x E K, 101 > Ck

There is a simple criterion in the case of classical symbols. Theorem 3.3. Let a(x, 0) be in S,-, (Sl x RN) with principal part ao (x, 0). Then a(x, 0) is elliptic if and only if we have

ao(x,0)#0, xESt, 101=1.

3.2B Phase Functions. Let 9 be an open subset of R'. A function o(x, 0) in Co" (S2 x (RN \ {0})) is called a phase function on St x (RN \ {0}) if it satisfies the following three conditions: (a) ep is real-valued. (b) cp is positively homogeneous of degree 1 in the variable 0. (c) The differential dcp does not vanish on St x (RN \ {0}).

Example 3.4. Let U be an open subset of RP and S2 = U x U. The function p(x, Y, ) = (x - y) - e

is a phase function on the space Sl x (RP \ {0}) (n = 2p, N = p). The next lemma will play a fundamental role in defining oscillatory integrals.

III. L' THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

100

Lemma 3.5. If co is a phase function on Sl x (RN \ {0}), then there exists a first-order differential operator ai (x, 0) a9, + ) , bk (x, 0)

L

c(x, 0)

k=1

such that

L(etiw)

= e' `',

and its coefficients aj, bk, c enjoy the following properties: aj E S° 0 ; bk, c E Si,o

Furthermore, the transpose L' of L has coefficients a', bk, c' in the same symbol classes as aj, bk, c, respectively. For example, if cp is a phase function as in Example 3.4,

(x,y)EUxU, E(R"\{0}), then the operator L is given by the following: P

L

22+Jx_ylz

+k

6j

P

19

-6j

I2aak+k-1 IS I2

k

+p(0' where

is a function in C0 00(11P) such that p = 1 for

3.2C Oscillatory Integrals. We let

S' (S2 x RN) _ U S a (St x RN)

.

mER

If cp(x, 0) is a phase function on SZ x (RN \ {0}), we wish to give a meaning to the integral (3.2)

I,(au) _ ff

e2w(X'°)a(x, B)u(x) dxd8,

u E Co (S2),

2x RN

for each symbol a(x, 0) E S° (S2 x RN). By Lemma 3.5, we can replace dc in formula (3.2) by L(ets). Then a formal integration by parts gives us that

I,(au) =

JfL'(a(x, B)u(x)) dx d9. 2xRN

But the properties of the coefficients of L' imply that L' maps S" b continuously into Sp b° for all r E R, where q = min(p, 1 - b). Continuing this process, one can reduce the growth of the integrand at infinity until it becomes integrable, and give a meaning to the integral (3.2) for each symbol

aESP (cxRN).

More precisely, we have the following:

3.2 FOURIER INTEGRAL OPERATORS

101

Theorem 3.6. (i) The linear functional

S-°° (S2 x RN) 3 a i- I,(au) E C extends uniquely to a linear functional f on the space S° (S2 x RN) whose restriction to each space SP (S2 x RN) is continuous. Further, the restriction to SPb (S2 x RN) of f is expressed as

f(a) = Jf

e

txRN

° (L)k(a(x, O)u(x)) dx d9,

where k > (m + N)/q, q = min(p, 1 - S). (ii) For any fixed a E SPb (S2 x RN), the mapping Co (S2) 3 u f-- I,p(au) = f(a) E C

(3.3)

is a distribution of order < k for k > (m + N)/q. We call the linear functional f on Sv an oscillatory integral, but use the standard notation as in formula (3.2). The distribution (3.3) is called the Fourier integral distribution associated with the phase function po and the amplitude a, and is denoted by

J

eiW(x,e)a(x, 0) dB.

RN

If u is a distribution on S2, the singular support of u is the smallest closed subset of f2 outside of which u is of class C°°. The singular support of u is denoted by sing supp u. The next theorem estimates the singular support of a Fourier integral distribution.

Theorem 3.7. If y' is a phase function on the space S2 x (RN \ {0}) and if a is in SP (S2 x RN), then the distribution

A = J ei`°(x'a)a(x, 0) d9 E D'(S2) N

satisfies

sing suppAC {x ES1: de(p(x,0)=Ofor some 0ERN\{0}).

III. LP THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

102

3.2D Fourier Integral Operators. Let U and V be open subsets of RP and R9, respectively. If cp(x, y, 9) is a phase function on U x V x (RN\{0})

and if a(x, y, 9) E So" (U x V x RN), then there is associated a distribution K E D'(U x V) defined by the formula

K= JRN

Applying Theorem 3.7 to our situation, we obtain that sing supp K C {(x, y) E U x V : dgcp(x, y, 0) = 0 for some 0 E RN \ loll.

The distribution K defines a continuous linear operator

A:Co (V)-) D'(U) by the formula

(Av, u) = (K, u ® v)

,

u E Co (U), v E Co (V ).

The operator A is called the Fourier integral operator associated with the phase function cp and the amplitude a, and is denoted by

''Ba(x, y, 9)v(y) dy d9, v E C(V).

e

Av(x) = ff XRN

The next theorem summarizes some basic properties of the operator A. Theorem 3.8. (i) If dy,a
0 on U x V x (RN \ {0}), then the operator A

(ii) If dx,ecp(x, y, 0)

extends to a continuous linear operator on £'(V) into D'(U). (iii) If dy,eW(x, y, 0) 0 0 and dx,e
and some 0ERN\{0}}. 3.3 Pseudo-Differential Operators Let fl be an open subset of R" and m E R. A pseudo-differential operator of order m on Q is a Fourier integral operator of the form (3.4)

Au(x) = ff 2XRn

e

a(x, y, )u(y) dy d,

u E C(),

3.3 PSEUDO-DIFFERENTIAL OPERATORS

103

with some a E Sp b(SZ x S2 x R"). In other words, a pseudo-differential operator of order m is a Fourier integral operator associated with the phase

(x - y)

function cp(x, y, We let

and some amplitude a E Sp b(S2 x S2 x R").

Lp a(i) = the set of all pseudo-differential operators of order m on Q. Applying Theorems 3.7 and 3.8 to our situation, we obtain the following: (1) A pseudo-differential operator A maps the space C0 (S2) continu-

ously into the space C°°(c) and extends to a continuous linear operator A : £'(S2) --> D'(c). (2) The distribution kernel KA of a pseudo-differential operator A satisfies the condition sing supp KA C {(x, x) : x E S2},

that is, the kernel KA is of class C°° off the diagonal {(x, x) : x E St} in

c>S2. (3) sing supp Au C sing supp u, u E £'(S2). In other words, Au is of class C°° whenever a is. This property is referred to as the pseudo-local property. We set

fl Lp b(ci) mER

The next theorem characterizes the class L-°°(S2).

Theorem 3.9. The following three conditions are equivalent: (i) A E L-°°(S2).

(ii) A is written in the form (3.4) with some a E S-°°(9 x S2 x Rn). (iii) A is a regularizer, or equivalently, its distribution kernel KA is in the space C°°(c x S2).

We recall that a continuous linear operator A : Co (S2) - D'(c) is said to be properly supported if the following two conditions are satisfied:

(a) For any compact subset K of fl, there exists a compact subset K' of S2 such that

supp v C K ; supp Av C K'. (b) For any compact subset K' of S2, there exists a compact subset K of 52 such that

supp v f1 K = 0

supp Av n K' = 0.

If A is properly supported, then it maps Co (S2) continuously into £'(S2), and further it extends to a continuous linear operator on C°°(52) into D'(S2). The next theorem states that every pseudo-differential operator can be written as the sum of a properly supported operator and a regularizer.

III. L" THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

104

Theorem 3.10. If A E LP 6(52), then we have

A=Ao+R, where Ao E LP 6(52) is properly supported and R E L-°°(52).

If p(x, ) E S,7,6 (52 x Rn), then the operator p(x, D), defined by (3.5)

p(x, D)u(x) = (21

fn e`x

n

u E Co (St),

p(x, e) ft (e)

is a pseudo-differential operator of order m on 52, that is, p(x, D) E LP 6(52). The next theorem asserts that every properly supported pseudo-differential operator can be reduced to the form (3.5).

Theorem 3.11. If A E LP 6(52) is properly supported, then we have p(x, ) _

-ix.

and

A = p(x, D).

Furthermore, if a(x, y, ) E SP 6(1 x Q x Rn) is an amplitude for A, we have the following asymptotic expansion: p(x,

)->-

as Dy (a(x, y, 6)) y=x

«>o

The function p(x, ) is called the complete symbol of A. We extend the notion of a complete symbol to the whole space LP 6(1). If A E LP 6(52), we choose a properly supported operator Ao E LP 6(52) such that A - Ao E L-°°(52), and define o(A) = the equivalence class of the complete symbol of A0 in

Sp 6(52 x Rn)/S-°°(52 x R'). In view of Theorems 3.9 and 3.10, it follows that o(A) does not depend on the operator A0 chosen. The equivalence class o(A) is called the complete symbol of A. It is easy to see that the mapping

Lp (Sl) E) A -- o(A) E S,', 6(52 x Rn)/S-°°(52 x R') induces an isomorphism

LP6/L

SP6/S

3.3 PSEUDO-DIFFERENTIAL OPERATORS

105

We shall often identify the complete symbol Q(A) with a representative in the class SP b(St x R') for notational convenience, and call any member of Q(A) a complete symbol of A.

A pseudo-differential operator A E Li 0(1) is said to be classical if its complete symbol Q(A) has a representative in the class Sm(Sx R"). We let

(SZ) = the set of all classical pseudo-differential operators of order m L '(Q) on SZ.

Then the mapping

L '(S2) D A t -> o(A) E S,', (Q x R")/S-°°(1l x R') induces an isomorphism

Li/L-°°

Sm/S-°°.

Also we have

L-°°(S2) _ fl Li(2). mER

If A E L '(Q), then the principal part of Q(A) has a canonical representative QA(x, ) E C°°(1l x (R" \ {0})) which is positively homogeneous of degree m in the variable . The function cA(x, ) is called the homogeneous principal symbol of A. The next two theorems assert that the class of pseudo-differential operators forms an algebra closed under the operations of composition of operators

and taking the transpose or adjoint of an operator.

Theorem 3.12. If A E LP 6(1l), then its transpose A' and its adjoint A* are both in LP',6(Q), and the complete symbols o,(A') and a(A*) have respectively the following asymptotic expansions: o,(A') (x, ) ^ a>O

a

(o,(A)(x, - )) ,

(a(A)(x, ))

a(A*)(x, ) ^

.

«>0

Theorem 3.13. If A E LP ,6,(S2) and B E P

where 0 < S' <

P" < 1 and if one of them is properly supported, then the composition AB is in Lp (SZ) with p = min(p',p"), 6 = max(6', S"), and we have the following asymptotic expansion: o,(AB)(x, ) ^ a>0

a! a, (o,(A)(x, 6)) .

Dx (o,(B)(x, ))

III. LP THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

106

A pseudo differential operator A E Lp b(SZ) is said to be elliptic of order m if its complete symbol a(A) is elliptic of order m. In view of Theorem 3.3, it follows that a classical pseudo-differential operator A E L m(1) is elliptic if and only if its homogeneous principal symbol UA(x, ) does not vanish on the space Sl x (R" \ {0}). The next theorem states that elliptic operators are the "invertible" elements in the algebra of pseudo-differential operators.

Theorem 3.14. An operator A E LP 6(52) is elliptic if and only if there exists a properly supported operator B E LPb (S) such that AB BA

I mod L-°°(52), I mod L-°°(52).

Such an operator B is called a parametrix for A. In other words, a parametrix for A is a two-sided inverse of A modulo L-°°(5?). We observe that a parametrix is unique modulo L-°°(1). The next theorem proves the invariance of pseudo-differential operators under change of coordinates. Theorem 3.15. Let Q1, 5l2 be two open subsets of R" and x : 521 -+ 522 a C°° diffeomorphism. If A E LP b(Sl1), where 1 - p < 6 < p < 1, then the mapping

Ax : Co (522) -' C°°(c12) V 1) A(v o x) o x-1 is in LP 6(Q2), and we have the asymptotic expansion (3.6)

o,(Ax)(y,77) a>O

a.

(a, a(A)) (x,' x'(x) . 7) - D"

(ei.r(=,=,1,))

LX

with

r(x,z,i) _ (X(z) - X(x) -

x

x

x and tx'(x) its transpose.

Remark 3.16. Formula (3.6) shows that o,(Ax)(y,77) = a(A) (x,t x'(x) - 17) Note that the mapping

522 x R'

mod SP6 (n-b).

(y, 71) --' (x,t x'(x) ' 71) E S21 x R"

is just a transition map of the cotangent bundle T*(Rn ). This implies that the principal symbol o,,,,(A) of A E Lp b(R") can be invariantly defined on

T*(R")when 1-p<6
3.3 PSEUDO-DIFFERENTIAL OPERATORS

107

Theorem 3.17. Every properly supported operator A E Lm (1), 0 < S < 1, extends to continuous linear operators

A : H"P(Sl) -'

A:Blo

Hiocm,P(Q)

(1)-'Bincm'

(1)

for all s E R. Now we define the concept of a pseudo-differential operator on a manifold, and transfer all the machinery of pseudo-differential operators to manifolds. Let M be an n-dimensional compact C°° manifold without boundary. Theorem 3.15 leads us to the following:

Definition 3.18. Let 1 - p < S < p < 1. A continuous linear operator A : C°°(M) -> C°°(M) is called a pseudo-differential operator of order m E R if it satisfies the following two conditions: (i) The distribution kernel of A is of class C°° off the diagonal {(x, x)

xEM}inMxM.

(ii) For any chart (U, X) on M, the mapping

AX : Co (X(U)) - C°° (X(U)) belongs to the class Lp 6 (X(U)). We let

LP 6(M) = the set of all pseudo-differential operators of order m on M, and set

L-°°(M) _ fl Lp b(M) mER

Some results about pseudo-differential operators on R' stated above are also true for pseudo-differential operators on M. In fact, pseudo-differential operators on M are defined to be locally pseudo-differential operators on R. For example, we have the following results: (1) A pseudo-differential operator A : C°°(M) -> C°°(M) extends to a

continuous linear operator A : D(M) - D(M). (2) sing supp Au C sing supp u, u E D'(M). (3) A continuous linear operator A : C°°(M) -* D'(M) is a regularizer if and only if it is in L-°°(M). (4) The class LP 6(M) is stable under the operations of composition of operators and taking the transpose or adjoint of an operator. (5) A pseudo-differential operator A E Li b(M), 0 < b < 1, extends to continuous linear operators A : H9'P(M) -- H9'm'P(M) and A : B9'P(M) -

B'-',P(M) for all s E R.

III. LP THEORY OF PSEUDO-DIFFERENTIAL OPERATORS

108

A pseudo-differential operator A E Li o(M) is said to be classical if, for any chart (U, x) on M, the mapping AX : Co (x(U)) --> C' (x(U)) belongs to the class L"' (X (U)). We let

Lm (M) =the set of all classical pseudo-differential operators of order

monM. We observe that

L-°°(M) _ fl L- (M). mER

Let A E L m (M). If (U, x) is a chart on M, there is associated a homogeneous principal symbol °AX E Cc" (x(U) x (Rn \ {0})). In view of Remark 3.16, by smoothly patching together the functions TAX, one can obtain a C°° function QA(x, ) on T*(M) \ {0} = {(x, tj) E T*(M) : t; 54 01, which is positively homogeneous of degree m in the variable 6. The function CA is called the homogeneous principal symbol of A. A classical pseudo-differential operator A E Ln'(M) is said to be elliptic of order m if its homogeneous principal symbol aA(x, t;) does not vanish on the bundle T*(M) \ {0} of non-zero cotangent vectors. Then we have the following:

(6) An operator A E L '(M) (M) is elliptic if and only if there exists a parametrix B E Llm(M) for A: AB BA

I mod L-°°(M), I mod L-°°(M).

Let fl be an open subset of Rn. A properly supported pseudo-differential operator A on S2 is said to be hypoelliptic if it satisfies the condition

sing suppu = sing supp Au

,

u E D'(Il).

For example, Theorem 3.14 tells us that elliptic operators are hypoelliptic. We remark that this notion may be transferred to manifolds. The following criterion for hypoellipticity is due to Hormander (cf. [Ho, Chapter XXII, Theorem 22.1.3]):

Theorem 3.19. Let A = p(x, D) E Lp b(S2) be properly supported. Assume that, for any compact K C St and any multi-indices cx, Q, there exist

constants Ch,a,p > 0, CK > 0 and y E R such that we have for all x E K and 161 > CK

(3.7a) (3.7b)

D' Dap(x, 6)1 < CK,a,#Ip(x, 6)I(1 +

Ip(x,6)I-' < Cx(1 + IeUµ.

Then there exists a parametrix B E LP 6 (1) for A.

CHAPTER IV

LP APPROACH TO ELLIPTIC BOUNDARY VALUE PROBLEMS In this chapter we study elliptic boundary value problems in the framework of Sobolev spaces of LP-style, by using the LP theory of pseudo-differen-

tial operators. For more thorough treatments of this subject, the reader might refer to Seeley [Se3], Taylor [Ty], Chazarain and Piriou [CP] and also Taira [Ta2] (L2 version).

4.1 The Dirichlet Problem In this section, we shall consider the Dirichlet problem in the framework of Sobolev spaces of L' style which is a generalization of the classical potential approach to the Dirichlet problem.

Let Q be a bounded domain of Euclidean space R' with C°° boundary F. Its closure SZ = St u F is an n-dimensional compact C°° manifold with

boundary. One may assume that Sl is the closure of a relatively compact open subset S2 of an n-dimensional compact C°° manifold M without boundary in which St has a C°° boundary r (see Figure 3.1). Let p be a strictly positive density on M and w a strictly positive density on F. We let n

A

i=1

a (J1x i

)

a (ah3(x)) 3x.j + j=1

n

i=1

8

b'(x) 01x i + c(x)

be a second-order elliptic differential operator with real coefficients such that:

(1) a'.' E C°°(M), a'r = a"i and there exists a constant ao > 0 such that n

E a' (x)6i6i

ao I6I2

on T*(M).

iJ=1

Here T*(M) is the cotangent bundle of M. (2) bi E C°°(M). (3) c E C°°(M) and c < 0 in M. Further, for simplicity, assume that (4.1)

The function c does not vanish identically on M. Typeset by AMS-TIC 109

110

IV.

LP APPROACH TO ELLIPTIC BOUNDARY VALUE PROBLEMS

First we consider the Dirichlet problem: (D)

(Au=f

in Q,

Sl ulr = v

on r.

The next theorem states the existence of a volume potential for A, which plays the same role for A as the Newtonian potential plays for the Laplacian (cf. [Se2, Theorem 1], [Tal, Theorem 8.2.1]):

Theorem 4.1. (i) The operator A : C°°(M) -* C°°(M) is bijective, and its inverse Q is a classical elliptic pseudo-differential operator of order

-2 on M. (ii) The operators A and Q extend respectively to isomorphisms

A : H',P(M) -.+ Hs-2,P(M), Q

:

H9-2,p(M) -, H',P(M)

for all s E R and 1 < p < oo, which are still inverses of each other. Next we construct a surface potential for A, which is a generalization of the classical Poisson kernel for the Laplacian. We let Kv = Q(v 0 6)Ir, v E C°°(F), where v ® S is a distribution on M defined by

(v ®b,P - P) _ (v,PIr -W) ,

a E C°°(M).

Then we have the following (cf. [Tal, Theorem 8.2.2]):

Theorem 4.2. (1) The operator K is a classical elliptic pseudo-differential operator of order -1 on F. (ii) The operator K : C°°(F) -f C°°(F) is bijective, and its inverse L is a classical elliptic pseudo-differential operator of first order on F. Furthermore, the operators K and L extend respectively to isomorphisms

K : 13°,P(F) ., B°+1,P(F), B°+1,P(F) L: -> 13°,P(F) for all o E R and 1 < p < oo, which are still inverses of each other. Now we let

P,p = Q(Lw ® 6)in,

E C°°(r).

Then the operator P maps C°°(F) continuously into C00(1l), and extends to a continuous linear operator P : B'-"P,P(F) -; Hs,P(Il)

4.2 FORMULATION OF A BOUNDARY VALUE PROBLEM

111

for all s E R and 1 < p < oo. Further we have for all co E B3-1/P'P(I') APcp = AQ(LV ® S)IQ = (Ly 0 6)I Q = 0 in n, { Poor=KLcp=cp on T.

The operator P is called the Poisson operator. We let

N(A,s,p)={uEH''P(cz): Au=0inQZ}. Since the injection H''P(St) --> D'(SZ) is continuous, it follows that the space N(A, s, p) is a closed subspace of H''P(SZ); hence it is a Banach space. Then we have the following (cf. [Se3, Theorems 5 and 6]):

Theorem 4.3. The Poisson operator P maps the space Bs-1/P'P(F) isomorphically onto the space N(A, s, p) for all s E R and 1 < p < oo. We remark that the spaces N(A, s, p) and Bs-1/P'P(r) are isomorphic in such a way that N(A,s,p)°, B'-1/P,P(T),

13'-1/P,P(r).

N(A,s,p) FP

Combining Theorems 4.1 and 4.3, we can obtain the following (cf. [ADN], [LM]):

Theorem 4.4. Let s > 1/p where 1 < p < oo. The Dirichlet problem (D) has a unique solution u in the space H8'P(5l) for any f E Hs-2'P(Q) and any p E Bs-1/P,P(T). Next we consider the Neumann problem:

Au=f

(N)

1

in Q,

a r = `p

on r.

1

Then we have the following (cf. [ADN], [LM]):

Theorem 4.5. Let s > 1 + 1/p where 1 < p < oo. The Neumann problem (N) has a unique solution u in the space H''P(d) for any f E Hs-2'P(52) and any cp E 138-1-11"(F).

4.2 Formulation of a Boundary Value Problem Let s > 1 + 1/p where 1 < p < oo. If u E H''P(Q), we can define its traces you and rylu respectively by the formulas

('You=uIr,

1

8u

71u = 01

Ir

and let 'Yu = {'You, 'yl u l

Then we have the following (cf. [St]):

.

IV.

112

LP APPROACH TO ELLIPTIC BOUNDARY VALUE PROBLEMS

Theorem 4.6 (The trace theorem). The trace map , 89-1/P,P(I') x Bs-1-1/P,P(r)

y : H9,P(Q)

is continuous and surjective, for s > 1 + 1/p where 1 < p < eo.

We introduce a subspace of Bs-1-1/p,p(r) which is a suitable tool to investigate the boundary condition

Bu = a Ou + bu

(4.2)

r

= ayju + byou,

u E Hs,p(Sl).

We let

B*-1-11P,P(F)

_

B'-1-1/P,P(F)

{co = aCp1 + b(P2 : o1 E

V2 E BS-11"(F)1

and define a norm I0I9-1-1/p,p =

inf { Io1 Is-1-1/p,p + IV2Is-1/p,p :

= avl + b4P2 }

.

Then we have the following:

Lemma 4.7. The space B*-1-11"P(F) is a Banach space with respect to the norm I

' 1* s-1-1/P,P.

Proof. (i) First we verify that the quantity satisfies the axioms of a norm. (a) If Ipls-i-i/p,p = 0, then one can find a sequence {co } in the space B8-1-11P ,P(r) and a sequence {
with 1

ko(i) Is-1-1/p,p < 1

(e)

IA Is-1/P,P <

Hence we have with a constant C > 0 IVIs-1-1/P,P C I ap(')

I g-]-1/p,p + I b(p2e) I s-1-1/P,P


<

2C

I s-1/P,P)

-->0as £->oc,

4.2 FORMULATION OF A BOUNDARY VALUE PROBLEM

113

so that (b) We have for all A E C AV = C1A1p1 + bAlp2 }

IAVI3-1-1/P,P = inf { IAp1 Is-1-1/P,P + I AV2I s-1/P,P

= IAI inf {Ik1Is-1-1/p,p + Ic02Is-1/P,p : V = av1 +

= Al

bcp2}

IIwIs-1-1/P,P

(c) If we have

V = ail + bcp2 with p1 E

Bs-1-11P,P(r),

V2 E

0 = a 7k, + b'2 with 01 E Bs-1-11P,P(r), 02 E

B3-11P,P(r),

Bs-1/P,P(r)

then it follows that

+0=a(
1-1/P,P Ik1 + 01Is-1-11p,p + Ia2 + yb2I s -1/P,P l

(Ia1Is-1-11p,p +

Ik2Is-1/P,P) + (I01Is-1-1/P,P + 1021s-1/RP)

This proves that IV + 4 Is-1-1/p,p C

I

I s-1-1/p,p'

(ii) To prove the completeness of the space B;-1-1/P'p(P), it suffices to show that the condition 00

P=1

I

(P)I _

-1-1/P,P G 00

implies that there is an element cp E B*-1-1/p'p(P) such that

--) 0asN->oo. P=1

s-1-1/P,P

By condition (4.3), one can find a sequence {V(1)} in Bs-1-1/P'P(r) and a sequence {cp(t)} in Bs-1/1 1(r) such that P

= a`P1 + b4'2

IV. LP APPROACH TO ELLIPTIC BOUNDARY VALUE PROBLEMS

114

with 1

e

Io1 )Is-1-1/P,P + I'P2eIs-1/P,p C I'P(e)Is-1-1/p,P + 2e.

Then we have 00

00

00

e

Is-1/P,P

Io 1 Is-1-1/P,P e=1

Ir(e) Is-1-1/P,p +

e=1

2e

1

00

Is-1-1/P,p + 1.

Ir(e)

C e=1

B3-1-1/p,p(r)

Hence, since the spaces find two elements cpl E

and Bs-1/P'P(r) are complete, one can c02 E B3-1/P,P(r) such that

Bs-1-11P,p(r) and

->0as N->oo, s-1-1/p,p

->0as N->oo. 1=1

s-1 /p,P

Therefore, letting B*-1-1/P,P(r),

= ayl + bV2 E

we obtain that, as N -> oo, 1*

N I e-1

Is-1-1/P,p N 01e)

+

- 'P1

e-1

s-1-1/p,p

02e)

-4 0.

'P2

1=1

s-1/P,P

This completes the proof of Lemma 4.7. Furthermore it is easy to verify the following:

Proposition 4.8. The mapping B : H5,P(Q) -f B*- 1-1/RP(F) is continuous and surjective, for s > 1 + 1/p where 1 < p < oo. Now we can formulate our boundary value problem for (A, B) as follows:

Given functions f E H8-2'P(S2) and cp E B;-1-1/p'p(r), find a function u E H-,P(S2) such that (*)

(Au = f

in S2,

S` Bu=cp on r.

4.3 REDUCTION TO THE BOUNDARY

115

4.3 Reduction to the Boundary In this section, we shall show that problem (*) can be reduced to the study of a pseudo-differential operator on the boundary. Let f be an arbitrary element of Hs-2,P(SZ), and cp an arbitrary element of B*-1-11P'P(r) such that cp = aCp1 + bV2,

Bs-1-1/P,P(r),

o1 E

IP2 E

B8-1/P,P(r),

where 1 < p < oo and s > 1 + 1/p. First we consider the following Neumann problem:

Av = f (N)

(l

av lr

aV

in SZ,

= 1 on r.

By Theorem 4.5, one can find a unique solution v E H3'P(fl) of problem (N). Then it is easy to see that a function u in H''P(SZ) is a solution of problem

Au = f

(*)

in SZ,

Bu=cp our if and only if w = it - v E Hs P(SZ) is a solution of the problem

(Aw=O in Q, Bw=cp-Bv=b(cp2-vjr) on IF. But Theorem 4.3 tells us that the spaces N(A, s, p) and Bs-1/P'P(F) are isomorphic in such a way that N(A, s, p)

-vo. B'-1/"(r),

N(A,s,p) P

B'-1/P,P(r).

Therefore, we find that w E Hs,P(SZ) is a solution of problem (*') if and only if '0 E Bs-1/P'P(r) is a solution of the equation (t)

BPib=b(cp2--yov)

onr.

Here 0 = yow, or equivalently, w = Pib. Summing up, we obtain the following:

Proposition 4.9. Let 1 < p < oo and s > 1 + 1/p. For functions f E H-1 -2,p(Q) and cp E B*-1-11P'P(r), there exists a solution u E H',P(SZ) of problem (*) if and only if there exists a solution 0 E Bs-1/P'P(r) of equation (f). We remark that equation (f) is a generalization of the classical Fredholm integral equation.

116

IV. LP APPROACH TO ELLIPTIC BOUNDARY VALUE PROBLEMS

We let

C°°(r)

T : C°°(r)

'p H BPcp. Then we have by condition (4.2)

T=all+b, where 11'p = -11PP

a

=

F

It is known (cf. [Ho, Chapter XX], [Se3], [RS, Chapter 3]) that the operator 17 is a classical pseudo-differential operator of first order on r; hence the operator T is a classical pseudo-differential operator of first order on the

boundary P. We remark that the operator T : C°°(r) - C°°(r) extends to a continuous linear operator T : B°'P(r) ---> Bo-1,P(r) for all o E R. Consequently, Proposition 4.9 asserts that problem (*) can be reduced to the study of the pseudo-differential operator T on the boundary r. We shall formulate this fact more precisely in terms of functional analysis. We associate with problem (*) a continuous linear operator

A : Hs,P(1) -) H9-2,P(Q) x

B*-1-11P,P(r)

as follows.

(a) The domain D(A) of A is the space Hs>P(S2). (b) Au = {Au, Bu}, u E D(A). Similarly, we associate with equation (f) a linear operator Bs-1/P,P(r) -, Bs-1/P,P(r)

T: as follows.

(a) The domain D(T) of T is the space

D(T) = {

E

B3-11P,P(r)

:

Ty7 E Bs-1/P,P(r)}

(,Q) Tp = Tp,'p E D(T). We remark that the operator T is a densely defined, closed operator, since

the operator T : Bs-1/P,P(r) -* B9-1-1/P'P(r) is continuous and since the domain D(T) contains the space C°°(r). Then Proposition 4.9 can be reformulated in the following form (cf. [Ta2, Section 8.3]):

4.3 REDUCTION TO THE BOUNDARY

117

Theorem 4.10. (i) The null space N(A) of A has finite dimension if and only if the null space N(T) of T has finite dimension, and we have

dimN(A) = dimN(T). (ii) The range R(A) of A is closed if and only if the range R(T) of T is closed; and R(A) has finite codimension if and only if R(T) has finite codimension, and we have codim R(A) = codim R(T).

(iii) The operator A is a Fredholm operator if and only if the operator T is a Fredholm operator, and we have ind A = ind T.

Here we recall that a densely defined, closed linear operator T from a Banach space X into a Banach space Y is called a Fredholm operator if it satisfies the following three conditions: (a) The null space N(T) of T has finite dimension. (b) The range R(T) of T is closed in Y. (c) The range R(T) has finite codimension in Y. In this case, the index of T is defined by the formula

ind T = dim N(T) - codim R(T).

Furthermore the next theorem states that the operator A has the regularity property if and only if the operator T has.

Theorem 4.11. Let 1 < p < oo and s > 1 + 1/p. The following two conditions are equivalent: (4.4)

u E LP(Sl), Au E Hs-2'P(S2), Bu E B:-1-1/P'P(F) (4.5)

u E Hs'P(52);

p E B-1/P,P(F) T(p E B"-1/P,P(r) = (P E Bs-1/n,n(I').

Proof. (i) (4.4)

(4.5): Assume that

0 E B-1/P'P(F) and

Tcp E

Bs-1/P'P(F).

Then, letting u = PV, we obtain that u E LP(Tl), Au = 0

and Bu = Tcp E Bs-11P'P(F).

Hence it follows from condition (4.4) that u E H3'P(Sl),

IV.

118

LP APPROACH TO ELLIPTIC BOUNDARY VALUE PROBLEMS

so that by Theorem 4.6 0 = 'You E Bs-1/P,P(r).

(ii) (4.5)

(4.4): Conversely, we assume that u E LP(f2), Au E

Hs-2,p (Q), Bu

E B*-1-1/P'P(r)

where Bu = acpi + bcp2,

with

'1 E

Bs-1-1/P,P(r),


B8-11P,P(r).

Then the function u can be decomposed as follows:

u =V+w. Here V E H8,P(fl) is a solution of the Neumann problem

Av=Au v

in Q,

onr

and so

w=u-vEN(A,O,p). But Theorem 4.3 tells us that the function w can be written as

w = Py,

y ='Yow E B-11P,P(r).

Hence we have by Theorem 4.6

Tcp = BPy = Bu - By = b(y2 - vir) E B8-1/P'P(r). Thus it follows from condition (4.5) that V

E B8-11P,P(r)

so that again by Theorem 4.3 w = Py E Hs,P(f2).

This proves that

u = v + w E Hs,P(fl). The proof of Theorem 4.11 is complete.

The next theorem states that a priori estimates for A are entirely equiv alent to the corresponding a priori estimates for T:

4.3 REDUCTION TO THE BOUNDARY

119

Theorem 4.12. The following two conditions are equivalent: (4.6)

Hull-,P < C (IIAuIIs-2,P +

IBuI9-1-1/P,P +

U E D(A);

IIuIIP

(4.7)

IPE D(T). IcI-1/P,P) , C (ITpIs-1/P,P + Here and in the following the letter C denotes a generic positive constant. IPIs-1/P,P C

Proof. (i) (4.7) = (4.6): We decompose a function u E D(A) as in the proof of Theorem 4.11:

u =V+W, where Bu = a' 1 + bco2 and 1

Av=Au avI r

in Q,

a

on r.

Then we have by Theorem 4.5 (4.8)

IIvII8,P : C

(IIAuIis-2,p +

IVlls-1-1/P,P)

Further, applying estimate (4.7) to the function yow, we obtain that Iyowls-1/P,P

C (IT(yow)I s-1/P,P + yyowl-1/P,P) = C (I BwI s-1/P,P + 1yowl-1/P,P)

C

(1c021s-1/P,P + I yovl s-1/P,P + l yowl -1/P,P)

G C (I02Is-1/RP + IIv11s,P +

1yowl-11P,P)

In view of Theorem 4.3, this gives that (4.9)

IIwII,.,P < C

(1P2Is-1/P,P + IIvIIs,P + IIw1IP)

< C (Ik21s-1/p,P + IIuIIP + IIvIIs,P)

Thus, combining estimates (4.8) and (4.9), we obtain that (4.10)

IIwIIs,P < C

(IIAuII8-2,P + Ip1Is-1-1/P,P +

IP2Is-1/P,P + IIuIIP)

Therefore, the desired estimate (4.6) follows from estimates (4.8) and (4.10), since u = v + w. (ii) (4.6) = (4.7): Conversely, taking u = Pcp with cp E D(T) in estimate (4.6), we obtain that (4.11)

C

IIPSV1Is,P

(IT'PI9-1-11P,P +

IIP(PIIP)

< C (ITVI s-1/P,P + IIPwIIP)

But Theorem 4.3 tells us that the operator P maps the space B0 l h P(F) isomorphically onto the space N(A, o, p) for all a' E R. Thus estimate (4.7) follows from estimate (4.11).

120

IV.

LP APPROACH TO ELLIPTIC BOUNDARY VALUE PROBLEMS

4.4 Operator 17 The next theorem summarizes two important properties of 17 as a pseudodifferential operator:

Theorem 4.13. (i) The operator 17 is strongly elliptic, that is, there exist constants cl > 0 and c2 > 0 such that for all cp E C°°(r) Re

(4.12)

fr

-

Ilcp ip do > cl Ic012/2,2

1/2,2

(ii) Let be the principal symbol of the operator 17. Then there exists a constant co > 0 such that

pi(x',C') >

(4.13)

on T*(r) \ {0}.

is Here T*(r) \ {0} is the bundle of non-zero cotangent vectors, and the length of ' with respect to the Riemannian metric of r induced by the natural metric of R".

Proof. (i) By the divergence theorem, it follows that for all u E C2(Ii)

-Re fj Au u dx = i,7=1

L aii au a u dx - Be J au r aL ax axi

u do,

i

1

+

"

ab2 axi - cudx

A. " -2 jb2.niIuI2dcT.

12

I

1

=1

Hence, taking u = Pcp with cp E C°°(r), we obtain that (4.14)

Re

Jr

1750

do, =

i,i=1

f J ai.i a(Pcp) axi S2

1

+JJ

2

-2

"

1

a(Pcp)

,

"

ab'

axi - c

dx

7

IPcp12 dx

b`.niI40I2do,

r i=1 >aoEn

la(a)12dx

-Cififin IPcoI2dx-C2J IcoI2dv r > ao11pV1li,2 - (ao + C1)IIPVIlo,2 - C2

f

r

IVI2 da.

4.4 OPERATOR H

121

But it is known (cf. [Mi, Theorem 3.16]) that, for each e > 0, there exists a constant C, > 0 such that Jr

IVI2 do, < eIIPV IIi,2 +

G'EIIPVII2

0,2

Hence, carrying this interpolation inequality into inequality (4.14) with e = ao /2, we have (4.15)

Re

Jr .UV i7da > 2 IIP'PIIi,2 -

Therefore, the desired inequality (4.12) follows from inequality (4.15), since the operator P maps B 1/2,2(x) isomorphically onto N(A, a, 2) for all

aER.

(iii) It is known (cf. [Ho], [Ku], [Ty]) that the strong ellipticity (4.12) implies inequality (4.13) for the principal symbol of the operator 1I.

CHAPTER V

PROOF OF THEOREM 1 In this chapter we consider the boundary value problem

Au= f (*)

in S2,

{ Bu = a av + bu l r = cp on r,

and prove Theorem 1. We associate with problem (*) a linear operator

A = (A, B) : H',n(d) &--p Hs-2,p(d) x

B*-i-i/r,r(r).

In order to prove Theorem 1, it suffices to show that the operator A is bijective. Indeed, the continuity of the inverse of A follows immediately from

an application of Banach's closed graph theorem, since A is a continuous operator. Theorem 1 will be proved in a series of theorems (Theorems 5.1, 5.5 and 5.7) in the subsequent sections.

5.1 Regularity Theorem for Problem (*) The next theorem states that the operator A has the regularity property:

Theorem 5.1. Assume that conditions (H.1) and (H.2) are satisfied:

(H.1) a(x) > 0 and b(x) > 0 on r.

(H.2)b(x')>0onro={x'Er:a(x')=0). Then we have for ails ERandl
u E H8"(d).

Proof. (1) By Theorem 4.11, we know that the question of regularity for solutions of problem (*) is reduced to the corresponding question for the operator T, where

T=all+b,

and

H

a

T" (P4')

r Typeset by AMS-TFX

122

5.1 REGULARITY THEOREM FOR PROBLEM (*)

123

The operator 11 is a classical pseudo-differential operator of first order on the boundary 1', and its complete symbol is given by the following (cf. [Tal, Section 10.2] ):

(p1(x', C') +

q, (x', C')) + (po(x', C') + V_-__1 q0 (x', c'))

+ terms of order < -1, where (cf. (4.13)) (5.1)

pi(x',C') >

on T*(F) \ {0}.

IThus,

For example, if A is the usual Laplacian L = a2l8x2 + we have

+ a218x22 , then

Pi(x',C') = Cl. we obtain that the operator T = all + b is a classical pseudodifferential operator of first order on the boundary r, and its complete symbol t(x', ') is given by the following: (5.2)

t(x', ') =a (x') (P1(x',C') + V_-__1 gi(x',C'))

+ ([b(x') + a(x')po(x', ) ] + via(x')go(x', + terms of order < -1. (2) Therefore, in order to prove Theorem 5.1, it suffices to show the following:

Lemma 5.2. Assume that conditions (H. 1) and (H.2) are satisfied. Then we have for all s E R (5.3)

V E D'(F), Tip E Bs,P(F)

V E B",P(F).

Furthermore, for any t < s, there exists a constant Cs,t > 0 such that 1k19,P

C9,t (ITV19,P + IVIt,P)

Proof. (a) The proof of Lemma 5.2 is based on the following lemma:

Lemma 5.3. If conditions (H.1) and (H.2) are satisfied, then, for each point x' of F, one can find a neighborhood U(x) of x' such that:

For any compact K C U(x') and any multi-indices a, (, there exist constants CK,a,Q > 0 and CK > 0 such that we have for all x' E K and Ie'I> -CK (5.5a)

I D{,DX",t(x', C') C CK,a,# It(x', C')I (1 +

jj)-lal+(1/2)I)3I

V. PROOF OF THEOREM 1

124

It(x', ')I-1 < CK.

(5.5b)

Granting Lemma 5.3 for the moment, we shall prove Lemma 5.2.

(b) First we cover the boundary r by a finite number of local charts {(Uj,Xj)}'1 in each of which inequalities (5.5a) and (5.5b) hold. Since the operator T satisfies conditions (3.7a) and (3.7b) of Theorem 3.19 with y = 0, p = 1 and 6 = 1/2, it follows from an application of the same theorem that there exists a parametrix S in the class L° 1/2 (Uj) for T. Let {cp j } ' be a partition of unity subordinate to the covering 1, and choose a function Oj E Co (Uj) such that Oj = 1 on suppcpj, so that cpjzb = cpj. Now one may assume that cp E Bt,P(r) for some t < s and that TV E

B8,P(r). We remark that the operator T can be written in the following form:

m

m

jTtbj + 1: ypjT(1 - Oj)

T=> j=1

j=1

But the second terms pjT(1 - tj) are in L--(r), since Vj(1 - Oj) = 0. Hence we are reduced to the study of the first terms cpjTOj. This implies that we have only to prove the following local version of assertions (5.3) and (5.4):

(5.3')

(5.4')

'/'jV E Bt,P(Uj), TOjw EB8,P(Uj) ==> Vjp E B9,P(Uj); j0jpjs,P

- Cs,t (lTojp18 +

l

jV1t,P)

But, applying Theorem 3.17 to our situation, we obtain that the parametrix S maps Bo'P(Uj) continuously into itself for all a E R. This proves assertions (5.3') and (5.4'), since we have ST - I mod L-°°(Uj). Lemma 5.2 is proved, apart from the proof of Lemma 5.3. (c) Proof of Lemma 5.3 (c-1) First we verify condition (5.5b): By assertions (5.2) and (5.1), it follows that we have for

large enough

b(x')

2 b(x')

a(x') if

0,

so that (5.6)

lt(x',C')I ?

1),

since b(x') > 0 on Fo = {x' E F : a(x') = 0}. Here and in the following the letter C denotes a generic positive constant.

5.1 REGULARITY THEOREM FOR PROBLEM (*)

125

Inequality (5.6) implies condition (5.5b): (5.7)

1 t(x', C1)1 > C.

(c-2) Next we verify condition (5.5a) for Jal = 1 and Iii = 0: Since we have for 16'1 large enough

D.,t(x', ')j

C (a(x') + 1C'J-1),

it follows from inequality (5.6) that

t(x', ')I < C(1 +

1)

< C(1 + I10-1 It(x',C')I This inequality proves condition (5.5a) for Jal = 1 and I/31 = 0. (c-3) We verify condition (5.5a) for 101 = 1 and Ial = 0: To do so, we need the following elementary lemma on non-negative functions.

Lemma 5.4. Let f be a non-negative C2 function on R such that for some constant c > 0

sup if"(x)I < c.

(5.8)

xER

Then we have (5.9)

If'(x)l <

2c

f(x) on R.

Proof. In view of Taylor's formula, it follows that

0 < f(y) = f(x) +

x) +

f,( 2

(y -

x)2

where C is between x and y. Thus, letting z = x - y, we obtain from estimate (5.8) that

0 < f (X) + f'(x)z +

z2


0

Since we have for C' large enough C (I Da,a(x')

1C'i +

1) 1

V. PROOF OF THEOREM 1

126

it follows from an application of Lemma 5.4 and inequalities (5.6) and (5.7)

that D",t(x',

C [( a(x')
[IC11112

(a(x')IC'I +

C It(x',C')I C

1) + (a(x')I e'I + 1)] 1)1 2

+

1)]

(Ie111/2 It(x',e')I-1/2

+ 1)

(1 +

This inequality proves condition (5.5a) for 101 = 1 and IaH = 0. (c-4) Similarly, we can verify condition (5.5a) for the general case: Ial +

101=k, kEN. Now the proof of Lemma 5.2, and hence that of Theorem 5.1, is complete.

0 5.2 Uniqueness Theorem for Problem (*)

The next theorem asserts that the operator A is injective:

Theorem 5.5. Let 1 < p < oo. Assume that conditions (H.1) and (H.2) are satisfied:

(H.1) a(x') > 0 and b(x') > 0 on r.

(H.2)b(x')>0onI'o={x'EI':a(x')=0}. Then we have (5.10)

uEH2''(Q),Au =0in Sl,Bu =0on IF

u=0in ft

Proof. In view of Theorem 5.1, it follows that

uEH2'P(Q),Au=0inSt,Bu=0onI'

uEC°°(Q).

Therefore, the uniqueness result (5.10) is an immediate consequence of the following maximum principle:

Proposition 5.6. Assume that conditions (H.1) and (H.2) are satisfied. Then we have

vEC2(1l),Av>0inQ,Bv<0onI' z v<0on ft Proof. If v is a constant m, then we have

0
5.3. EXISTENCE THEOREM FOR PROBLEM (*)

127

Now we consider the case when v is not a constant. Assume to the contrary that

m=maxv>0. a Then, applying the strong maximum principle (cf. Appendix, Theorem A.1) to the operator A, we obtain that there exists a point xo of r such that

v(xo) = m, {

v(x)
Furthermore, it follows from an application of the boundary point lemma (cf. Appendix, Theorem A.2) that

av(xo) > 0. Since we have

Bv(xo) = a(xo)n- (xo) + b(xo)v(xo) < 0,

it follows from condition (H.1) that

a(xo) = 0,

b(xo) = 0.

This contradicts condition (H.2).

5.3. Existence Theorem for Problem (*) The next theorem asserts that the operator A is surjective:

Theorem 5.7. Lets > 1 + 1/p where 1 < p < oo. Assume that conditions (H.1) and (H.2) are satisfied:

(H.1) a(x') > 0 and b(x') > 0 on r.

(H.2)b(x')>0onI'o={x'er:a(x')=0}. Then, for any f E Hs-Z'P(SZ) and any cp E B*-1-"P'P(F), the boundary value problem

(Av=f (*)

inQ,

jl aav+bvlr=y on r

has a solution u in the space H''P(SZ).

V. PROOF OF THEOREM 1

128

5.3A Proof of Theorem 5.7. First, by Theorem 4.10, we know that ind A = ind T,

where the operator

T : BB-1/P,P(r) _, Bs-ilr,n(r) is defined as follows:

(a) The domain D(T) of T is the space

D(T) =

l`,

E

BB-1/P,P(r)

:

Tip E B3-11P,P(r)I .

(b) Tip = Tip, V E D(T).

But Theorem 5.5 tells us that the operator A (or equivalently the operator T) is injective. Hence, in order to prove the surjectivity, it suffices to show the following:

Proposition 5.8. The index of the operator T is equal to zero. Proof. (1) We replace the operator A by the operator A - A with A > 0, and consider instead of problem (*) the following boundary value problem:

(A - A)u = f

in Sl,

Bu=aau+bulr=yo on r. We associate with problem (*)A a linear operator

A(A) = (A - A, B) : H',P(cZ) ,)

H'-2,P(c) x

B:-1-1/P,P(r).

We remark that the operator A(A) coincides with the operator A when A = 0.

We reduce the study of problem (*)A to that of a pseudo-differential operator on the boundary, just as in the proof of Theorem 5.1. We can prove that Theorem 4.1 remains valid for the operator A - A. That is, we have the following: (a) The Dirichlet problem

( (A - A)w = 0 in St,

wIr=p

onr

has a unique solution w in Ht'P(1l) for any a E Bt-1/P,P(r), where t E R. (b) The Poisson operator P(A) :

Bt-1/P,P(r)

-> Ht,P(1),

5.3. EXISTENCE THEOREM FOR PROBLEM (*)

129

defined by w = P(A)cp, is an isomorphism of the space Bt-1L"(F) onto the space N(A - A, t, p) = {u E Ht,P(52) : (A - A)u = 0 in Q} for all t E R; and its inverse is the trace operator on r. Let T(A) be a classical pseudo- differential operator of first order on the boundary r defined as follows:

T(A) = BP(A) = an(A) + b,

A > 0,

where 17(A) : C°°(r)

C°°(r)

gyp' BP(A)cp.

Since the operator T(A) : C°°(r) -> C°°(r) extends to a continuous linear operator T(A) : Bt,P(r) > Bt-',P(r) for all t E R, one can introduce a densely defined, closed linear operator T(A) : Bs-1IP,P(r)

, B`11P,P(r)

as follows.

(a) The domain D(T(A)) of T(A) is the space D(T(A)) =

E B'- /P,P(r) : T(a)i E Bs- /P,P(r)}

.

(Q) T(A)co = T(A)p, V E D(T(A)). We remark that the operator T(A) coincides with the operator T when A = 0. Then we can obtain the following results (cf. Theorem 4.10): (I) The null space N(A(A)) of A(A) has finite dimension if and only if the null space N(T(A)) of T(A) has finite dimension, and we have

dim N(A(A)) = dimN(T(A)). (II) The range R(A(A)) of A(A) is closed if and only if the range R(T(A)) of T(A) is closed; and R(A(A)) has finite codimension if and only if R(T(A)) has finite codimension, and we have codim R(A(A)) = codim R(T(A)).

(III) The operator A(A) is a Fredholm operator if and only if the operator T(A) is a Fredholm operator, and we have ind A(A) = ind T(A).

(2) To study problem (*)A, we shall make use of a method essentially due to Agmon (cf. [Ag], [LM] and also [Ta2, Section 8.4]). This is a technique of

V. PROOF OF THEOREM 1

130

treating a spectral parameter A as a second-order differential operator of an extra variable and relating the old problem to a new one with the additional variable.

We introduce an auxiliary variable y of the unit circle

S = R/21rZ, and replace the parameter -A by the second-order differential operator a2

aye'

That is, we replace the operator A - A by the operator 2

A + aye,

and consider instead of problem (*)A the following boundary value problem:

Au := (A+a )u= f inQxS, 1 Bu = as +bulrs = `P on r x S.

(*)

We can prove that Theorem 4.1 remains valid for the operator A = A + 52 /8y2.

(a) The Dirichlet problem

in52xS, wlrxs = b on r x S

( Aw=O Sl

has a unique solution w in Ht>P(f2 x S) for any cp E Bt-1/1'P(r x S), where t E R. (b) The Poisson operator P : Bt-1/P,P(r x S) ---) Ht,P(12 x S),

defined by w = Pcb, is an isomorphism of the space Bt-lIP°P(r x S) onto the space N(A, t, P) = {u E Ht,P(1 x S) : Au = 0 in ci x S} for all t E R; and its inverse is the trace operator on IF x S. We let

T:c°°(rxs)-ic°°(rxs) BPS

.

5.3. EXISTENCE THEOREM FOR PROBLEM (*)

131

Then the operator T can be decomposed as follows:

T=a17+b, where

III

av

rxs

The operator II is a classical pseudo-differential operator of first order on the boundary r x S, and its complete symbol is given by the following: [Pi (x', C', y, 77) + V_-_1 41(x', 6', y,11)]

+ [Po(x', 6', y, 77) + / 4o (x', e', y, y)] + terms of order < -1, where (cf. (4.13)) (5.11)

P, W, C', y, y) > co

on T*(r x S) \ {0}.

I'12 + y2

For example, if A is the usual Laplacian 'A = a2/ax2 +

+ a2/Ox2, then

we have

Pl(x',S',y,17) = Thus we find that the operator

IS'I2 +712.

T=alt+b is a classical pseudo-differential operator of first order on the boundary r x S and its complete symbol is given by the following: (5.12)

a(x) [Pi (x', C', y, 77) +

41(x', ', y, q)]

+ [(b(x') + a(x')po(x', e', y, 71)) + ia(x')4o(x', C', y, ii)]

+ terms of order < -1. Then, by virtue of assertions (5.12) and (5.11), it is easy to verify that the operator t satisfies conditions (3.7a) and (3.7b) of Theorem 3.19 with t = 0, p = 1 and S_= 1/2, just as in the proof of Lemma 5.2. Hence there exists a parametrix S in the class L° 112(F x S) for the operator T. Therefore we obtain the following result, analogous to Lemma 5.2: Lemma 5.9. Assume that conditions (H.1) and (H.2) are satisfied. Then we have for all s E R

E D'(r x s), Tip E B''P(r x s)

E B3'P(r x S).

Furthermore, for any t < s, there exists a constant Cs,t > 0 such that (5.13)

PI9,P < O9,t OT'I9,P + I'It,P)

.

V. PROOF OF THEOREM 1

132

We introduce a densely defined, closed linear operator

T : B'-i"P,P(r x s)

B3-,/P,P(r x s)

as follows.

(&) The domain D(T) of T is the space

D(T) = {, E Bs-i/p,p(r x s)

:

Be-'/P,P(r

T

x S)} .

D(T) Then the most fundamental relationship between the operators T and T(A) (A > 0) is the following:

Proposition 5.10. If ind T is finite, then there exists a finite subset K of Z such that the operator T(A') is bijective for all A' = £2 satisfying

2EZ\K. Granting Proposition 5.10 for the moment, we shall prove Theorem 5.7.

(3) End of Proof of Theorem 5.7 (3-1) We show that if conditions (H.1) and (H.2) are satisfied, then we have

ind T < oo.

(5.14)

To this end, we need a useful criterion for Fredholm operators (cf. [Ta2, Theorem 3.7.6]):

Lemma 5.11 (Peetre). Let X, Y, Z be Banach spaces such that X C Z is a compact injection, and let T be a closed linear operator from X into Y with domain D(T). Then the following two conditions are equivalent: (1) The null space N(T) of T has finite dimension and the range R(T) of T is closed in Y. (ii) There is a constant C > 0 such that IIxIIx < C(IITxIIy + IIxIIz),

x E D(T).

Now, estimate (5.13) gives that (5.15)

I;)I9-1/p,p

Cs,t (IT;aI8-11P,P + PIt,p)

ED(T),

where t < s - 1/p. But it follows from an application of Rellich's theorem that the injection B3-I/P,P(P x S) -> Bt'P(P x s) is compact (or completely continuous) for t < s - 1/p. Thus, applying Lemma 5.11 with X = Y = B9-i/P,P(r x S),

5.3. EXISTENCE THEOREM FOR PROBLEM (*)

133

Z = B',P(r x s),

T=T, we obtain that the range R(T) is closed in Bs-I/P,P(r x S) and dim N(T) < oo.

(5.16)

On the other hand, by formula (5.12), we find that the symbol of the adjoint T* is given by the following (cf. Theorem 3.12): a(x') (Pi (x', 6, y, 77)

- / 41(x', C, y, ii)) 2

+ ([b(x') + a(x')o(x', ', y,

-2axs

(a(xe)-5E; 41(x',E1,y,71))

j=1 2

[ax'qox'

axj (a(x')-Oi11(x',e',y,rl)) 11

]/

j=1

+ terms of order < -1. But, by virtue of Lemma 5.4, it follows that

C9xia(x)=0onro={x'EI':a(x')=0}. Thus one can easily verify that the pseudo-differential operator T* satisfies conditions (3.7a) and (3.7b) of Theorem 3.19 with a = 0, p = 1 and 6 = 1/2. This implies that estimate (5.15) remains valid for the adjoint operator of T: 101-.'+1/P'P' C Cy,T

I

I T* 4' I -s+11P,P' + I W I

r,P),

Y E D(T * ),

where r < -s + 11p and p' =\ p/(p - 1), the exponent conjugate to p. Hence we have by the closed range theorem (cf. [Yo, Chapter VII, Section 5]) and Lemma 5.11 (5.17)

codimR(T) = dimN(T*) < oo,

since the injection B-s+1/P,P'(r x S)

-s+1/p.

Br,P'(r x S) is compact for r <

Therefore, assertion (5.14) follows from assertions (5.16) and (5.17). (3-2) By assertion (5.14), we can apply Proposition 5.10 to obtain that the operator T(Q2) : B9-1/P,P(r) , B9-1/P,P(r) is bijective if f E Z \ K for some finite subset K of Z. In particular, we have (5.18)

indT(Ao) = 0 if ao = 22, £ E Z \ K.

V. PROOF OF THEOREM 1

134

But it is easy to see that the symbol t(x', e'; A) of the operator

T(A)=all(A)+b, A>0, has the following asymptotic expansion: (5.19)

q, (x, c')]

t(x', e'; A) = a(x') [P1(x',C') +

+ [(b(x') + a(x')Po(x', )) + a(x')go(x', ')] + terms of order < -1 depending on A. Thus we can find a classical pseudo-differential operator K(Ao) of order -1 on the boundary r such that

T = T(Ao) + K(Ao) Furthermore, Rellich's theorem tells us that the operator K(Ao) :

Bs-1/p,p(r)

Bs-1/p,p(r)

is compact. Hence we have ind T = ind T(Ao ).

(5.20)

Therefore, Proposition 5.8 follows from assertions (5.18) and (5.20).

Theorem 5.7 is proved, apart from the proof of Proposition 5.10.

5.3B Proof of Proposition 5.10. (i) First we study the null spaces N(T) and N(T(A')) when A' = 0 with .£ E Z: N(T) = {So E Bs-1/p,p(r x s) : Ty, = 0}

N(T(A')) = {Ip E

Bs-lip,p(F)

:

,

T(A')sv = o}

.

Since the pseudo-differential operators T and T(A') are both hypoelliptic, it follows that

N(T)_{y1EC°°(rxS):T, =o}, N(T(A')) = {gyp E C°°(F) : T(A')

Therefore, we can apply [Ta2, Proposition 8.4.6] to obtain the following most

important relationship between the null spaces N(T) and N(T(A')) when A' = £2 with .£ E Z:

5.3. EXISTENCE THEOREM FOR PROBLEM (*)

135

Lemma 5.12. The following two conditions are equivalent: (1) dim N(T) < oo. (2) There exists a finite subset I of Z such that dim N (T(22)) < oo

if f E I,

dimN(T(f2)) = 0

if2¢I.

Moreover, in this case, we have

N(T) _ ®N(T(/2)) 0 eily, 1Ez

dim N(T) = E dim N(T(t2 )). 1EI

(ii) Next we study the ranges R(T) and R(T(A')) when A' = f2 with f E Z. To do so, we consider the adjoint operators T* and T(A')* of T and T(A'), respectively. The _next lemma allows us to give a characterization of the adjoint op-

erators T* and T(A')* in terms of pseudo-differential operators (cf. [Ta2, Lemma 8.4.8]):

Lemma 5.13. Let M be a compact C°° manifold without boundary. If T is a classical pseudo-differential operator of order m on M, we define a densely defined, closed linear operator

T : B9,P(M) -f B9-m+1,P(M) (s E R) as follows.

(a) The domain D(T) of T is the space

D(T) = {cp E H',P(M) : Tcp E H'-'+',P(M)I. (b) Tip = T
B-'+m-1,P'(M)

: T*b E B-",P (M)}

,

where p' = p/(p - 1) and T* E L i (M) is the adjoint of T. (d) T*4 = T*,b, 0 E D(T*). We remark that the pseudo-differential operators T(A)* and t* also sat-

isfy conditions (3.7a) and (3.7b) of Theorem 3.19 with u = 0, p = 1 and S = 1/2; hence they are hypoelliptic. Therefore, applying Lemma 5.13 to the operators T and T(7'), we obtain the following:

V. PROOF OF THEOREM 1

136

Lemma 5.14. The null spaces N(T*) and N(T(A')*) are characterized respectively as follows:

N(T*)={bEc-(rxs): N(T(A')*) = { V ) E C°°(r) : T(A')*b = 01.

By Lemma 5.14, we find that Lemma 5.12 remains valid for the adjoint operators T* and T(A')* (cf. [Ta2, Lemma 8.4.10]):

Lemma 5.15. The following two conditions are equivalent: (1) dim N(T*) < oo. (2) There exists a finite subset J of Z such that

( dimN(T(f2)*) < oo

if f E J,

t dimN(T(t2)*) = 0

if 2 V J.

Moreover, in this case, we have

dim N(T*) =

dim N (1-(f2)*). eE J

Hence, combining Lemma 5.15 and the closed range theorem, we obtain

the most important relationship between codimR(T) and codimR(T()')) when .A' = f2, f E Z (cf. [Ta2, Proposition 8.4.11]):

Lemma 5.16. The following two conditions are equivalent: (1) codim R(T) < oo. (2) There exists a finite subset J of Z such that codim R (T(f2)) < 00 { codim N (T(f2)) = 0

if f E J,

if f J.

Moreover, in this case, we have

codimR(T) =

codimR (T(f2)) QE J

(iii) Proposition 5.10 is an immediate consequence of Lemmas 5.12 and

5.16, with K=IUJ. Now the proof of Proposition 5.8, and hence that of Theorem 5.7, is complete.

CHAPTER VI

PROOF OF THEOREM 2 In this chapter we prove Theorem 2. More precisely, we prove a generation theorem for analytic semigroups (Theorem 6.8) for the operator 2t from LP(Sl) into itself defined by the following: (a) The domain of definition D(2t) of 2t is the set

D(2t)_{uEH2°p(St):Bu=a1%+bu (b) 2tu = Au, U E D(2t). Once again Agmon's method plays an important role in the proof of the

surjectivity of the operator 2t - AI (Proposition 6.7).

6.1 A Priori Estimates In this section we study the operator 2t, and prove a priori estimates for the operator 2( - AI (Theorem 6.3) which will play a fundamental role in the next section. In the proof we make good use of Agmon's method (Proposition 6.4). First, we have the following:

Lemma 6.1. Assume that conditions (H.1) and (H.2) are satisfied: (H.1) a(x') > 0 and b(x') > 0 on r.

(H.2)b(x')>0on I'o={x'EI':a(x')=0}. Then we have the a priori estimate IIu1I2,p < CIIAuIIp,

U E D(21).

Proof. Estimate (6.1) follows immediately from Theorem 1 with s = 2

andcp=0. 0

Corollary 6.2. The operator 2t is a closed operator. Proof. Let {uj} be an arbitrary sequence in the domain D(21) such that

Jui-;u Auk -> v

inLP(1l), in LP(fl). Typeset by AMS-TIC

137

VI. PROOF OF THEOREM 2

138

Then, applying estimate (6.1) to the sequence {uj}, we find that {uj} is a Cauchy sequence in the space H2'P(fl), so that u E H2'p(S2) and

uj

in H2''(1).

)U

Hence we have

Au = lim Au j = v in LP(1), j 00 and also by Proposition 4.8

Bu = jlim00 Buj = 0 in B*-11P'P(I') This proves that u E D(2t) and 2(u = v.

0

The next theorem is an essential step in the proof of Theorem 2:

Theorem 6.3. Assume that conditions (H.1) and (H.2) are satisfied. Then, for every -7r < 9 < ir, there exists a constant R(8) > 0 depending on 8 such that if A = r2eie and IAl = r2 > R(8), we have for all u E H2'P(Sl) satisfying Bu = 0 on P (i.e., u E D(21)) (6.2)

1u12,P + IA1112 . lull,P + IAl - IlullP

C(8) II(A - A)ullP,

with a constant C(O) > 0 depending on 0. Here I

1j

P

(j = 1, 2) is the

seminorm on the space H2'P(Q) defined by 1/p

lulj,P = f E IDau(x)IPdx I1=j Proof. (1) We replace the operator A - A by the operator z

A + e 'B y2

,

-7r < 0 < 7r,

and consider instead of the problem

( (A - A)u = f (*)''

in S2,

Bu=as--+buI=O onl'

the following boundary value problem:

_ (*)

A(6)u:_ (A+ezeaye)u= f inSlxS, Bu:=aa-.+bulrxs=0 onl'xS.

We remark that the operator A(8) is elliptic for -7r < 8 < 7r. Then we have the following result, analogous to Lemma 6.1:

6.1 A PRIORI ESTIMATES

139

Proposition 6.4. Assume that conditions (H.1) and (H.2) are satisfied.

Then we have for all ii EH2'P(SlxS)satisfying Bu=0on rxS IIuII2,P <- c(8)

(i(o)

+ IIuIIP)

,

with a constant C(O) > 0 depending on 0.

Proof. We reduce the study of problem (*) to that of a pseudo-differential operator on the boundary, just as in problem (*). We can prove that Theorems 4.3 and 4.4 remain valid for the operator A(0) = A + e:ea2/8y2, -ir < 0 < ir: (a) The Dirichlet problem

(A(0)iv=0 inQxS, wlrxs=gyp on rxS has a unique solution w in Ht,P(1 x S) for any cp E Bt-1/P,P(r x S), where t E R. (b) The Poisson operator P(0) :

Bt-i/P,P(r x S) ---> Ht,P(S2 x S),

defined by w = P(0)ep, is an isomorphism of Bt-1/P,P(r x S) onto the space

N(A(0),t,p)={uEHt,P(S2xS):A(0)u=0in SlxS}for all tER;and its inverse is the trace operator on r x S. We let

T(0) : C°°(r x S) -i C°°(r x S) BP(e)(p.

Then the operator T(0) can be decomposed as follows: T(0) = a17(0) + b

where

n(B)t

a (P(Bv)

rxs

The operator 17(6) is a classical pseudo-differential operator of first order on the boundary r x S, and its complete symbol is given by the following (cf. [Ta2, Section 10.2j):

(pi(xt,et,y,i1;0)+

4i(xt,et,y,l;e))

VI. PROOF OF THEOREM 2

140

+ (Po(x', ', y, z1; 9) + v I 1 qo(x', e', y, i; 8)) + terms of order < -1,

where (cf. (5.11)) Pi (x', i', y, 71; 0) ? cB

(6.4)

-F+ 77 2 Ib'

on T*(r x s) \ {0}.

For example, if A is the usual Laplacian A = 192/ax2 +

+ a2lax2, then

we have PI W, 6', y,'1; 9) ( [[hI2+coso.2)2+sin2.4J

1/2

1/2

2

Therefore, the operator t(O) = aft (0)+b is a classical pseudo-differential operator of first order on the boundary r x S and its complete symbol is given by the following: (6.5)

a(x') (PI (x', ', y,71; 9) + v 91(x', C', y,71; 8))

+ ([b(x') + a(x')Po(x', 6, y, y; B)) + \Ta(x')4o(x', ', y, 71; B))

+ terms of order < -1.

Then, by virtue of assertions (6.5) and (6.4), one can verify that the operator T(O) satisfies conditions (3.7a) and (3.7b) of Theorem 3.19 with

p = 0, p = 1 and 6 = 1/2, just as in the proof of Lemma 5.3. Hence we obtain the following result, analogous to Lemma 5.2:

Lemma 6.5. Assume that conditions (H. 1) and (H.2) are satisfied. Then we have for all s E R

E D'(r x s), T(e)d E B9'P(r x s) ==*

E B8'P(r x s).

Furthermore, for any t < s, there exists a constant Cq,t > 0 such that I'Is,P <- Cs,t (IT(evI9,p + I'vlt,P) .

The desired estimate (6.3) follows from estimate (6.6) with s = 2 - 1/p and t = -1/p, just as in the proof of Theorem 4.12. (2) Now let u be an arbitrary function in the domain D(2t):

uEH2''(1) and Bu=0 on r.

6.1 A PRIORI ESTIMATES

141

We choose a function (E C°°(S) such that

0<(<1 onS, supp(C [3 sr ' ((y)=1 f o r 2
3

t and let

vn(x,y) = u(x) ®((y)eZny,

y > 0.

Then we have vn E H2'p(1l X S), 492

A(9)vn = CA

e:e

2

vn

y

_ (A - ?12de)u ® (e:ny + 2(i,y)e:eu ® ('e'ny + eieu ®("eFey,

and also

Bvn=Bu 0Ce'ny=0 on F x S. Thus, applying inequality (6.3) to the functions vn = u ®(e'ny, we obtain

that (6.7)

IIu ®(etny112,p C C(O) (11x9u®(e'ny)

We can estimate each term of inequality (6.7) as follows: (6.8) 1/ p

IIu ® (e`nyllp =

Uf xS

ju(x)IP (y)jp dx dy)

_ I1(IIp -

HIulGp.

(6.9)(

A(9)(u ® (etny)ll p C II(A - 772e:e)u ®(e`nyll p +2iIIu0('e'nyllp+llu®("e:nyllp

II(Ilp . II(A - ri2ete)ullp + (2zII('Ilp + II("Ilp) Ilullp. (6.10)

ll u ®(e'ny II2,p

Icr2

JxS I Dx,y(u(x) ®((y)e'ny)Ip dxdy 37r/2

fJD,y(u() ® e)I dxdy /2

dl<2

3 1r/2

= k+E

2

LL,2

ykD'eu(x) l p dxdy

VI. PROOF OF THEOREM 2

142

>

7r E f0 IDOu(x)I' dx I,a1=2

E J I DQu(x)IP dx 1131=1

+ 772pJ u(x)Ipdx) UuI2,P

+77Plull,p +?12Pllullp)

Therefore, carrying these inequalities (6.8), (6.9) and (6.10) into inequality (6.7), we have with a constant C'(9) > 0 independent of i > 0: Iu12,P +

IUI1,P + g2IIujl P < C'(8) (II(A - i72eie)ul )P + 71 IlullP) .

If 71 is so large that 77 > 2C'(9),

then we can eliminate the last term on the right-hand side to obtain that Iu12,P + 77 Iu11,P +77211 UII P

< 2C'(0) II (A -

7I2eie)uII

P.

This proves inequality (6.2) if we take A_772eie

R(9) = 4C'(9)2, C(O) = 2C'(9).

The proof of Theorem 6.3 is now complete.

6.2 Generation of Analytic Semigroups In this section we prove that the operator 2( generates an analytic semigroup on the space LP(SI). First we prove part (i) of Theorem 2:

Theorem 6.6. Assume that conditions (H.1) and (H.2) are satisfied: (H.1) a(x) > 0 and b(x') > 0 on r.

(H.2)b(x')>0onI'o={x'Er:a(x')=0}.

Then, for every 0 < E < it/2, there exists a constant r(e) > 0 such that the resolvent set of 2t contains the set E(e) = {A = r2eie : r > r(e), -ir+e < 0 < 7r - e}, and that the resolvent (2t - AI)-1 satisfies the estimate (6.11)

II(2(-

AI)-1II

<

)A E E(E),

1 Al'

6.2 GENERATION OF ANALYTIC SEMIGROUPS

143

where c(e) > 0 is a constant depending on s.

Proof. (1) By estimate (6.2), it follows that if A = r2e'B, -7r < 9 < 7r and 1Al = r2 > R(9), then we have for all u E D(2t) lul2,p +

JAl1"2

- lull,p + lA - Ilullp < C(9)II(2t - AI)ullp.

But we find from the proof of Theorem 6.3 that the constants R(9) and C(9) depend continuously on 0 E (-ir, 7r), so that they may be chosen uniformly in 9 E [-7r + e, it + e], for every e > 0. This proves the existence of the constants r(E) and c(s), that is, we have for all A = r2e'B satisfying r > r(e)

and-7r+E<9<7r+E (6.12)

lul2,p + 1x1112 .

lull,p +

JAI

Ilullp S c(E)Il(2t - AI)ull p.

By estimate (6.12), it follows that the operator 2t - AI is injective and its range R(21 - Al) is closed in LP(Sl), for all A E E(E). (2) We show that the operator 2t - AI is surjective for all A E E(E): (6.13)

R(2t - AI) = LP(Sl),

A E E(e).

To do so, it suffices to show that the operator 2t - Al is a Fredholm operator with (6.14)

ind (2t - AI) = 0,

A E E(e),

since 2t - AI is injective for all A E E(E). (2-1) We reduce the study of the operator 2t - Al (A E E(E)) to that of a pseudo-differential operator on the boundary, just as in the proof of Theorem 1.

Let T(A) be a classical pseudo-differential operator of first order on the boundary r defined as follows:

T(A) = BP(A) = an(A) + b,

A E E(E),

where

17(A) : C°°(r) - C' (IF)

r Since the operator T(A) : C°°(r) -a C°°(r) extends to a continuous linear operator T(A) : B°'P(r) -* B°-1°P(r) for all a E R, one can introduce a densely defined, closed linear operator

7(A) : Bs-l/P'P(r)

> B9-11p,P(r)

VI. PROOF OF THEOREM 2

144

as follows.

(a) The domain D(T(A)) of T(A) is the space D(T(A)) = {V E

Bs-'Ip,p(r)

: T(a)i E

Bs- /P,P(r)} .

(a) T(A)MP = T(a)w, W E D(T(A)) Then we can obtain the following results (cf. Theorem 4.10):

(I) The null space N(2( - AI) of 2t - AI has finite dimension if and only if the null space N(T(A)) of T(A) has finite dimension, and we have

dimN(21- Al) = dimN(T(A)). (II) The range R(2( - AI) of 2t - AI is closed if and only if the range R(T(A)) of T(A) is closed; and R(21- AI) has finite codimension if and only if R(T(A)) has finite codimension, and we have codim R(2l - AI) = codim R(T(A)). (III) The operator 21-AI is a Fredholm operator if and only if the operator T(A) is a Fredholm operator, and we have

ind (2t - AI) = ind T(A). Therefore, assertion (6.4) is reduced to the following assertion:

indT(A) = 0, A E E(e). (2-2) To prove assertion (6.14'), we shall make use of the method of Agmon just as in Section 5.3. Let T(9) be the classical pseudo-differential operator of first order on the boundary r x S introduced in Section 6.1: (6.14')

T(O) = BP(O) = all(O) + b,

-1C < 0 < 0,

where

c°°(r x s)

17(o): c°°(r x s)

a (P(e)g) rxs We define a densely defined, closed linear operator Bs-1/P,P(r x S)

7(9) :

-> Bs-'IP,P(r x S)

as follows.

(a) The domain D (T(9)) of T(9) is the space D (T(9))

E

B'-1/v,n(r x S) : T(9)y3 E

B9-1/P,P(r x

S)} .

(Y(0)). Then we can prove the the most fundamental relationship between the (Q) T (B)HP =

ED

operators T(9) and T(A), analogous to Proposition 5.10:

6.2 GENERATION OF ANALYTIC SEMIGROUPS

145

Proposition 6.7. If indT(8) is finite, then there exists a finite subset K of Z such that the operator T(A') is bijective for all A' = 22eie satisfying

2EZ\K.

(3) End of proof of Theorem 6.6 (3-1) We show that if conditions (H.1) and (H.2) are satisfied, then we have ind T(8) < oo.

(6.15)

Now, estimate (6.6) with s = s - 1/p gives that (6.16)

PI s-1/p,p

cs,t

(IT(0)so1s-1/p,p +

Iwlt,p)

,

E D(7(e)),

where t < s - 1/p. But it follows from an application of Rellich's theorem that the injection Bs-1/p'p(I' x S) -> Bt'p(I' x s) is compact for t < s -1/p. Thus, applying Lemma 5.11 with

X = Y = B9-1/p,P(F x S),

Z = Bt'p(P x S), T = T(8),

we obtain that the range R (T(8)) is closed in Bs-1/1 1(P x S) and dim N (T(B)) < oo.

On the other hand, by formula (6.5), we find that the complete symbol of the adjoint T(8)* is given by the following (cf. Theorem 3.12): a(x') (Pi (x', ', y, 71; 8) - / 91(x', 6', y,17; 8)) n-1

+ ([bx1) + a(x')Po(x', C', y, q; 8) - E axj (a(x') . afj 41(x', C', y, ?I; 0))] j=1

n-1

axj (a(x') . afj P, (x', 6', y, 71; e))1

la(x')4o (x', 6', y, 71; 8) +

j=1

+ terms of order < -1. But, by virtue of Lemma 5.4, it follows that

axja(x')=0onFo={x' EF:a(x')=0}. Thus one can easily verify that the pseudo-differential operator T(0)* satisfies conditions (3.7a) and (3.7b) of Theorem 3.19 with y = 0, p = 1 and 6 = 1/2.

VI. PROOF OF THEOREM 2

146

This implies that estimate (6.16) remains valid for the adjoint operator T(9)* of T(9): W I-s+1/p,p,

< Cs,r

(JT(9)*n/,I-s+11p,p'

+JWI

r,p)

E D (T(0)*)

,

where r < -s + 1/p and p' = p/(p - 1), the exponent conjugate to p. Therefore, assertion (6.5) can be proved just as in the proof of Proposition 5.8.

(3-2) By assertion (6.15), we can apply Proposition 6.7 to obtain that the

operator T(t2e`e) : Bs-1/p,p(r) -+ B3-1/P,P(h) is bijective if f E Z \ K for some finite subset K of Z. In particular, we have indT(f2eiO) = 0 if .£ E Z \ K.

Thus, just as in the proof of Proposition 5.8, we can prove assertion (6.14') and hence assertion (6.13). (3-3) Summing up, we have proved that the operator 2( - AI is bijective for all A E E(c) and its inverse (21- AI)-1 satisfies estimate (6.11). The proof of Theorem 6.6 (part (i) of Theorem 2) is complete. Part (ii) of Theorem 2 may be proved as follows. Theorem 6.6 tells us that, for pE > 0 large enough, the operator 2I - ,u I satisfies condition (1.1) (see Figure 6.1).

Figure 6.1

Thus, applying Theorem 1.2 (and Remark 1.3) to the operator 21 - ttj, we obtain part (ii) of Theorem 2:

Theorem 6.8. If conditions (H.1) and (H.2) are satisfied, then the operator 2l generates a semigroup U(z) on L'(1) which is analytic in the sector

z#O,1 argzJ <7r/2-e}

6.2 GENERATION OF ANALYTIC SEMIGROUPS

147

for any 0 < e < ir/2, and enjoys the following properties: (a) The operators 2IU(z) and J (z) are bounded operators on LP(1) for each z E and satisfy the relation

dU(z) = %U (Z),

z E OE.

(b) For each 0 < e < r/2, there exist constants Mo(e) > 0, Mi(e) > 0 and a, > 0 such that IIU(z)II <_

II2U(z)II <_

M0(e)eµ`-Rez

M1(e)eµ6.Rez

II

zEA,, zELX.

(c) For each uo E LP(Sl), we have as z -* 0, z E zA,

U(z)uo --p uo

in LP(1l).

The proof of Theorem 2 is now complete.

CHAPTER VII

PROOF OF THEOREMS 3 AND 4 This chapter is devoted to the semigroup approach to the following semilinear initial boundary value problem: Given functions f and uo defined in S x [0, T) x R x RN and in S2, respectively, find a function u in St x [0, T) such that (**)

(at - A)u(x, t) = f (x, t, u, grad u)

in S2 x (0, T),

Bu(x',t) = a(x')a--(x',t)+b(x')u(x',t)lrx[o,T) =

0

u(x, 0) = uo(x)

on r x [0,T), in Q.

By using the operator 2l, one can formulate problem (**) in terms of the abstract Cauchy problem in the Banach space LP(S2) as follows: (**,)

L = %u(t) + F(t, u(t)), 0 < t < T, u1t=o = u0.

Here u(t) = t) and F(t, u(t)) = f t, u(t), grad u(t)) are functions defined on the interval [0, T), taking values in the space LP(S2).

Our semigroup approach can be traced back to the pioneering work of Fujita and Kato [FK]. For detailed studies of semilinear parabolic equations, the reader is referred to Friedman [Fr] and Henry [He].

7.1 Fractional Powers and Imbedding Theorems By Theorem 6.6, one may assume that the operator 2l satisfies condition (1.17).

(1) The resolvent set of 2( contains the region E shown in Figure 7.1.

(2) There exists a constant M > 0 such that the resolvent R(.\) = (2l AI)-' satisfies the estimate (7.1)

JIR(A)I)

M 1+JAI

A E Z.

Thus we can define the fractional powers (-21)" for 0 < a < 1 on the space LP(52) as follows (cf. formula (1.19)): sin a7r

j Typeset by AM5-TX 148

7.1 FRACTIONAL POWERS AND IMBEDDING THEOREMS

149

and

a > 0.

(-2t)a = the inverse of

Recall that the operator (-2t)a is a closed linear operator with domain D((-2t)a) D D(2t).

E

0

Figure 7.1

In this section we study the imbedding characteristics of the spaces D((-2t)a), which will make these spaces so useful in the study of semilinear parabolic differential equations. We let

Xa =the space D((-2t)a) endowed with the graph norm

lla of

(-2t)a Here

1/2

Hull. = (Ilullp + II(-2t)aull P)

,

u E D((-2t)a).

Then we have the following (cf. Proposition 1.17): (1) The space Xa is a Banach space. (2) The graph norm Ilulla is equivalent to the norm II(-2t)auIIp (3) If 0 < a < ,Q < 1, then we have Xp C Xa with continuous injection. The next theorem gives the imbedding properties of the spaces Xa into the Sobolev spaces (cf. [He, Theorem 1.6.1]):

Theorem 7.1. Let 1 < p < oo. Then we have the following continuous injections:

1
The proof of Theorem 7.1 is based on the following two results which characterize the imbedding properties of Sobolev spaces:

VII. PROOF OF THEOREMS 3 AND 4

150

Theorem 7.2 (Sobolev). Let SZ be a bounded domain in R' with boundary r of class C2. Then: (1) If 1 < p < n, we have 1

H2,p(Q) c Hl,q(Q),

1 p

-n<

1

q

< 1, p

with continuous injection.

(ii) If n/2 < p < oo and p # n, we have

0 < v < 2 - n,

H2,p(1l) C C'(1l),

p

with continuous injection. Part (i) of Theorem 7.2 follows by using Corollary 2.22 and Theorem 2.25

with m = 2, j = 1 and r := q, while part (ii) of Theorem 7.2 follows from Theorem 2.18 and Theorem 2.25 with m = 2 (cf. [Ad, Theorem 5.4]).

Theorem 7.3 (Gagliardo-Nirenberg). Let 52 be a bounded domain in R" with boundary of class C2, and 1 < p, r < oo. Then:

(i) If p #n and if

=-+6

--nJ+(1-9)-,

2
then we have for all u E H2,p(Sl) n Lr(1?) (7.2)

IIuI11,q < C1IluII2,pllullr_0,

with a constant C1 = C1(f , p, r, 0) > 0.

(ii) If n/2 < p < oo, p#n and if

0
1,

then we have for all u E H2'p(Sl) n Lr(1l) (7.3)

C2llull2,pllullr

with a constant C2 = C2(Sl, p, r, 8) > 0. Part (i) of Theorem 7.3 follows by using from Theorem 2.15 and Theorem

2.25 with m = 2, j = 1, q := r and r := q, while part (ii) of Theorem 7.3 follows from Theorem 2.18 and Theorem 2.25 with m = 2, j = v, q := r and r := oo (cf. [Fr, Part I, Theorem 10.1]).

7.1 FRACTIONAL POWERS AND IMBEDDING THEOREMS

151

Proof of Theorem 7.1. First, by estimate (7.1) with A = 0, it follows that (7.1')

IIUII2,p -< Mll2tullp,

U E D(2t).

(i) For each 9 E (1/2,1), we let

11 r and 1

1

q

p

e

0<e<29-1

1-0

p

n

29-1+e- 191 121+(1-9)1 n n \ p n /// r

Then it follows from inequalities (7.2) and (7.1') that C1llull2,pllullr-e

Ilulli,q <

MC1II2uIIp'IIuIIP

Hence we have for all b > 0 (7.4)

Ilulll,q s MC1(b-Bll ull p +

bl-bll2tullp) ,

u E D(2t).

But, if we let

B1 = the identity operator on the space H1'q(Sl), then, by part (i) of Theorem 7.2, it follows that D(2t) C H2,p(1) C Hlq(Q) = D(B1), since we have 1

29-1

p

n

<

1

1

1

q

p

2

<9<1.

Further we can write inequality (7.4) as follows: (7.4')

IIBlulll,q <_ MC1 (b-BIIuIIp

+bl-eli2tullp)

U E D(2t).

Therefore, applying Lemma 1.13 to our situation, we obtain that

X, = D((-2t)«) C D(BI) = Hl,q(c ),

9 < a < 1,

and

Ilulll,q

This proves part (i).

K. II(-2()"ullp ,

u E X.

VII. PROOF OF THEOREMS 3 AND 4

152

(ii) For each 0 E (n/2p,1), we let

r=p, and

0
\\.

Then it follows from inequalities (7.3) and (7.1') that C

C2IItII2,pIlullp-B

Ilull1-B

< MC2II2tnII Hence we have for all S > 0 (7.5)

II UII c-(i) <- MC2

(S-elluII

+

b1-bll

u E D(2t).

ullp)

But, if we let

B2 = the identity operator on the space C"(S2), then, by part (ii) of Theorem 7.2, it follows that D(2t) C H2,p(SI) C C°(n) = D(B2),

since we have

0
IIB2uIIc-(ff) <- MC2 (6-0IIuII p + 51-0II2uIIp)

,

u E D(2t)

Therefore, applying Lemma 1.13 to our situation, we obtain that

X. = D((-2t)') C D(B2) = C°(SZ),

0 < a < 1,

and

K' II(-2t)'ullp This proves part (ii). The proof of Theorem 7.1 is complete.

UEX"'.

7.2 SEMILINEAR INITIAL BOUNDARY VALUE PROBLEMS

153

7.2 Semilinear Initial Boundary Value Problems This final section is devoted to the proof of Theorems 3 and 4.

7.2A Proof of Theorem 3. We verify that all the conditions of Theorem 1.18 are satisfied; then Theorem 3 follows from an application of the same theorem. Since p > n, one can choose a constant a such that 2

so that

(n+1)
1<2a-n. p

Then, by part (ii) of Theorem 7.1 with v = 1, we have (7.6)

Xa C C1(S2)

and X,,, C H1'P(Q),

with continuous injections. Thus we find that the function F(t, u) := f (x, t, u(x), grad u(x))

is well defined on [0, T] x X. Furthermore, since the function f (x, t, u, ) is locally Lipschitz continuous, in view of assertion (7.6) it follows that, for all t, s E [0, to ] and for all u, v E Xc. with Il u - uo II c. < R, 11v - uo 11 ,. < R, (7.7)

IIF(t, u) - F(s, v)llp < II F(t, u) - F(t, v)II P + II F(t, v) - F(s, v)IIP


n

x. (u - v)

+ It - sI


Here C = C(to, R) > 0 is a (local) Lipschitz constant for the function f, and C' > 0 is an imbedding constant for the imbedding Xa. C H1'P(SZ). By inequality (7.7), we obtain that the function F(t, u) is locally Lipschitz continuous in t and u. The proof of Theorem 3 is complete. 0

7.2B Proof of Theorem 4. The proof is similar to that of Theorem 3; we verify that all the conditions of Theorem 1.18 are satisfied.

Since n/2 < p < n and 1 < ry < n/(n - p), one can choose a constant a such that (7.8)

max(n, 2p

1+ n y-1))
VII. PROOF OF THEOREMS 3 AND 4

154

so that 0<2a-n

and

1-2a- 1< 1 <1

P p n p'Y P Then, by Theorem 7.1 with v = 0 and q = pry, we have

Xa C L-(S2) and Xa C H1°P-I(SZ),

(7.9)

with continuous injections. We let

F(t, u) := f (x, t, u(x), grad u(x)),

t E [0, T], u E Xa

Then we have by condition (a) of Theorem 4 IIF(t, u)II P < 2P-1P(t, IIuII.)P

j(i + gradu) dx 1

< 2P-1P(t, IIuII.)P (I1I + IluIli n7)

Here and in the following, IS2I denotes the volume of the domain Q. By assertion (7.9), it follows that the function F(t, u) is well defined on [0, T] xXa for all a satisfying condition (7.8). (1) First we verify the local Lipschitz continuity of F(t, u) with respect to the variable t. By condition (b) of Theorem 4, it follows that II F(t, u) - F(s, u) IIP

=

J

If (x, t, u(x), grad u(x)) - f (x, s, u(x), grad u(x)) IP dx

s1

< 2P-1P(t, IIuII-)PIt - SIP

f(i + graduI) dx z

< 2p-1P(t, IIkII-)P (III + It - SIP. Ilull1P7) In view of assertion (7.9), this proves that

IIF(t,u) - F(s,u)IIP

(7.10)

C1(Ilulla) It - sI,

where C1(I I u I l a) > 0 is a constant depending on the norm I I u I I a

(2) Next we verify the local Lipschitz continuity of F(t, u) with respect to the variable u. To do so, we remark that (7.11)

II F(t, u) - F(t, v) II P

_

if (X, t' U(x), grad u(x)) - f (x, t, v(x), grad v(x)) IP dx SZ

r <2P-1J If(x) t,u(x),grad u(x))- f (x, t, u(x), grad v(x))IP dx sI

+2P-1

J sI

f(x,t,u(x),gradv(x)) - f (x, t, v(x), grad v(x))IP dx.

7.2 SEMILINEAR INITIAL BOUNDARY VALUE PROBLEMS

155

We estimate the two terms on the right of inequality (7.11). (2-1) By condition (c) of Theorem 4, it follows that (7.12)

I f (x, t, u(x), grad u(x)) - f (x, t, u(x), grad v(x))IP dx 3P-1p(t,

11U11-)P

f (1 +

IgradvIP(1-1))Igrad(u

IgraduIP(7-1) +

- v)IP dx.

But, by Holder's inequality, it follows that (7.13)

r

J

1dx\

Igrad(u - v)IPdx <

(y-1)/Y

I

sz

/

(fa

<

IQI(ry-1)/-YIIu

f

\1/'Y

(J

I

sz

- vlll,Pry

and also - v)IP dx

IgradulP('r-1)Igrad(u

(7.14)

inst

/f


Jsl

\ (7-1)/7 / r (graduIP-1 I

I

\ Jsz

/

<- IIUII11'Pti 1)IIu -

\ 1/.Y

Igrad(u-v)IP-1 dx J

vIl1,P'Y

and

(7.15)

IgradvlP(^'-1)

Igrad(u - v)IP dx < IIvll1(P- 1) IIu - vlll,n7

Thus, carrying inequalities (7.13), (7.14) and (7.15) into inequality (7.12) we obtain that (7.16)

if (x, t, u(x), grad u(x)) - f (x, 1, u(x), grad v(x))IP dx < 3P-1P(t, 11U11-)P

1) + ilvIl1'l ry 1)) IIu - vlip1,P7

(2-2) By condition (d) of Theorem 4, it follows that (7.17)

J

If(x,t,u(x),gradv(x))- f (x, t, v(x), gradv(x))IP dx

< 2P-1P(t, Hull.

+ 11v11-)p f (1 +

IgradvlP7)

lu - vlP dx

z

< 2P-1p(t, Hull. + 11v11-)Pllu - vIIp.

f

(1 + 1gradvlP-1) dx

< 2P-1 P(t, Hull. + 11v11 -)p (lcl + Ilvlli P-) Ilu - vll..

VII. PROOF OF THEOREMS 3 AND 4

156

Therefore, combining inequalities (7.11), (7.16) and (7.17), we obtain that JIF(t, u) - F(t, v) II y C

6p-1P(t

11 U11 o)P

+ Hull 'p7

1)

+ 11VII

Y

1)) flu - v111,

+4P-1p(t, Hull. + hull-)r (1921 + Ilvlli P') Hu - vllP.

In view of assertion (7.9), this proves that (7.18)

JI F(t, u) - F(t, v)llp C C2(IIuIla, Ikvlla) 1ju - v1Ja,

where C2(jjujj., Ilvll ) > 0 is a constant depending on the norms llull& and ll'IIaSumming up, we find from inequalities (7.10) and (7.18) that the function

F(t, u) is locally Lipschitz continuous in t and u. The proof of Theorem 4 is now complete.

APPENDIX: THE MAXIMUM PRINCIPLE Let SZ be a bounded domain of Euclidean space Rn, with boundary r, and let A be an elliptic second-order differential operator with real coefficients such that 2

a''(x)

A=

axiaxj +

i, 7=1

i=1

V(x)

a axe

+ c(x)

where:

(1) a'3 E C(R'), a'3 = a)', 1 < i,j < n and there exists a constant ao > 0 such that n

Ea

CC

lS l2,

x E Rn,

i,j=1

r , 52, " . _ (1 tt

Sn) E R'Z. rr

,

(2) bi c C(Rn).

(3) cE C(Rn) andc<0inQ. First we have the following strong maximum principle:

Theorem A.1 (The strong maximum principle). Assume that

(uEC2(Sl), Au>0 in St, m=max?u>0. If the function u takes its maximum m at some point xo of the interior St, then u - m in the connected component containing xo. Now assume that S2 is a domain of class C2, that is, each point of the boundary r has a neighborhood in which r is the graph of a C2 function of n-1 of the variables x1, x2i , xn. We consider a function u E C(St)f C2(1 ) which satisfies the condition

Au>0 inQ, and study the exterior normal derivative au/av at a point where the function u takes its non-negative maximum. The boundary point lemma reads as follows: Typeset by AA4S-TX 157

158

APPENDIX: THE MAXIMUM PRINCIPLE

Theorem A.2 (The boundary point lemma). Let S2 be a domain of class C2. Assume that a function u E C(S2) fl C2 (f2) satisfies the condition

Au>0 in Q, and that there exists a point x'0 E I' such that u(xo) = maxxEff u(x) > 0, { u(x) < u(xo), x E Q. Then the exterior normal derivative an (x'o) of u at x'o, if it exists, satisfies

a

n(xo) > 0.

For a proof of Theorems A.1 and A.2 and a general study of maximum principles, the reader might refer to [PW, Chapter 2] and [Ta2, Chapter 7].

REFERENCES

Adams, R.A., Sobolev spaces, Academic Press, New York, 1975. Agmon, S., Lectures on elliptic boundary value problems, Van Nostrand, Princeton, NJ, 1965. [ADN] Agmon, S., A. Douglis and L. Nirenberg, Estimates near the bound[Ad] [Ag]

ary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623-727. [BL]

Bergh, J. and J. Lofstrom, Interpolation spaces, an introduction,

[Bo]

Springer-Verlag, Berlin, 1976. Bourdaud, G., LP-estimates for certain non-regular pseudo- differential operators, Comm. Part. Diff. Eq. 7 (1982), 1023-1033.

[CP]

Chazarain, J. et A. Piriou, Introduction a la theorie des equations

[Fr]

[FK]

[Gal [He] [Ho]

[Ku]

aux derivees partielles lineaires, Gauthier-Villars, Paris, 1981. Friedman, A., Partial differential equations, Holt, Rinehart and Winston, New York, 1969. Fujita, H. and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rat. Mech. and Anal. 16 (1964), 269-315. Gagliardo, E., Propriety di alcune classi di funzioni in pill variabili, Ric. di Mat. 7 (1958), 102-137. Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes in Math. No. 840, Springer-Verlag, Berlin, 1981. Hormander, L., The analysis of linear partial differential operators III, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. Kumano-go, H., Pseudodifferential operators, MIT Press, Cambridge, Mass., 1981.

[LM]

[Mi]

[Pa]

[PW]

Lions, J.-L. et E. Magenes, Pro blames aux limites non-homogenes et applications, Vol. 1, 2, Dunod, Paris, 1968; Non-homogeneous boundary value problems and applications, Vol. 1, 2, Springer-Verlag, Berlin, 1972. Mizohata, S., The theory of partial differential equations, Cambridge Univ. Press, London, New York, 1973.

Pazy, A., Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, Berlin, 1983. Protter, M. H. and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1967. Typeset by AMS-TEX 159

160

[RS]

[Sel] [Se2] [Se3]

[St]

[Tal] [Ta2] [Ty] [Tr]

[Um] [Yo]

REFERENCES

Rempel, S. and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982. Seeley, R.T., Extension of C°° functions defined in a half-space, Proc. Amer. Math. Soc. 15 (1964), 625-626. Seeley, R.T., Refinement of the functional calculus of Calderon and Zygmund, Proc. Nederl. Akad. Wetensch., Ser. A 68 (1965), 521-531. Seeley, R.T., Singular integrals and boundary value problems, Amer.

J. Math. 88 (1966), 781-809. Stein, E.M., The characterization of functions arising as potentials II, Bull. Amer. Math. Soc. 68 (1962), 577-582. Taira, K., On some degenerate oblique derivative problems, J. Fac. Sci. Univ. Tokyo Sec. IA 23 (1976), 259-287. Taira, K., Diffusion processes and partial differential equations, Academic Press, San Diego, New York, London, Tokyo, 1988. Taylor, M., Pseudodifferential operators, Princeton Univ. Press, Princeton, NJ, 1981. Triebel, H., Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978. Umezu, K., LP-approach to mixed boundary value problems for second-order elliptic operators, Tokyo J. Math. 17 (1994), 101-123. Yosida, K, Functional analysis, Springer-Verlag, Berlin, 1965.

INDEX a priori estimate 118, 137 abstract Cauchy problem 5, 148 adjoint operator 105, 133, 135, 145 Agmon's method 129, 137, 138 amplitude 101, 102 analytic semigroup 4, 12 Ascoli-Arzela theorem 78 asymptotic expansion 99 Banach space 2, 3, 8, 95, 112 Banach's closed graph theorem 122 Bessel potential 95, 96 Besov space 3, 96 Besov space boundedness theorem 107 bijective 122 boundary condition 2, 112 boundary point lemma 127, 158 boundary value problem 1, 114, 127, 138 Cauchy problem 31, 38 Cauchy's theorem 9, 16, 17, 18 classical pseudo-differential operator 105, 108, 116, 123, 129, 131, 139 classical symbol 99 closed graph theorem 27, 122 closed linear operator 8, 28, 116, 128, 129, 132, 137, 144 closed range theorem 133 coercive 2

compact operator 97, 132, 145 complete symbol 104, 123, 131, 145 completely continuous 97, 132 composition 105 conormal derivative 1 contraction mapping theorem 43 cotangent bundle 106 densely defined operator 8, 28, 116, 129, 132, 144 Typeset by AMS-TX 161

INDEX

162

density 93, 109 diagonal (set) 103, 107 Dirichlet condition 2, 109 Dirichlet problem 109, 110, 128, 130, 139 divergence theorem 120 domain of definition 4, 8, 128, 129, 132, 137, 144

of R- 86, 157 double 93

elementary symmetric polynomial 89 elliptic boundary value problem 109 elliptic differential operator 1, 109, 138, 157 elliptic pseudo-differential operator 106, 108 elliptic symbol 99 existence theorem 33, 127 existence and uniqueness theorem 31, 39 extension operator 91 fixed point 43 formulation of a boundary value problem 111 Fourier integral distribution 101 Fourier integral operator 102 Fourier transform 94 fractional power 19, 25, 148 Fredholm integral equation 115 Fredholm operator 117, 129, 132, 143, 144 Frechet space 94, 98 Fubini's theorem 10, 20, 24 function rapidly decreasing at infinity 94 function space 2, 93 Gagliardo-Nirenberg inequality 150 generalized Sobolev space 93, 95 generation of analytic semigroups 142 generation theorem for analytic semigroups 8, 146 graph norm 38, 149 Holder continuous 33, 36, 37, 39, 40, 43, 47 Holder space 47 Holder's inequality 49, 54, 58, 61, 63, 65, 67, 69 homogeneous principal symbol 105, 108 hypoelliptic 108, 134, 135 imbedding theorem 74, 148

INDEX

index of an operator 117, 128, 145 initial boundary value problem 5, 28, 148 injective 127 interpolation inequality 121 interpolation theorems 48 inverse Fourier transform 94 isomorphism 4, 95, 104, 110, 111 L"-space 2 Laplacian 110, 123, 131, 140 Lebesgue's dominated convergence theorem 16, 24 linear Cauchy problem 31 local existence and uniqueness theorem 39 locally Holder continuous 33, 36, 37, 39 locally Lipschitz continuous 5, 39

maximum norm 40 maximum principle 126, 157 moment inequality 26 Neumann condition 2, 111 Neumann problem 111, 115, 118 Newtonian potential 110 non-homogeneous Cauchy problem 32, 44 norm 2, 3, 46, 47, 48, 91, 95, 96, 112 normal coordinate 93 normal derivative 157 null space 117, 129, 132, 134, 144 oscillatory integral 101

parametrix 106, 124, 131 Peetre's lemma 132 phase function 99 Poisson kernel 110 Poisson operator 111, 128, 130, 139 positive density 93, 109 positively homogeneous 98, 99 principal part 99 principal symbol 120 properly supported 103 pseudo- differential operator 102, 107 pseudo-local property 103 range 117, 129, 132, 133, 144

163

INDEX

164

reduction to the boundary 115 regularity property 117 regularity theorem 122 regularizer 103, 107 Rellich's theorem 97, 132 145 residue theorem 10, 16, 22 resolvent 4, 8, 19, 23, 138, 142 resolvent set 4, 8, 19, 142 restriction 91, 96 restriction map 97 Riemannian metric 120 semigroup 9 semilinear Cauchy problem 39 semilinear initial boundary value problem 5, 148 semilinear parabolic equation 28, 148 seminorm 46, 47, 94, 98, 138 singular support 101 Sobolev imbedding theorem 74, 150 Sobolev space 2, 46, 47, 93 solution 31, 39 space of bounded linear operators 9, 13, 24 strictly positive density 93, 109 strong maximum principle 127, 157 strongly elliptic 120 surface potential 110 surjectivity 128 symbol 98 symbol class 97

tempered distribution 94 trace 111 trace map 112

trace operator 129, 130, 139 trace theorem 112 transpose 105 uniformly Holder continuous 47 uniqueness theorem 31, 39, 126

Vandermonde determinant 89 volume potential 110