Lectures on Elliptic Boundary Value Problems

; ~

Lectures on

ELLIPTIC BOUNDARY VALUE PROBLEMS

by SHMUEL AGMON Professor of Mathematics

The Hebrew University of Jerusalem

~

...

•

...• '.:.II!I

Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr.

D. VAN NOSTRAND COMPANY, INC.

TORONTO

PRINCETON, NEW JERSEY

NEW YORK

Univer~itv ('f HI~"l.;:"''Tfon

LONDON

Dtirary

Marsh R I!:e C(i1.IVtr'S/f~ fI()l.Isl-~YlI 1ex4sSummer Irlsfriu te.. '~r 'Ad vtlf"1C!.~d Graduate :S1i-1de;;;-f $) Ie; 68,]

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PREFACE

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This book reproduces with few corrections notes of lectures given at the Summer Institute for Advanced Graduate Students held at the William Rice University from July 1, 1963, to August 24, 1963. The Summer Institute was sponsored by the National Science Foundation and was directed by Professor Jim Douglas, Jr., of Rice University. The subject matter of these lectures is elliptic boundary valued problems. In recent years considerable advances have been made in .. developing a general theory for such problems. It is the purpose of these lectures to present some selected topics of this theory. We . consider elliptic problems only in the framework of the L 2 theory. This approach is particularly simple and elegant. The hard core of the theory is certain fundamental L 2 differential inequalities. The discussion of most topics, with the exception of that of eigen­ value problems, follows more or less along well-known lines. The' treatment of eigenvalue problems is perhaps less standard and differs iD some important details from that given in the literature. This approach yiel ds a very general form of the theorem on the asymptotic distribution of eigenvalues of elliptic operators. Only a few references are given throughout the text. The literature . on elliptic differential equations is very extensive. A comprehensive bibliography on elliptic and other differential problems is to be found in the book by J. L. Lions, Equations diffe'rentielles ope'rationelles, ~• . Springer-Verlag, 1961. 'li:i' These lectures were prepared for publication by Professor B. Frank ~ Jones, Jr., with the assistance of Dr. George W.Batten, Jr. I am " greatly indebted to them both. Professor Jones also took upon him­ \ self the trouble of inserting explanatory and complementary material t'-. in several places. I am particularly grateful to him. I would also ~ like to thank Professor Jim Douglas for his active interest in the \) publication of these lectures. ~

PRINTED IN THE 'UNITED STATES OF AMERICA'

~

Jerusalem

Shmuel Agmon

;c ~

CONTENTS

1'0

Section <5'

o1l,

(;)

:d-­

"',

....J)j

!ill!

I

\\\

o Notations and Conventions

1

Calculus of L 2 Derivatives-Local Properties 1

Calculus of L 2 Derivatives-Global Properties 11

Some Inequalities 17

Elliptic Operators 45

Local Existence Theory 47

Local Regularity of Solutions of Elliptic Systems 51

o Garding's Inequality 71

Global Existence 90

Global Regularity of Solutions of Strongly Elliptic Equations 103

Coerciveness 134

Coerciveness Results of Aronszajn and Smith 151

Some Results on Linear Transformations on a Hilbert Space 175

Spectral Theory of Abstract Operators 208

Eigenvalue Problems for Elliptic Equations; the Self-Adjoint

Case 230

15 Non-Self-Adjoint Eigenvalue Problems 261

16 Completeness of the Eigenfunctions 278

1 2 3 4 5 6 7 8 9 10 11 12 13 14

O.

Notations and Conventions

''i.;"

;,1'

~The following notations and conventions will be used. En will de­

¥i'..ote real n-space. For any points x = (x 1""'xn ) and y = (Y1'''''Yn)

C

:E .

\~(

:"::£.E

1,c'·,

1'0

n'

i.

Ixl =(X 1 2 +...+X n 2)'i2, x ·Y=X 1Y 1 +...+XnY n · For an index or exponent a = (a 1 , ... ,an ), whose components are ~integers, lal = a 1 +... + an; from the context it will be clear whether ~this norm or the euclidean norm is intended. Also, a! = a 1 !... a n !'; :;for f3 = (/3 1 ,· .. ,f3n)' ,

1

.j)

o"11

(a)f3

,

=~

QII-

o

a 1 !... a n !

~"

, For any x £: En' a. = x '" -_ d-­

f3 1 !. .. f3n! (a 1-{31)!. .. (an -{3n)!

(ap ••• ,an ),

X 1 '" 1 ••• X n "'"

In particular, this will be -used with the differentiation operator: DOC

"1­

a"'l'_~ ..__ a . = D1 "'1 ••• Dn '" n = __ ocn

~

dX 1

'" 1

u~ X n "'n

Arso a~ f3 ~ a 1 ~ f31.· ...a n ~ f3". Thes~ conventions greatly simplify many expressions. For example, Taylor's series for a function f(x) has the form

~

1 D~f

(0)

x~.

We will also find it convenient to have a special set inclusion:


01 CO 2 ,

For any function u, the not,ation supp (u) will be used to denote the support of u; i.e., the closure of the set Ix:u(x) ~ol.

1.

......p

Calculus of L 2 Derivatives-Local Properties

In this section

0'1 'ql!

°

is an open set in n-dimensional Euclidean spaCe £m (0) is the class of m times

En' For any non-negative integer m,

1

Ii

--

2

~

EIIiptic Boundary Value Prohlems

continuousl¥ differentiable (complex-valued) functions on U and Coo(O) = n cm(O). Also C ;'(0) is the subset of Coo(O) consisting m-O

??

~

sec. 1

of functions having compact support contained in U. The functions in C';(O) will be called test functions for U. Now some norms and semi-norms will be defined.

Definition 1.1. For u r;:: C Il1 (O) [lull U =[J I ID"'uI2dx]~.

'c5'

m,

U

(m

Suppose thatu has strong L 2 derivatives of order up to m, and let D"'u k -> u'" in L 2 (m, lal ~ m. If q; :€ C'; (m, then integrating by parts, using the fact that q; '" 0 in a neighborhood of U,

a

H~m

r q;D"'u k dx = (-l)iexI rUk D"'q;dx.

If IJullmu, Ilvl/m,u are both finite, then there exists

(u, v)

(;)

m,

U= f

U

U

I

H~m

D"'u D"'vdx.

lui U = [f

m,

I nH-m

(Ll)

,,,'

-> 00,

rq;u"'dx

=

(-l)iexI

U

ID"'uI2dx]'~

J uD"'q;dx,

lal·~

m.

U

Taking this last relation on its own merit, the following definition is made.

This latter quantity is of course only a semi-norm: lui m,H n =0 U

U

Then, letting k

Likewise, for u r;:: cm(O),

'>

3

said about this identification below. First, two kinds of derivatives will be defined. Definition 1.4. A function u € L 2 has strong L 2 derivatives of order up to m if there exists a sequence {uk I Cern * (U) such that tD"'u k 1is a Cauchy sequence in L 2 (U) for lal ~ m, and Uk -> U in L 2 (U).

o

"11:

Calculus of L 2derivatives-Local properties

-J

= O. The notation (u, v)m = (u, v)m.U: Iluli m = Ilui Im,u lul m =

lui m,H n may also be used if U is fixed during a discussion.

Definition 1.5. A locally integrahle function u on U is said to have the weak derivative u'" if u'" is locaIIy integrahle on U and (L2)

r q;u"'dx = (-1)iexI J uD"'q;dx,

Definition 1.2. Cm'!\O) is the suhset of cm(U) consisting of func­ U tions u with lIullm,u < 00. Definition 1.3. H m (0) is the of cm'!\O) with respect to _'" , completion .-. ,,' the norm II lin. . ... m.H

Since cm*(u) is obviously an inner product space with respect to the inner product (u, v) n, it follows that H m (0) is itself a Hilbert space.

all q; :€ C'; (U).

U

A few results are immediate. THEOREM L L If u € L 2 (U) has strong L 2 derivatives of order up to m, then u has weak L 2 derivatives (,)f order up to m. This result is an immediate consequence of (1.1).

m.~£

\'

),

--\fl,

\ll

a:I

According to the general concept of completion for a metric space, the members of Hm(U) c.an . be.assumed to be equivalence classes of Cauchy sequences of elements of C m1U). However, in the present case another characterization of Hm(O) is possible. Indeed, if {ukl is a Cauchy se­ quence in Cm*(O), then for any fixed a, lal ~ m, it follows from the in­ equality

[J \D"'u k - D"'u/dx]~

S;

llu k

-

uJJ Im.u

U

that {D"'Ukl is a Cauchy sequence in L 2(0). As L 2 (0) is complete, there exists u'" .€ L 2 (U) such that D"'u k -> u'" in L 2 (U). Therefore, every ele­ ment of H m(0) can be considered to be a function u r;:: L 2 (0) with the pro' perty that there exists a sequence {Uk 1c Cm*(O) such that {D"'u k 1is a Cauchy sequence in L/O), lal ~ m, and Uk -> U in L 2 (0). More will be

THEOREM 1.2. Weak derivatives are unique. That is, if u has the weak derivative u'" and also the weak derivativev"', then u'" = v'" a. e. Proof. From (1.2) it follows that

(m

r q;(u'" -

v"')dx = 0 for all

U

¢ E:: C';. (U)..As C'; is dense in L 1 (C) for any compact subset C of U, u'" - v'" = 0 a.e. Q. E. D. CoroIIary. Strong derivatives are unique. This theorem justifies the following notation for strong and weak derivatives. If u has the weak derivative u"', write u'" = D"'u. Like­ Wise, if u has strong L 2 derivatives, and if {ukl CCrn" (U),

4

Elliptic Boundary Value Problems

u i'n L (0), and D"'u k -> u'" in L 2 (0), define u'" = D"'u. Accord­ 2 ing to the above corollary, D"'u is independent of which sequence {uk} is chosen. Also, it is seen that if u has strong L 2 derivatives DlXu, then u also has the weak L 2 derivatives D"'u.

Uk -+

.f 'Q

o'!I,

'0

(u, v)m, 0

=I

I

o loci

2

(0) of order up to m. If u, v €W m (0),

~m

DlXu + DlXV , = CDlX U, C constant. =

Proof. It· suffices to show that if {Uk} C Wm (0) and if Uk -> U, D"'u k -> u'" in L 2 (0), then u € Wm (n) and D"'u = u lX, la\ .::; m. Bl)t this is immediate upon writing (1.2) for u = uk and letting k -> 00. Q. E. D. For reference we display the following fact, which has been es­ tablished.

(x) s 0 for Ixl ~ 1, f E

r i( (x - y)u(y)dy n

a

THEOREM 1.5. If u is locally integrable in 0 and also integrable then l(u C € OO (E n)' If, in addition, the support of u is contained in K, a compact subset of 0, and if ( < dist . (K, 0), then I (u € C';;' (0). Proof. Continuity of l(u follows from the continuity of i (' Differ­ entiation can be carried under the integral sign, so that the differ­ tmtiability properties of 1(u follow from those of ie The last state­ riient is obvious. Q. E. D.

on bounded open subsets of 0,

a

THEOREM 1.6. If u € L 2 (0), then

II(ul 2 = I n r u( (x -

.: ; r i( (x i(x) dx

=

Ill(ull o. 0

~

Ilull o. O'

Proof. By the Cauchy-Schwarz inequality,

o

In order to treat local properties we now introduce the idea of

mollification. Let i(x) f: Coo (E n) satisfy

0, i

=

One sees readily that I (u(x) is defined at all points x with dist (x, 0) > (. If u is also integrable on bounded open subsets of O. then l(u(x) is defined for all x. The importance of 1( arises in the fact that l(u behaves much like u, but it is very smooth. This is stated precisely in the following theorems.

THEOREM 1.4. Hm (0) C Wm (0).

i (x) ~

i( (x) dx = 1.

:lor any locally integrable function u in O.

THEOREM 1.3. W", (0) is a Hilbert space. t,'

and that

n

l(u(x)

D"'u D"'v dx.

Since strong derivatives are unique, we will identify the class H m (0) with the class of functions in L 2 (0) which have strong L 2 derivatives of order up to m. It is clear that if a function has continuous pointwise derivatives of certain order, these derivatives are also weak and strong deriva­ tives. Clearly, W", (0) and H m (0) are linear spaces of functions, and in either class DlX (u + v) D'" (cu)

IE

Ixl 2: (

Definition 1.7. The mollifier 1( is defined by

Definition 1.6. W'" (n) is the class of functions in L 2 (0) which have weak derivatives in L

s

(Note that i( (x) vanishes for

.

5

Calculus of L 2 derivatives-Local properties

sec. 1

y)dy

o

I

I [j( (x - y)]'i2 u(y) ldy 12

i( (x - y)

I

u ty) 1 2dy

0

::;; I i( (x -y)

1.

y)]'i2

I u (y)

12dy.

n

For example,

i (x)

=

Hence, by Fubini's theorem,

i can be the function

c exp (-

1 ) for 1- Ix12

Ixl < 1, i (x)

== 0 for

[xl

2:

II1(u115

1

1- i (n

0::;; =

where c is a suitable constant. Let

i( (x) =

.

(~).

I

0

[r0 i( (x -'y) I u (y)

r

[Iu (y) 12 I i( (x -y)dx] dy

o

0

(m

12dy] dx ~

Il u115. o.

(m

Q. E. D.

THEOREM 1.7. If u €L 2 then l(u -> u in L 2 as ( - > O. 11 u is continuous at a point x, then U(u) (x) -+ U (x), the convergence being uniform on any compact set of continuity points.

(

~,.'r

6

:~

~

'0

EIIiptic Boundary Value Problems

Proof. Extending u to En' lettering u == without loss of generality that n = En. Clearly u (x)

IE i{ (x -

=

Calculus of L 2 derivatives-Local properties

sec. 1

°

outside

n,

we can assume

= (-

010:1 r D; i{ (x -

n

7

y) u (y) dy

= I i{ (x - y) DO: u (y) dy

n

y) u (y) dy;

n

V{ DO:u)

=

(x),

hence' .;5>

I V{u) (x) - u (x) I ~

i{ (x -y) I u (y) - u (x) \ dy

~ n

~

~. .J

o

I
sup

Iy - xl < {

lu(y)-u(x)\.

J.l

THEOREM 1.9. Let F CE n be compact, and let F C U 0;,

Since the last term tends to zero with { at each continuity point x, the convergence to zero being uniform for any compact set of con­ tinuity points, the last part of the theorem follows. Let now T/ be an arbitrary positive number. Choose v.,., (C 0' (En} such that

Ilu - v.,.,llo, En <.,.,. IIJ{u-J{vTJllo, En~ Ilu-v.,.,llol

En'

Moreover, if the support of v.,., is contained in the sphere Ixl < R, then by the first part of the proof V{v.,.,) (x) ----> vTJ (x) uniformly for Ixl ~ 2R, while J{vT/ = v.,., '" for Ixl > 2R and {< R. Thus for { sufficiently small

°

IIJ{vTJ-v.,.,llo, En

u in L 2 (En) as

{---->

0. Q. E. D.

THEOREM 1.8 If u (W m (!l), and if Ial ~ m, then (DCK. J (u) (x) = provided dist (x, a!l) > {.

V ~CK.u) (x) for x E: n,

Proof. Under the conditions of the theorem, we have (Do:J{u) (x)

= DO: =

I i{

n

where each 0; is open. Then there exist functions ~; (CO' (0;) J.l

sl!ch that

•

I

~; (x)

= 1 for x ( F.

1= 1

Proof. First we will choose Ic; I, a collection of compact sets such that C; cO/ and Feu C;- This can be done as follows. For x~ 0;, let S (x, 2r x) be the sphere centered at x and hav:'g radius 2r x > chosen so that S (x, 2r x) CO;, Since F is compact and covered by ts (x, rxL x (FI, it is covered by a finite number of these spheres, say by Is (x k ' r x ) : k = 1, ... ,ml. Let k C; =

U

xkE:O;

S (x k' r x

)

•

k

Then C; has the desired properties.

whence

That is, J{u

;=1

°

By the preceding theorem, t'

the fourth equality following from the definition of the weak deriva­ tive. Q. E. D. Now a theorem will be stated to guarantee the existence of a ·partition of unity."

(x - y) u

(y) dy

r D';j{ (x -y) u (y) dy n

Now let Ic;* I be a collection of compact sets satisfying CjCint C;* CC;*,cO/. Let tP;* be the function which equals 10n C/* and vanishes elsewhere. Choose {> less than dist (CI' C;*) and dist (C;*, 0/). Let

°

a

tP; = J{tP;*· tP/ (COCO (0;) and tP;

Then

(x)

=

1 for x

(C;'

a

Finally, let

~l=tPl ~; Then

=

(l -

tP 1)

. (1 -

t; ( CO' (0;) and

tP 2)'"

(l -

tP; _ l)tP i'

i>1.

-----._----_._~

- ----

----

-

---

~--~----

---

8

Elliptic Boundary Value Problems

v 2 1

= 1

~I = 1- (1- !f1).. ·(l-o/v )

1 I>;:{\

<;)

=

f

n

1

unity subordinate to the open covering 1011 of F. Definition 1.8. u f: Wm10C(0) [H m1o c(0)] if any x f:

°en

n has an open

THEOREM 1.10. It u E: Wm (0) [H m (nn then for any

o."

.1"

'"

-+

u in Wm (n1) [H m (n1)] as

f -+

;

'v

°

n

Proof. Clearly J fU is well defined in 1 .for f > sufficiently sma11 and it is enough to show that DocJf u -+ Docu in L 2 (n1) for O.s lui .s m.

This however follows immediately from Theorem 1.8 and Theorem 1. 7 since for f sufficiently small u (x)

=

JfDocu (x)

-+

Doc u

Proof. Suppose u E: Wm loc (0). Since

1

n !s compact, it can be 1

°

weak derivative Docu in Wm(OI)' ju] ~ m and i

=

k

1011; 2 ~I

_

=

1 on

n1. For any ¢

1

is a test function on k

¢ =

1,oo.,k. After summing on i, we have

uU. ¢dx

(-1) loci

=

1

f

n

u. DU¢dx,

lui ~ m.

1

2 ¢I on I = 1

n1·

° I

f: Co (n1)' let ¢I = ¢ ~I' i = 1, ... ,k;

and

1.10, Jfu Q.E.D.

m

n 1 CC n2 CC n.

f: Wm (n2) with

Then by the corollary to Theorem

u in Wm(n1)' Since Jfu f:

-+

n 1 CC n,

Corollary. It

n1' are in H

m

c oo(n1)'

this implies u f: H m(n1)'

then the functions in W~OC(O), restricted to

(n1)" i.e., u f: W~OC(O) implies u f: H m (n1)' -+

u in Wm(n 1 ) as

f -+

°

and Jfu f: Coo

by Theorem 1.5. Q.E.D. THEOREM 1.12. H m1oc (0)

Wloc(O).

'=

m

Proof. This follows immediately from the above corollary. Q.E.D.

THEOREM 1.13. (Leibnitz's rule). If u f: Wm (0) [H m (0)], and if v f: cm(O) has bounded derivatives of all orders ~ m, then uv f: Wm to) [H m (nn and

(*)

DU(uv)

(~) Df3 uD u-f3 v .

2

=

f3~ \1

.

I

1, .•. ,k.

Let now 1~11 be a partition of unity subordinate to

I

=

Proof. If u f: W~OC(O), then Jfu

covered by a finite number if open subsets 0l' ... ,Ok of n, with u E: Wm (01)' i = 1, ... ,k. Thus the restriction of u to 01 adm-its weak derivatives Docu E: L 2 (° 1 ) , lul.s m, which we denote by u~. From the uniqueness of weak derivatives it follows further that u~ = uj almost everywhere in 01 n OJ' 1,.s i, j .s k. Thus after correction on OC null-sets we .obtain functions UOC E: L 2 ( ~ I) such that U = the

then ¢

i

m

U

0.

1= 1

~J)

n

Suppose u f: H1oc(0) C W1oc(0). Then by the first part of the proof,

n 1 C en

in L 2 (n1)' Q. E. D. THEOREM 1.11. It u E: Wm loc (0) [H m loc (nn, then u E: Wm (n1) [H m (n1)] for any n 1 ccn. --;;

u uD ¢ Idx

f

Thus, u f: Wm (n 1 ).

DOCJ f

l'

lui ~ m,

n

_ L 1oc (0) 2

J(u

~lt,.

1

f

such that u f: Wm (0) [H m (0)].

-

, ~~

for

Note that H;:C(O) C W~oc (0). For m = 0: H~ oc(O) = W~OC(O)

o

')

uU. ¢ Idx = (-1) loci

n n 1,

Terminology: The collection of functions ~I is called a partition of

neighborhood

"

Since u f: Wm(O), and since the support of ¢I is contained in 01

v U C1::J F. Q. E. D.

-= 1 on

9

Calculus of L 2 derivatives-Local properties

sec. 1

Proof. If u f: Hm(n), then there is a sequence lukl

that DUu k

-+

DUu in L 2 (0) for

derivatives in DU(ukV)

=

°

~

lui ~ m.

n, 2 f3~u

(~) Df3ukDu-f3v .

ccm"\m such

Since v has bounded

=;;;I~§~~~~~";;';~~~~==~~:;'~~:;f~~~

10

sec. 2

Elliptic Boundary Value Problems

(~) Df3uD a-f3v

-> I

;~

~

II Dau N

Calculus of L

-

Dau II 0 0 ~

•

f3Sa

2

a\ 1

I

flSa

11

derivatives-Global properties

(

(3) N max x

IDf3~{x) I IIDa-f3u IIoJ}

f3f,0 in

L/m for 0~ lal ~ m,

+[

Hence the theorem is proved for Hm(n)·

Now suppose u r;: Wm (n). For any test function ¢ on 0, let 0 1 CC 0 be a set containing the support of ¢. By the corollary to

f

Ixl>N

ID Q u\2dx]'Ii

->0 as N ->

00.

Q.E.D.

Theorem 1.11, u r;: Hm(Ol)' Thus by the result for strong derivatives uv r;: H m (Ol) cW m(Ol) and (*) holds on 0 1, Thus uv r;: W (01) for every 0 1 CC 0 and (*) holds on each 0 l' But since Da-Sv is bounded

o

and Df3 u r;: L 2 (O), DOC(uv) r;: L 2 (n) by (*). But any function in W~oc(m

whose derivatives up to order m are square integrable over 0 is in Wm(n). Q.E.D.

Corollary. Functions u r;: Wm (0) having compact support in E n are dense in Wm (n). Proof. Let (x) r;: COO(E) satisfy 0 ~ (x) ~ 1 and 1 for

v

(x) =

o for

Ixl ~ 1, Ixl L 2.

Let uN(x) = (~) u(x). By Leibnitz's rule UN r;: Wm(n), and UN has compact support in En' Moreover, for

DQu

N

=

I

f3~a

(~) Df3[(!.)] Da-f3 u -> Dau N

in L/O) as N ->

_ Dau =

Dau N

00.

I f3~a

(~).l NIf3\

We wish to extend the local property that H 1 oC(n) = WJ oC(n) to a m

m

global property; i.e., to establish that Hm(O) = Wm(n). This result which is true in general we shall prove only with some restrictions on

o.

Definition 2. 1. 0 has the ordinary cone property if there is a cone C such that for each point x r;: 0 there is a cone C i cO with vertex x congruent to C. 0 has the restricted cone property if ao has a locally finite open covering 10.1 and corresponding cones IC.I with vertices at I I the origin and the prnperty that x + C leo for x to: 0 n 0 i' 0 has the and _corsegment property if ao has a locally finite open covering 10.1 I responding vectors Iyil such that for 0 < t < 1, x + tyi to: 0 for x to: 0 no;. If 0 is bounded and has the restricted cone property, then 0 has the ordinary cone property. It is also easily seen that if 0 has the re­ stricted cone property, then 0 has the segment property. We will show that Hm(O) = Wm(n) if 0 has the segment property. First, however, we prove the following approximation theorem) which is of in­ terest by itself. THEOREM 2.1. If 0 has the segment property and if u to: Wm (0), then there is a sequence lukl C C';(E) such that Uk -> U in Wm(n).

For

(Df3()(!.) Da-f3 u + [(!') _ 1] Dau;

f3~0

\\\\11111

lal S m

2. Calcu Ius of L 2 Derivatives-Global Properties

therefore by the triangle inequality

N

N

Proof. Since by the corollary to Theorem 1.13 f~nctions with compact support are dense in Wm(n), assume, without loss of generality, that u has its support in a compact set C. There are two cases: either i) C cO, or ii) C n 0 f, O. In case i) we obviously have lfu -> u in Wm(n)· Case ii) is more difficult. Let 10;I be the locally finite_open covering of. ao guaranteed by the segment property. Le t F = C n CO - U 0.). I Then F is compact and Fe O. Choose 0 0 so that Fe 0 0 cc O. As

a

•

12

sec. 2

Elliptic Boundary Value Problems

J

·~

En

~

13

Calculus of L 2 derivatives-Global properties

-r

uDO<¢dx

=

J

uDO«(¢)dx

K

I~

=

J

=

(-I)H

J

DO
= (-I)H

J

DO
o

uDO«(¢)dx

n K

o

=

(-I)H

r En

n

-r

DO
where in the last integral we define DO
n.

an

no

n

X.l

C n an is compact, a finite collection 10 lIt covers C n an. Then I0,.l~ is an open covering of C no, hence of the support­of u. As in the proof of Theorem 1.9, let 10:1~ be a covering of en n such that 0: ce 0I for i = 0, ... ,k. Let I';) be a partition of unity subordinate to 10:1~ . f gl = 1 on C no, and let u l = ugr Then u = !ou p the sup­

an-

,- 0

Port of u., is contained in 0'.,, and u l ~ Wm (0). Thus it is sufficient to prove the theorem for the functions u., individually. Since 0 0 ee n, case i) takes care of u . o Thus it is sufficient to prove the theorem under the assumption that the support of u is contained in 0; ee 0 1 , Extend u to En' defining u to be zero outside 0'1' Note that 8 = dist (O~, 0 1) > O. Let = 1' n Then u ~ Wm (E n For let ¢ be any test function on En Then K = supp (¢) n supp (u) cO. Let (~C~(O) be 1 in K. Then (¢ is a test function on and we have

r

° •

ao. r.

n. n

a

for if z ~ r T no, then z ~ 0 n Oland z + ry 1 ~ r, so that z + ry 1 ~ n, contradicting the segment property. Since rr is compact, dist (rr ' 0) ;

14

sec. 2

Elliptic Boundary Value Problems

If u E: C m (0), then integration by parts shows that

Let u T = u(x + ryl). Then u T ( Wm(En-r T); since 0 eE n - rT>we have, a fortiori, u T (W m (0). Since (D''V) (x) = (D"'u)(x + ryl), D"'u T --+ D"'u on L 2 (O) by a familiar theorem on Lebesgue integration. Hence, u T --+ u in Wm (0) as T -+ O. Thus it is sufficient to approxi­ mate u T by functions u~ ( C';;(E n)' To obtain such a sequence we simply take the mollified sequence Uk = ] 11k U, k = 1, 2,.... Clearly u~ ( C':;'(E n) (Theorem 1. 5). Since'dist (r T' 0) > 0 it follows from Theorem 1:8 that

~I

I~

(D"'u~)(x) = (j IlkD"'uT)(X) for x

o'II

i

(0,

r uA¢ dx = r Au¢ dx

o

Definition 2.2. Let u be a locally integrable function on O. If there exists a locally integrable function f on 0 such that

J

lal ~ m

THEOREM 2.2. If 0 has the segment property, then H m (0)

o

=

Wm (0).

Let uic = ] Ilk

if dist (x,

, 1.1 l

f exists weakly and

f.

(0), and if u is a weak solution

loc

-+

u and A.J uk =

-+

f.J in L 2 (0) as k 2

-+

0 outside O.

U.

As in Theorem 1.8, we see that

U

(x)

=

J1/k A j u (x)

a0) > 11k.

It follows from Theorem 1.7 that Ai

Uk

=

AjJl/ku=Jl/kAju-+Aju=fjinL2'(OI).ask-+00. Q.E.D.

Let A'" (t) = De- A (t). We will call A'"

a",D'"

=

A'" (D) a subordinate of A.

We have the following analog of Leibnitz's rule. THEOREM 2.5. Let A be a linear differential operator of order m with constant coefficients. Suppose that u E: L loc (0) and that 2 Au and A'" u exist weakly for luI,::;; m. If t; E: C m (0), then A (t;u) exists weakly and

be a linear differential operator with constant coefficients. Let

l

=

WimOC(O) for any 0 (Theorem 1.12), Ai J llk

=

=

Proof. Extend u to En' defining u

locI~m

liD)

then we will say that Au

such that for any 0 I ceo, u k

We will now extend the idea of generalized derivatives to differential operators. Let

'0'

C':;' (0),

of A.J u = f.,J i = 1, ... ,m, on 0, then there is a sequence lukl CC oo (E n )

THEOREM 2.4. For any open set 0, if u (H:oC(O), and if D"'u ( Hi oc(O) for lal = j, then u ( H:~~(O).

l

f¢ dx

THEOREM 2.4. If u, f E: L 2 .

we have

=

I 0

Note that the weak existence of Au does not imply the existence of any weak derivatives of u. However, weak existence does imply strong existence in the sense of the following theorem.

Proof. It is sufficient to prove the theorem for H replaced by W. This is trivial. Q.E.D.

A(D)

=

that u is a weak solution of Au

THEOREM 2.3. If 0 has the segment property, if u ~ H/O), and if D"'u (H k(O) for lal = j, then u (Hj+k(O).

=

uA¢ dx

for all ¢ E:

This theorem is important for obtaining properties of H m (0), for fre­ quently it is easier to obtain them for Wm (0). As an example, we prove

Likewise, using the fact that Hloc(O) m

0

for any ¢ (C:;"(O). More generally we make the following definition.

for all k > [dist crT' O)r I. From this and from Theorem 1. 7 we have: D"'u~ -+ D"'u T in L 2(0) for lal ~ m. This shows that u~ is the desired approximating sequence of u T• Q.E.D. As a corollary we have

0_

15

Calculus of L 2 derivatives-Global properties

a",(-I)I"'1 D"'.

l"'l~m

...,jl:,'..

A «ul

~ . I 1"'1 ~

D'( A' u. m

a!

00.

16

Elliptic Boundary Value Problems

sec. 3

Some Inequalities

Proof. First assume that u E: Coo (n). Then by Leibnitz's rule we

3.

have

''0

A «(u) =

loci

I

~

1f31

a",

I

~

f3

ttl

ttl

I 5, '"

(~) Df3 OJoc-f3 u

THEOREM 3.1. If f E: C 2 (a, b), then

f3 /a) D , f3 ~ ",a", \ f3 D",-f3 u

«>

Df3,

I

1f31 ~

ttl

73!

A

r

f3

Joel::; I

~

a oc

ttl

f3::;'" \acI ::; m

a

r

b

If

(X)!2 dx].

II

a

~ C l (a, b), then b

(3.2)

b

If (x)12 dx 5, 6 [(b - a)2

[ "

= (+2(b-a)/3

"

a+(b-,,)/3'

I; a-f3.

THEOREM 2.6. Let a be a collection of linear differential operators with constant coefficients, such that if A E: a then for any a, A oc E: a. Assume that has the se~ment property, that u E: L (0), and that Au exists weakly for every A E: a. Then there 2 exists a sequence lukl (En) such that uk -> u and AUk --. Au in

n

f If (x)!2

dx],

a

the mean value theorem gives f' (1])

= f (x 2 ) X

for some

ceo

repetition of that of Theorem 2. 1. Q. E. D.

b

If' (x)12 dx +

Proof. It is clearly sufficient to consider f a real function. Con­ sider the case a =0, b = 1. Let 0 < a < 1/2. For xl < a, 1 - a < x

Ifl

L 2 (n) for every A E: a. Proof. Theorem 2.5 can be used to prove a density theorem analogous to the corollary to Theorem 1. 13. The rest of the proof is just a

J a

where /

role (Theorem 1.13), using Theorem 2.4. Q.E.D. The following approximation theorem is analogous to Theorem 2.1­

CO,

If (xW dx

u,

The proof for arbitrary u ~ L;OC(n) is similar to that of Leibnitz's

ill"~

b

+ (b _a)2

Df3 eoc

a! '" (a - (3)!

If' (x)1 2 dx:s 54 [_1_ (b - a)2 "

"

since

I

r

b

(3.1)

If f

Af3 «) =

Some Inequalities

In this section we will derive some basic inequalities. We begin

with some inequalities for functions on E 1 •

H~m

=

17

1]

2

E: (Xl'

(7)) I .: ;

-

f (Xl) X

1

x);

hence

1 (1 _ 2a)

(If (x ) I + If (x )/). 2

1

Now f' (x) = f' (7)) +

J" f

lf

(~

dlj,

1]

so that

\f

l

(x)\ ::; _1_ 1 - 2a

(If (x

)\ + 2

If (x

)\)+ [1

l·o

If" «)I

dlj.

2

,

18

sec. 3

Elliptic Boundary Value Problems

Integrate this inequality with respect to x lover (0, a), and with re­

''';

a 2 If' (x)1 :s; _a_ (f 1 - 2a 0

00

~llill

o

1

+.r ) If (~I d'; + a

2

1-00

1

I If"

(_1_)2 a I-2a

If' (x)\2 dx

«( + II ) If (~12 d';'+ 2 / 0

1-00

(~I d';.

\f"

(';)12

d';.

0

(_1_) 2 fl if (~12 d'; + 2 (1 If II (~12 d';. al-2a 0 ' 0

s!

The maximum of the first coefficient occurs at a 1

( \f'

o

(xW dx

s 54' I

(.;)1 2 d';.

If'

0

Thus,

s!

o

1

0

Thus, /

J

'I,

Squaring and using the Cauchy-Schwarz inequality twice, we obtain \f' (xW

If (x 1 )1 2 dx 1 + 4/9

1/3 If (x)12 :5 2 (

spect to x 2 over (l - a, 1):

19

Some Inequalities

1

If (xW dx + 2

0

I

=

1/6; hence

Ii

I

f If (x) \ 2 dx:s; 6 I

l

~

If(x)12 dx + 4/3

~

0

1

I

If'

(,;)1 2 d';'

0

From this inequality (3.2) follows by the transformation'; = (b - a) x + 8 Q. E. D. Corollary. If f E: C2 (8, b), and if 0

< (s 0,

then

1

\f" {xW dx.

0

b

(3.3)

(

The transformation'; = x (b - a) + a yields (3.1). Again consider a = 0, b = 1. For any x E: (0, 1) and" 1 € (1/3, 2/3),

If' (X)[2 dx ~y [

(I

b

b

n

If (X)!2 dx +

(-1

8

8

(

If (x)!

2

dx },

8

we have x

f(x)=f(x 1 )+(

where y depends only Qn b - a. If f E: C 1 (a, b), and if 0 there is an interval [a', b' ] C(a, b) such that

f'(~d';,

< ( :s; 1,

then

Xl

b

(3.4)

whence, by the Cauchy-Schwarz inequality,

(

If (x) 12 dx

8

If (x)j2 S

21 f

(x1 )12 +

21 x - XII J

X

If'

1

,\\

r

If' (.;)p d';.

o

Integrate over (1/3, 2/3) with respect to xl:

(I

b'

If' (x) \ 2 dx +

8

I,

if (x) 12 dx ],

8

(.;)1 2d';

J[1

S 2 \f(x 1)\2 + 4/3

b

:s; Y [

where

y depends only on b - a, while [a', b'] depends only on (a, b)and

Proof. Let a = a o < a 1 <. .. < a k = b be a partition of (a, b) such that 1/2 Yl (b - a) < a i - a/ _ 1 < (b ..:. a) for i = I, ...,k. If fE: C2 (a, b) then by (3.1),

vr

I

8/

8/ -

If' (x) 1

j2

dx,:s; 54 [ ( (b - a)2

I

8/

8, _

If" (x) 12 dx 1

-,-,_. ..., - -_.

===""'-~""='=---==--==---====-=--=='=""""-====--=~---==--~="==--=---===--=---==-~---'=========.===-""'-=''''''

. '­

"!'V_

20

sec. 3

Elliptic Boundary Value Problems

t!

+ __4 ( (b - a)2

I~t.

r

81

If

81 -

(X) \ 2 dx )

° If U has the restricted cone property,

2

(3.5)

~54max(b-a)2.,.

21

Some Inequalities

THEOREM 3.2. Let U be a bounded open set. then there is a constant y, depending on only U, such that for any u E: H (n).

1

1

"~="===---=-'-==='''''-''=''===~'~==-=--=---=='

4,? ][(/;

lul 21, ra;::;Y «( lul 2.2 ra ~£

~£

+ (-

1

lul O.2 ra) ilL­

W(xWdx

8/_ 1

if 0 < (~ 1.

o"111

+

(-1

f 8

2° If U has the segment property, then there is a constant y, de­ pending on only U, such that for 0 < ( .::;: 1 there is a domain U cc U for which

8/

If(xW dx], ;-1

(

(3.6)

whence (3.3) follows by addition. If f E: C 1 (a, b), (3.2) yields

lu\~. U;::; y

«( lui:. U + lul~. U}

I'

for any u E: H (n). 1

Proof. Since any function in H (n) [H 8/

r

8;-1

8/

If (x)

12

dx ;::; 6 (( (b - a)2

J

If' (x)

2 1

dx

8/-1

+f

~

.

forf~~ctions u in cl:"'ss

C 2 (n) for 1 ° and in class C 1 (n) for 2°.

a

81

\f (x)

81 -

12

dx ]

1

For 1°, let 10'} and Ic'} be the covering of U and the set of / i corresponding cones, respectively, as guaranteed by the restricted cone property. Since U is bounded 10'1} is finite. Let h; denote the height of C'.,I and let h

~ 6 max ( (b - a)2, 1 ) [ (

(U)] can be approx,im,ated, 1

by functions in C (n) [C 1 (U)], it is sufficient to p;~~~theTheorein 2

81

r

"'I -

If' (x) 12 dx

81 _

min h~.1 Let

I

a finite open covering

aU with spheres 0" whose diameters are less than h;2. /

oI= i 0' no", assign the cone C k ;

a

1 \f

lo'.'} be

Let 10 } ;

OJ n O'~; to any set Cl. Then 10 }, Ic I is a covering J / /

be the collection of all sets of the form

1

a,i

+f­

of

=

(X)

12

dx ],

=

of U, together with the set of corresponding cones as in the defi­ nition of the restricted cone property, and 10 ; } has the additional property that the diameter of 0; is less than the height h. of C.' Let I

IQ ; } be a finite collection of cubes such that U - U 0 / C U Q/ and (3.4) follows, if we take a' = a + (a 1 - a) /3 and b'=b-(b-a k _ 1 )/3. Q.E.D. This theorem can be extended to functions on En; the following form will be convenient for our purposes.

I

CC

u,

and the edges of each cube are parallel to the coordinate axes. Let y be a generic constant depending only on the U. Consider a cube Q; = la < x k < bkl. For any k = 1, ... n, we have k

by the corollary to Theorem 3.1 that

22

Elliptic Boundary Value Problems

J B

bk

\DkU\2dxk~y[(r

b

k

Bk

k

sec. 3

Some Inequalities

23

bk

\D;U\2dx k +(-lr lu\2dx k ], Bk

(V)I

=

v in

of; (v).;: 0 in

E -

l I n

of. J

whence Denote by L y a line in the direction of ~ passing through a point y.

r

(3.7)

QI

IVkuI2dx~y[(r !D k 2 U\2dx+(-lr QI

It follows from the observation concerning

\u\2dx].

0 ~ and from the corollary

to Theorem 3.1 that

QI j.':

rL

Summing over k we get

lul~ ,

Q

~y[(\uI22 1

Q II

.,

2 +(-1\uI 2 '

Q

I

1(D~u)/12 ds ~ y [( Y

rL

1(D~U)112 dx + (-1 (

L

Y

r

on o.

ID~u\2dx~y[(r IV!U\2 dx +(-1

0

I

lul~.u QI~Y[(lu\~.O+(-1ju\~.O]·

Since 0 - U QI CU 0 1, it is sufficient to consider 0 1 no.

e

the inequality above holds with ~ replaced by for k = 1, ... n. Since any differentiation operator can be written as a linear combina­ tion of D , ...,V {: , it follows on adding that ~1 sn .

•

Let ~ be a unit vector which is a positive multiple of some vector in the cone C/' We will establish an inequality for the directional derivative in the direction of~. However, we must be sure that the domain of integration is connected along each parallel to~. To this Then

n 0 1,0.::;; t
en n 0/) cO 1CO (recall the cone property).

easily tha t the intersection with 0

r

One verifies

of any line parallel to

~ is either

void or is a line segment of length d satisfying: hi'::;; d.::;; 211 1, Let D ~ be the operation of differentiation in the direction of~. If ,\\

v is a function in

0 define

(v)1 by

r juI2dx].

0

Let 1~1, ... ,~nl be a linearly independent set of vectors, each of which is a positive multiple of some vector in the cone C I' Then

Thus, it is sufficient to consider the boundary region 0 - U QI'

= y + t~, Y (0

Y

Letting y vary in a n - 1 -space orthogonal to ~, integrating the last inequality with respect to y, we find that

],

and summing this over i, we get

end introduce the set Or = Ix: x

[(u)/1 2 ds].

lul~. 0 n 0 '::;; Y [( lui;. 0 + (-1 lul~. 0 ]. 1

This completes the proof of part 1°. The proof of part 2° is similar, using (3.4) instead of (3.3), but needing only the segment property since (3.4) is applied in only one direction for each 0 I" In proving 2 the role of ~ is played by the

°

segment guaranteed by the segment property. Q.E.D.

More generally, we have the INTERPOLATION THEOREM:

24

Elliptic Boundary Value Problems

sec. 3

THEOREM 3.3. Let n be a bounded open set with the re­ stricted cone property, and assume 0 < f ..:s; 1. If u ~ Hm
lui]. n ~y (fm -J

(3.8)

lul~. n +f -

IUI~.

j

IUI;_l.n~Y(flu\tn+f-(k-dlul~.n)

holds for k

,I

rY

= 2. Assume that (3.9) holds for

Since u E: H m (n), for

lal

~

k

lui;::; Y1 f lui; + 1+ 1/2 !ukl2 + 1/2 [) - k f - k !ulg

< m and all

f

~

lui; ~ 2 Y1f lui; + 1+[) - k f -

lui; _ 1~ Y(f lui; +f -

(0, 1].

ID'" ul ~ lal

and in particular, taking

[\-'\.

+f

=

\D'" ul ~).

-1

k - 1, we obtain

d lul~)

replaced by [)

f

Y[f}' (fm - k lui;' +f -

k

lul~)

lui;' +f - (k -

+f 1)

(k -

1)

lul~

]

lul~),

and the theorem follows by induction. Q.E.D. As a corollary we have THEOREM 3.4. If n is bounded and has the restricted cone property, if f > 0, and if u E: Hm (n) for some m 2 2, then there exists a number C f

f

(k -

~ Y2 (fm -(k - 1)

IU!;.:s;Y1 (f lu!;+ 1+f- 1lul;_ 1)'

Hence, using (3.9) with

lul~

m - 2,D'" u E: H 2 (n); hence, by .:s;

~ y 1 (f

k

so that (3.9) holds for k = 2, ... ,m. Now we will show that if (3.8) holds for j = k > 1, then it holds for j = k - 1. Under the hypothesis that (3.8) holds for j = k .:s; m we have by (3.9) that

Theorem 3.2

\D'" ul;

25

or

n)'

where y = y
Proof. First we will show that (3.8) holds for j = m - 1. Observe

that, by Theorem 3.2,

(3.9)

Some Inequalities

for 0

< [) .:s;

=

Cf


m), which depends only on .

n,

m, and (, such

that

1,

\ I~!

lui; ~ Y1 f lui; + 1+YY1[) Let [)

=

min (1,

----1.-). 2y y 1

Then

!

uk l 2 +

YY1[)

-(k-1)

f -k lul~.

lI ullm- 1. n ~ f lul m. n + Cf lulo, n· For convenience, we now collect some facts about Fourier series. We will use the notation Q = Ix: Ix k I < 1/21. The cube Q will serve as a fundamental domain for various classes of periodic functions.

'"

26

Elliptic Boundary Value Problems

~finition 3.1.

sec. 3

cm rcooJ is the class of functions u (x) # #

which are m times continuously differentiable [infinitely differenti­ able]. Also, H#m is the completion of Cm# (or COO) with respect to the #

II 11 m •

c:;

fined on 0, and, with this identification H m# E:

Ilull~ ~

1

Iluli m •

IIUI/:'

Q;;; Cm

Q'

We may obviously identify the elements of H#m with functions deU

27

THEOREM 3.5. There is a constant C m > 1 such that

defined on En which are periodic of period 1 in each variable and

norm

Some Inequalities

C H

m

holds for all u E: Ca;.

(0). Let

Proof. Consider the ratio IfI - 2m - I

H~",

Ca;. Then u has a Fourier series expansion:

(21T)2 H eo< for fan

n-tuple of integers. f f- O. This ratio is bounded above for all such ,. Therefore, since by (3.12) (3.10)

u (x)

=

I b c e 2TTi f·

f

x

e;,

•

Ilu[1

2 m. Q

=

H;;;I m

jDO

o.

=Ilbel 2 - I

f

Q

H ~ m (21T)2Heo<,

<,

We use the notation Ito stand for summation over all f = (fl, ... ,f ),

f

n

the result follows immediately. Q.E.D. Consequently, the completion of in the norm

where f1, ... ,fn are integers.

Ca;

This series and the derived series

II II:.

same as its competion in the norm (3.11)

DO< u (x) = (2TTi)1cxI I bf f"'e2TTif .

such that the expression (3.13) is finite. Now that the formal series I bf e

are uniformly convergent, and, by Parseval's identity

IDO<

ul 0,2

Q

=(2TT)2\oc!I Jb c\2 f20<.

f

<,

:\0\

(3.13)

II u 1\m#

= [ Ib 0, •••• 0 \ 2 + fI

(3.13)

is finite, and let

•

f

f

suppose

f is such that the expression

Uk

(x) =

I IfI ~

be e 2TTix k

•

f. Then

<,

is a Cauchy sequence in L 2 (0) for lal ~ m, so that, by the Riesz-Fischer theorem, DO< Uk converges in L (0), lal ~ m. Thus, 2 Uk E: Ca;, Uk -> U in L 2 (0), and DO( Uk converges. la/:.:; m. Hence,

DO(

Uk

in H

'"

(0), so that u E: H#. m

Therefore, we may identify

.H~ with the collection of all formal Fourier series

Ib<,c\ 2 Ifl 2m ] '(

u = I b e 2TTix

f f

(IfI Euclidean norm.)

2TTix

•

f

Uk -> U

We will use the notation

'Q)

11

identi fy H~ with a class of formal Fourier series I bf e 2TTix x

f

(3.12)

• Q is the m Therefore, we may II

SUch that

~

Ilull:

<

00,

•

f

Moreover, if u E: H~ has the Fourier series

i I

28

I,

'~

Elliptic Boundary Value Problems

sec. J

g

C

=~-1

then for

la\ s: m1

Since

~

bg (2TTi.f)0< e 2TTix . (,

U (x) = I bt:

g

S. J

IluJ I\~

liP

r

U

is bounded,

11 < const

J m-1.Q--r­

where by const we denote constants depending only on m and on the bound for

THEOREM 3.6. If IUJI is a bounded sequence in H~ with the

j

.

DO< u has the Fourier series

formal Fourier series

i

IIUJII~

r

the series converging in L 2 (Q).

j

29

u = I bt e 2TTix . (

DO< u =

/1'1

Some Inequalities

Ilull:.

IIU I - ukll m -

e 2TTig .

1. Q

s: liS, (U r - uk)ll m_ 1.

Q

+

IIP r u/lf m_ 1.

+

IlPr ukll m -

•

g,

IUJ I converges in H m

(Q) as j

!bg,J I converges to bg as j

-> 00,

1. Q

then S C m-

-> 00.

1

liS

r

(u.J -

U

k

)11 m-l + Const ­ r

Proof. Let P u. = ,

1

S U ,

J

=

Igl

I

L.,

I

\g\ <,

bt:. e 2TTig .

s·

~ C (r, m) max Ibg. J - bg.

x

1

bt:. e 2TTig .

+ const , r

where C(r, m) depends only on rand m and the maximum is taken

over Igi ( r only. Since bg, J - bg. k -> 0 as j, k -> 00,

x •

S. 1

lim sup

j. k -> 00

1. Q

kl

•

By Theorem 3.5

\IP, uJ\l m _

s: C m -

1

I\P, uJ\I~

-

IIu J -

U

k

\

Im

-

1, Q

s: const r- .

1

Since r is arbitrary,

=C m

-

1

I

(

\2Ig\2(,m-l))'1,

Ibt:

\glL.,

s.

J

limsup I,

C

~~( I \bt: r 19l L. , s·

C

~~ (Ilbt: r

Q

x

if m ~ 1, and if for fixed _ l

Thus

g

S. J

J

12 \g1 2m )'1,

.

12 Ig\2m)'I,

k -+

00

\\uJ-ukllm_l

Q=O. Q.E.D. '

THEOREM 3.7. Let IUJI be a bounded sequence in H~. If III

~ 1, then IUJI has a subsequence which converges in H

Proof. Since u J E:

~

H~ ' u J has

m

_ 1

(Q).

a formal Fourier series expansion

30

{uil cC~
u.<x)=! b e .e 21Ti';·x· ~

I

"'I

31

Some Inequalities

sec. 3

Elliptic Boundary Value Problems

•

First, it is sufficient to sh?w that some subsequence !u j I conr

I

verges in Hm

Since lu.1 is bounded, there is a number C such that for every j I

_

1

(0') for ev~ry U' ceo. For suppose that such a

subsequence has been found.) Since U has the segment property, for 0 < f ~ 1 there is a Uf SC 0 such that

Ilujll~ < C.

"-'",

Ilu i

Since Ib~jl ~ IIUjll~

t

u i 11 m

_

I.

s

O~ ydl ui

'­

,-

-

u i 11 m , s

r

n + Y Ilu l r

-

ui 11 m

-

s

1,0; f.

this inequality is derived by applying Theorem 3.2 (2 0) to derivatives DOC of the functions considered, for lal ~ m - 1. Since u j converges

< c,

r

in Hm _ I (Oe)' it follows that

it follows that for fixed ~, lb ~j I is a bounded set of complex numbers. Let ~1'~2"" be some enumeration of the ~'s. We will ~se the

limsup Ilu i -u j 11 m - I u~fsuplluj r. 8~OO r s ' r, s l'

diagonal process to extract a convergent subsequence form lu .1.

-U i

8

11 m U.s 2Cf. '

I

Since lb e

"'I'}

I is bounded, it has a subsequence lb e

converges as j

->

'" 2'

2

. I of lb e

"'k,lk

00.

Then lb e

.

"'k' I k _

f

I which converges as

. I is a subsequence of Ibe.1 converging for

"'I

"'j,l j

every

1

~I

(,N

which converges in H.
~.

I

11.11 m _

1

for j S m - 1, it is sufficient to

,.., s C for some constant

Prove that if Iu I I c H m 0, then there is a subsequence lUi I such that

Il u.f - u I r

s

II m

-

,.., -> 0 as 1, ~£

r

r,s

->

00.

We now perform several re­

ductions to show that it is sufficient to prove the theorem for

s

_

I U •

->

O. Note that

=

a subsequence 1(,2Ui I which converges in Hm_l(U). Proceeding in

THEOREM 3.8. Let U be a bounded domain having the segment \ property. Then every bounded sequence in H m
II.II} s

r

1 on UN' By Leibnitz's rule, for fixed N, !(,Nu;l is bounded in Hm
I

Proof. Since

Ilu j - U111 m

N->oo

By Theorem 3· 6 the corresponding subsequence of lu.1 con­

verges in Hm-l
f,

only at this stage of the argument is the segment property needed. Next, it is sufficient to prove the theorem in the case that there exists a U I ceO such that each u j has its support in UI . For sup­ pose the theorem to have been demonstrated in this case. Take an expanding sequence IU NI such that lim UN CC U; let (,N ~ C';
"'2 1 1

} I. Continuing in this manner, for each k we

obtain a subsequence lb e ->

Since this holds for every

Since !be . I is bounded, it has a convergent

00.

1

subsequence lb e

jk

I which

""'}l

2

this manner, for each N we obtain a subsequence {(,NU, I of !(,NUI 'N

N-I

which converges in Hm-I
r

r

for each N. Since (,NUI' r

=

u l ' on UN' it follows that Iuj,l converges in r

r

Hm_/U') for every 0' ccU. By the first reduction, {UI" converges j,n Hm_l
r

I

32

Elliptic Boundary Value Problems Finally, suppose the theorem has been proved for a sequence in

~,

i~!

33

Some Inequalities

sec. 3

unit vector which is a pOsitive multiple of some vector in Gh o

,>

0

•

Let

C':;' Cn). If lUI I is a bounded sequence in Hm Cn) and if supp (u.)

Ivil cC';;'Cn) can be Then Iv;l is bounded in

COl ceo, then by Theorem 1.10 a sequence

chosen such that

Ilu I - vilim. 0 < i -

Hm Cn), so that some subsequence Thus lUI r

I converges

1.

Iv I converges 0

J

r

in H m _ 1 (U) since

in H m

-

JJ.ll m - 1 :':: 11.ll m •

1

Cn).

f (t)

= u Ct~,

O:s;; t:s;; h o' I, 'I,

Then, by Taylor's formula we have

Therefore.

by the second reduction it is sufficient to prove the theorem for

k U

lUll CC':;' (n).

J

lu;1 CC':;' (n). By a suitable change of assumed that fi C Q. Extend u to Q, de­

1

~

(0) =

=0

t J f (t) + k / r k J o

1

-

f k (r) dr,

Therefore, assume that coordinates it may be

0

z

fining u; = 0 outside 0, and then consider u; to be extended by

Periodicity to E n . Then Theorem 3.7

lUll

luol Z

hence, a fortiori in Hm

_ 1

f (t) '" (-})) l/) (t).

J

}.

is a bounded sequence in H#. By m

has a subsequence which converges in

Hm

_ 1

Cn),

Cn). Q.E.D.

The following theorem is one of the versions of the important Sobolev inequality (reference: Ehrling. Math. Scand. vol. 2 (1954) p. 267). THEOREM 3.9. (Sobolev's Inequality). Let 0 be a domain hav­ ing the ordinary cone property, let u E:: Wm (n) [H m (n)] with m > 0/2,

e

let be the largest integer less than m- n/2 ce = m- [n/2] - 1). Then u can be modified on a set of measure zero so that u E: (0). More­ over, for any r ~ 1, lui:::; xE: 0,

ee

e,

ID'" u(x)1 ::; Y r -

Cm

-~n

-

H) (Iul m • 0 + r lui 0, m

where

0),

We shall assume n/2

lu (O)\2:s;; (k + 1) I

I rt r k o

1

t

t 21 tf) Ct)1 2 + k 2 I r k -

1

0

Cr) drl2 =

1f /r

I(

rk

-

0

t

::;.{ r 2k -

f k Cr) dr1 21.

'~ (n + 1) [r'~ (n - d f (r)] drl2 k

n

-

1

t

dr'[ r n

o

'" _1_ t 2k 2k - n

'h o and vertex at the origin. Let Gh denote the cone obtained by

to height h. Assume that u E: C m * Cn). Let ~ be a

~

By the Cauchy-Schwarz inequality

Now since k > 0/2.

Proof. First, assume that 0 contains a closed cone Gh of height o

o

k -

m.

J= 0

where y is a constant that depends only on 0 and m. (Indeed, y depends only on m and the dimensions of the cone associated with 0.) If, in addition, U is bounded and has the segment property, then u can be modified on a set of measure zero so that u E: Cf (rn.

truncating Gh

< k :s;;

Thus

-

0

t n

[

0

rn -

1

Ifk

1

Ifk

Cr)1 2 dr

(r)\2 dr.

34

EIIiptic Boundary Value Problems k -

lu
1

t 2J

i = 0

II

(tW + k

J

~

r

t

2k

2

t 2k - n

"

r" -

sec. 3

r G

1

35

Some Inequalities t 2 k -"

r" -

Ilk (r) 2 drdx

1

0

h

o

h

r

\

• Ilk (r)1

2

=

drl.

=

o of Gh is wh", where w is a constant. Since 0 < t < h, we have

wh "

(3.14)

lu (0)1

k -

.::;;

(k + 1) I I

1

h 2i

i = 0

+

~

2k - n

G

r G

J If. (t)\2 h

Let y be a generic constant depending only on w, m, and h o' Since lJ is a linear combination of derivatives of u, the first term on the right in (3.14) is bounded by k -

Y

1

I J=O

h 2i

lui?J,G

h

r gdx = Ar r

Gh

-"

I

II

1

r

G

(r)l1 dT] t" - Idtda k

h

1

dt)

(r r

r" -

A0

Ilk (rW drda

1

Ilk (rW dx h

~k '::;;Y2J(!U

1

2 k ,G

h

'

Hence, (3.14) implies

(3.16)

lu (OW.::;; y i

h"

k

I

.

= 0

lui 1.~

h 21

Gh

.

By the interpolation inequality, Theorem 3.3, with (

To estimate the second term, we introduce polar coordinates. Let a be the ordinary surface measure on the surface of the unit sphere, and let A be that part of the surface of the unit sphere subtended by the cone G h • We then have the formula for integration using the o spherical measure a

(3.15)

.....-.-

(f -t 2k -

2k

Ilk (rW drdxl.

r" -

0

o

dx

0

"J

0;

h 2k

1

h

[t 2k -

1

t 2k -" ( r" h

A

(-- h

Now integrate this inequality over Gh for some h < h ' The volume

2

I h

rr

lul J,2

G

'::;;y(h dm h

gt" -

1

G

+h- 2i

h

lula '

G

h

),O.::;;j::;,m,

so that (3.16) implies

lu (0)\ .::;; y h -

h

2 - i )\uI m,

h 2)

=

(11," - m)

(lui m,

G

(lui

G

dtda .

+h - m h

lui O.

G

0

For h

=

h0

Ir.

this gives

Using (3.15), the second term on the right side of·(3.14) becomes (3.17)

\u(O)\ .::;; y rlJ," -

m

m,

+ r m lui 0 h

G

'

).

h

). h

36

Elliptic Boundary Value Problems

If lal (3.17),

::;; eand

if u E: C m * (0), then D'" u E: C<m -

I'" 1)* (11),

and by

37

Some Inequalities

sec. 3

in Wm (11'). Then by (3.17) applied to uk - u J• lukl converges uni­ formly on U (to a continuous f~nction u o). As Uk --> U in L 2 (11), U o equals u almost everywhere pn U. Moreover by (3.18) applied to

\D'" u (0)1 ::;; y r~n

-

r~ n

-

::;; y

m

m

+

+

H (ID"'ul m _ 1"'[ •

H (lui m, G . h

+

rm

H [D'" ul 0,

+ rm -

G

h

I'" I Iu II'" I'

-

o

By the interpolation inequality applied to the region Gh with o

lull"'l ,

G

::;;

h

Y (r - (m - H)

o

lui m.

G

+ rH h

lUI O.

G

0

h

Gh

f

G

e.

Uk - u j' D'" uk converges uniformly on U for lal ::;; It is easily seen then that lim D'" Uk = U'" Uo. _By the argument leading to (3.18), for

) h

x(U

).

ID'"

0

=

r-

In the limit as k

),

[D'" U o (x)1

I

ID'" u (0)1 ::;; y r~n -

m

+ H ([ul

+ r m lu[o

G m"

h

o

G '

), h

0

e.

for lal ::;; This inequality has been derived for the cone Gh

r~n

--> "",

H (Iukl m, 11'

- m+

2,

0

so that

uk (x)!::;; y

la I :; ; ewe

for

:;;; Y r~n -

m

lukl o• 11')'

obtain .

(Iul m , 11'

m + \ex\

+r

+r

m

lui 0,11')'

and the first part of the theorem is proved. For the second part of the theorem, by Theorem 2.1 there is a sequence

lu k Icc"" (E)

part of the proof, if

ID'" (!1k -

such that Uk

U in Wm (0). By the first

-->

lal :;;;e,

u) (x)1 ~Y r -

~n

(m -

-

H) (Ju k - uj!m, 11 + r m IU k - UJ

' not for 11. o However, since 11 has the ordinary cone property, each point in 11 lies at the vertex of such a cone lying entirely in 11. Thus, since the norm over a cone in 11 is nO greater than the norm over 11, for any

10

for x E: 11; by continuity, this inequality holds for all x E:

xE:11

u0

ID'"

(3.18)

u (x)1 ::;; y

r~n

- m+

m

H (!u·[m. 11 + r !ul 0, 11)

n

e.

of xo' Let 11' cc11 be a domain such that the cones (of the ordi­ nary cone property) with vertices in U are contained in 11'. By Theorem 1.10, choose a sequence lu k Icc"" (E) such that Uk --> U

Let

lim uk; by the inequality above D'" uk converges uniformly on k-->"" for lal ::;; Hence, Dex U o is uniformly continuous on 11 and thus has =

e.

a continuous extension to for lal :; ; Thus the theorem is proved for u E: C m (11). For any u E: Wm (0), let x0 E: 11 and let U CC 11 be a neighborhood

n.

.11)

Corollary 1. If u E: H

m

ft,

10 C

tion u 0 E: C (11) such that u 0 Corollary 2. If uE: H m

10c

if

la\ :; ; e.

Q.E.D.

(0) and if m =

> n/2,

then there 'is a func­

u almost everywhere in 11.

(11) for all positive integers m, then there

is a function u 0 E: C"" (11) such that u 0

=

u almost everywhere in 11.

1

11

1[1

I I11 : I I1

IIII

III

II

III

II '/li

Iii

II

38

Elliptic Boundary Value Problems

Let M be a smooth (n-l) -manifold in a domain a. If u is a func­ tion defined on a, then the restriction of u to M is the trace of u. For an arbitrary function in L 2 (0), the trace defined in this manner

39

cone property. Let 0'; cc 0; be chosen such that

O~ = a

-

U 0';. If

aa C U 0';, and let

1 0 is any sufficiently small cone, then for x E: O~

,e

we have x + I o ca. Let O.J = 0'.I n II, i = 0, I,oo.,N, and let S/:11, ... n .' I be linearly independent vectors in 1/. By rearrangement (if necessary),

has no significance since the measure of M is zero. If, however, u E: HI (0), a more satisfactory definition can be given as follows:

-

let IUkl be a sequence of funcetion, Uk E: C 1 * (0), such that Uk

assume that the projections of n-v on the orthogonal com­ plement of E v are linearly independent. Let I; be the convex hull of

->

u

in HI (0); if the restriction of Uk to M (that is, the trace of Uk) converges in L 2 (M), then we will call the limit function the trace of u on M. Of course, M could just as well be a lower dimensional manifold. The following theorem (also a version of a Sobolev theorem) shows that the trace is well defined under suitable conditions. E will be used to denote a v-dimensional subspace of En for v VS:vS:n-l.

THEOREM 3.10. Let a be a bounded domain with the restricted cone property. Let x O E: a, let n = x O + E v and let II' = II (l a. Let

eiI, ...,e;·

I,u, ... ,e·

if m > 1/2 (n - v), then the trace u 0 of u on II' is a well defined L 2 (II') function~ If, in arjdition

lal < m-

1/2 (n - v), and

r ~ 1,

then the trace DO< u 0 of DO< u on II' exists, u E: HI' I (II'), and 0 0 < (3.19)

«( IDO< II'

U

o

(x)12 da)~ ::;; y r - (m - 'I, (n-v) -

n-VI; that is,

n-v.

Ij=lx:x= I

k=l

n-v

0, 1

C k ,;'k,C k2

CkS:ll.

k=l

,

Then

1/ C I., so that x + 1. c a for any x E: 0.. Furthermore, I. spans 1

I

I

I

a complementary space to E v : this is evident from the fact that the projections span the orthogonal complement.

If u E: C

a be the v-dimensional Lebesque measure on II. If u E: H m (0), and ,t'

Some Inequalities

sec. 3

m

(0), then for any point x E: 0., u (x) can be estimated as in 1

the proof of the Sobolev inequality, except that now the cone used is the n-v dimensional cone x + 1/. From relation (3.16), if hi is the height of 1; and if 1/2 (n -v) < k ::; m, then for x E: 0/

\0(1).

h~-v lu (xW s: y 1=0 I

hi i

lui:,

x+1.

I

'(\ul m , a + r lui 0, a),

I

ll1

i\."

where y is a constant depending only on

"

a.

Proof. First note that in II there is a finite open covering

where y is a, generic constant depending only on over O. nIl, we have

10.1 of

II' such that to each OJ there is associated an (n - v) dimensional "~1

'Ill

,'\

1/ which has the following properties: the vertex of 1/ is at the origin, 1 1 spans a complementary space to Ev ' and x + 1 j c a for each x E: OJ" For let 10; 1~ = 1 be a finite covering of an and

cone

II: I~=1

the set of corresponding cones guaranteed by the restricted

On integrating

I

h7-v

I

a.

k

( 0;

nil

,

lu (x)1 2 das: y I hi J ( 1=0

0/

nII

,

lui? x+I J.

da /

k

s: y 1=0 I hi J IUlf, a the last inequality arising from the fact that the integration over

°1 n II' is over an open set in Ev' and lul/. ><+1/ contains an integra­ ~

40

EIIiptic Boundary Value Problems

41

Some Inequalities

1\

tion over a domain in E n-J.1 . Since the finite collection 1011 covers I II , a summation gives

II

Weak convergence will be denoted Uk -

The following WEAK COMPACTNESS THEOREM is well known.

THEOREM 3.11. If {ukl is a sequence of functions in L 2 (E)

'1

,II

h -J.1 J n

2

lu (x)1 da::;; y

II'

I

i"'O

h 2i

lui ~

u.

n

such that Ilukllo. E is bounded, then {ukl has a subsequence which

I.

converges weakly in L 2 (E).

'III

Now we can prove where h ::;; h 1111\

==

min (h;). Upon using the interpolation theorem

n

THEOREM 3.12. Let be a domain having the segment property, (n). If there is a sequence lukl CH m (ll) which is 2

o (Theorem 3.3) and replacing h by h o Ir, we obtain as in the proof of

atJd let u E: L

Sobolev's inequality

bounded in H m (n) and converges weakly to u in L

(r,

lu (xW da]'Ii s; y r',L, (n-J.1) -m
uE: H m

n + r m lui o. n)'

if

III'I

(x)1 2 da S; y rv' (n-J.1) -m

f IDO< U I

+1"'-1

(lui mil ~, n

lal S; m.

the weak limit of DO< Uk in L

2

(n) for

lal : ; m.

I converges

that {D 0< uk

weakly in L (n) if 2

I

+ r m lui 0, n) ~11

DO< u

II

k

2

laI::;; m,

say

--:>. uO< l

II

for 0 S;

ilml

\a\ S; m y

From (3.20) it follows that if Uk E: I

I~II

L 2 (n). Thus, for any ep E: C'; (n),

1/2 (n - J.1).

cm

(n), and if Uk converges in

I

H m (n), then Uk converges in L 2 (II). Hence the trace is well de-

I uO< epdx n

fined. For an arbitrary u E: Hm (n), u can be approximated by functions in

v

Definition

3.2.

==

lim k; -->

==

Coo (0) as in the proof of the Sobolev inequality. Inequality (3.19) follows immediately. Q.E.D. Now we will collect some simple lemmas which will be useful later. First, some definitions.

00

lim

I DO< Uk epdx

n I

(-l)H

kl-->oo

==

Let E be a measurable subset of E n . If

(-1)H

I

n

f

h

DO< epdx

u k;

uDO< epdx.

u E: L 2 (E), a sequence {Uk I C L 2 (E) is said to converge weakly to

<.?

co

I I I I

i

u in L

2

(E) if and only if for every v E: L 2 (E)

lim f Uk vdx k->oo E

==

f

uvdx.

Thus u has the (uniquely determined) weak derivative DO< u == uO<; it . follows that u E: Wm (n) == Hm (n), and that

E (3.21)

(n)

By the weak compactness theorem, lukl has a subsequence

I such

For the derivatives of u, we have as in (3.18) that (3.20)

u is

(n), then

Proof. Since {uJ is bounded in H m (ll), {DO< ukl is bounded in L

II

1.li

(fl), and DO<

2

DO< urDO< u in L (n) for I

2

H .: ; m.

42

sec. 3

Elliptic Boundary Value Problems

Since every subsequence of

Iu) has

a subsequence

Iuk.1

Some Inequalities

43

such that

1

rb+h If' (,;}I

:;. h 2

(3.21) holds, it follows that Doc uk~Doc u in L 2 (n) for lal'::;; m.

2

d~.

B

Q.E.D.

Definition 3.3. If

e l is the unit vector (01l.021 .....onl) and if h

Hence

is a real number, then the difference operator o~ is defined'by o~

u

=

h

-I

[u (x + he

l

(3.22) ) -

u (x))

(/

for any function u. THEOREM 3.13. If u E: H m (11), where m L 1, if

dist

0' en,

(0', am> h > 0, then I\o~

e

Now if

Uk

E:

em

riDe< 0 1

(U), then on integrating, using iterated integrals,

U

h

I

2 dx

=

k

r

Q'

WDOC u h

.: ; I ID Q

f (x + h) - f (x) == .r

J<\-h

x+h

If(x+h)-f(xW.::;;h[ x

r

b

If (x + h) -f (x)!2

dx

Ilo~ ukll~I' Q'.:s: Ilukll~. Finally, if uk

:s h.r

b

B

x+h

r

If' W\2 df dx

"

r. .r.. f If' (,;) a+h

=

II

h

r

b

+h

B+h

II 1)11

Iii

el

DOC ukl2 dx for

lal'::;; m-1,

Q'

1f'(,;}1 2df·

B

~

2 dx

>so that

whence, by the Cauchy-Schwarz inequality

Thus

1

k

df

f' (,;)

"

\;

If' (,;}1 2 d~ .

a

(a, b + h), we have

I

rb+h

and applying (3.22), we see that

Q'

u\Im_l. Q''::;; \Iull m. Q.

Proof. For any function f E:

and if

dx.::;;

h

I

a

2

f (x +h) - f (x)

+h

I b

I

f

~-h

if'

b+h ,P J

f-h

2

1

dxdf

(,;}1 2 dxdf

If' (,;}1 2 dxdf

-->

118~ ull~_I.

u in H m (n), then we see that

Qr'::;;

Ilull';'.

Q' Q.E.D.

In the periodic case we can do even better. THEOREM 3.14. If

u E: H~

, then lIo~

aU h > O.

ull m_ l •

Q

S

lIullm.

Q

for

Proof. Let h be fixed and let r be a fixed integer larger than 2h. Now apply Theorem 3.13 to the case Q = NQ == Ix: Ixkl < NI2l and {}, == (N - r) Q, where N

> r. For x E:

an and y E: ij',

~I III

44

Elliptic Operators

Elliptic Boundary Value Problems

Ix -

yl

~

THEOREM 3.16. Let ~t

r/2 > h, so by Theorem 3.13

1\8~ ull';'_l,

III1

(N-t)

O~ \Iull';'.

\18~

ull';'_1,

0

~N

n

Ix: Ixl < r, xn > 01, r> O. If

\\ull';'.

ll8 h ull 0. ~ R' ~ C for i

i

=

1, ... ,n ­ 1,

0'

then the weak derivatives V.I u, i = n

Now simply divide by N and let N

,..-> 00

m

\18~

ull m • 0'

IIV I

ull. 0 ~ C for i = 1.... ,n.

~ C for all h sufficiently small, then

u (H m + 1 (m

IIVluli o, ~ R

and

~C,

12

k __ oo,i=l, ... ,n. Clearly Ilu.ll r\'~C, I o, ~£

r

0'

u i ¢ dx

=

r

lim

0'

k->oo

4. Elliptic Operators

u...:> u in L (0') as k

I

2

Let A (x, V) be a linear differential operator of order E, that is

Forany¢(COO(O'), 0 A (x, V) =

8~ u¢ dx

~ 0 a (x) Vex \cxiS;L ex

k

.where the coefficients =-lim k->oo

r

0'

u8~h

¢ dx

r

0'

uV I ¢ dx.

'

with

lal

u E:: W

lac

1

(m = H 1

lac

NoW proceed by induction. Suppose the theorem is true if m is

replaced by m - 1. As above. there is a sequence Ih k I such that

I

\;...

,·1,

I

o·h u ~ VI u in L 2 (0'), i = 1, ... ,n. By Theorem 3.12 V. u E H (0'), , m k

i ~

1, ... ,n. Hence u ( Wm+ 1 (0') by Theorem 2.4. That \ IV i Ul 1m • 0 '

C is immediate, Therefore u E: Wm+l (m and Theorem 2.2 implies

=

u (H m + 1

(m.

Q.E.D.

=

Evanish indentically.) We associate with A (x, V) the

A I (x, ~

(m.

(x) are complex valued functions defined

h()mogeneous polynomial in

As this relation holds for any 0' CC 0, u l is uniquely determined in

. 0 and is equal to the weak derivative V.I u there. Hence

B ex

in some open set 0 in En' (It is assumed that not all coefficients

k 0<

=-

i=1,,,.. ,n-1,

-""",

h __ 0 and functions u. E:: L (0') such that 8 hi k

1, exist as functions in

Proof. The proof is the same as that of Theorem 3.15. Q.E.D.

Proof. First we prove the theorem with m = O. By the weak com­ pactness theorem there is a sequence Ih k I of real numbers with ~~I ~I

t ... ,n ­

L 2 (~R) and

to obtain the result. Q.E.D.

THEOREM 3.15. Assume that 0 has the segment property. If (m, and if there is a number C such that for every 0' ceo,

u (H


and all h sufficiently small

By periodicity this inequality may be written

r)n

=

u E: L 2 (~), Emd if there is a number C such that for every R' NO'

I

(N -

45

=

~

lcxI- E

a ex (x)

e= (e .....e) of degree E: 1

n

f' .

,The corresponding differential operator A' (x, V) is called the prin­ cipal part of A (x, V). We now introduce the notion of an elliptic perator.

Definition 4.1. A

=

A (x, V) is said to be elliptic at a point

e.;,

.10 if and only if for any real 0, A' (xO, ~ f o. A is uniformly elliptic in a domain 0 if and only if there is a constant C such that

C-

1

lei ES; lA'

(x,

~\ ~ C lei E

sec. 5

Elliptic Boundary Value Problems

46

Local Existence Theory

r

n.

be a contour in the upper half plane containing all of the zeros of P (~~. ~n) which lie in the upper half plane. Then P (~~. ~) does

for all real ~ 10 0 and all x E:: A is strongly elliptic if and only if there is a function C (x) such that

not vanish on

:R (C

(x) A I (x, ~ ) > 0

for all real

~

i

IP (1;'0' ~) - P (~', ~)I

eat x 0.

number of zeros within

If .

=

(~I""'~n-l)' and let P (I;', ~n) bo

~! + b 1

(I;')

~!-1

=

A' (xO,

0·

fore, N+ (~') is a continuous function of ~' for ~' f O. Since n L 3, E:: E n -1 : Sc' r-L 01 is a connected subset of E n-l ; N+ (CI) is a con­ S

e= N+ (~') + N- (~') = 2N+ (~') must be even.

=

1, ... ,n.

5.

on ~n' there is a real value of ~n for which P (~', ~)

=

O. This con­

e

tradicts the ellipticity of A. Hence is even. In case 2°, for I;' 10 0 let N+ (~') [N- (1;')] be the number of complex zeros ~of P (I;', sn C ) = 0 having positive [negative] imaginary part. n

111

I

2

assumptions are made on the coefficients in the differential operator ':< A. First we treat the case which A has constant coefficients and coin­ :1 'cides with its principal part. We shall also momentarily restrict our Considerations to periodic f. Recall that Q is the cube Ix: IXkl < 1/21. THEOREM 5.1. Let A (D)

Since A is elliptic, there are no real zeros, so that N\~')+ N-(~') == e. We will show that N+(~') == N-(~'); from this it follows that is even.

e

Now we use Rouche's theorem to show that for 1;'0

.ems N' (tl in the npp"' half plane is constant £0'

=

Q

N- (-1;').

f 0, the number of

f'

nea,

f;.

Let

=

.

~

a

H = e ""

D"" be an elliptic operator

order Ehaving constant coefficients. Then for every f E:: L (Q) such 2 that J fdx = 0 there is a unique u satisfying

Note that P (-I;', - ~n) = ( - 1)e P (I;', ~n)' Thus, any zero of P (1;', ~n) is also a zero of P (- ~', - ~n); hence N+ (~')

Local Existence Theory

In this section it will be extablished that the elliptic equation Au = f always has weak solutions in small neighborhoods, if f E:: L and mild

f O. If e is odd, P (~', ~n ) has the same sign as ~n for suf­

ficiently large I~n\' Since P (I;', ~) is real and depends continuously

,I

Q.E.D.

Y2 (D ; + iD 2)'

°

ii'll

(~'); it follows

!:J. = D ;+...+D ~ and, for n = 2. the Cauchy-Riemann operator

Since A' (xO, en) = P (0, 1) = b o and A is elliptic at xO, bolo O. Clearly, in case 1° it is sufficient to assume that b > O. Let I;' be

ltl

t') = N-

Two classical examples of elliptic operators are the Laplacian

Then

+... + be (~'),

r

"
N+ (~') for~' close to ~~. There­

Ic, s

that

where b. (~') is a homogeneous polynomial in ~' of degree i, i

fixed, ~.

=

must be constant. Therefore, N+ (~') = N+ (-

e

I

Applying the same technique to the lower

tinuous:, integer-valued function on a connected set, and therefore

or if 2° n L 3, then is even.

(~' ~n) =

r.

half plane, it follows that N+ (~~)

~

P

< \P (~~. ~n)1 on r.

Hence, by Rouche's theorem, P (~', ~n ) and P (~O' • ~n ) have the same

the coefficients of A' are real, "

Proof. Let ~'

Since P (~', ~n) is a continuous function of ~',

°

O.

THEOREM 4.1. Let A be an elliptic operator of order

/'

r.

for ~ I sufficiently near ~'

Note that uniform ellipticity and strong ellipticity are distinct pro­ perties of differential operators. In discussing strongly elliptic operators we will assume that a normalization has been made so that C (x) can be taken to be identically constant. Note that strongly elliptic operators are necessarily of even order. The first result concerns the order of an elliptic operator.

e

47

1

u E::

He'

48

Local Existence Theory

Elliptic Boundary Value Problems

Wm and II II m,

49

Au = I,

Since"

r udx = O.

on TN I is proved. Q.E.D. It will also be necessary to solve the equation Au = I for 'unctions I which do not have mean value zero. Since in this section we wish only to state that at least one solution u exists, it will be sufficient here to construct a single particular solution of the equation Au = 1. This can be accomplished very simply in the present case of A having constant coefficients and A = A '. For since A is elliptic,

Q

Denote this uniquely determined u as TN I. Then there is a constant N depending only on A such that 11 TN 111e. Q .:s: N 1\111 0 , Q' Moreover, il I has the Fourier series expansion (5.1)

I (x) =

I' c~ e 2TT1X {

E

f.'

cannot vanish: if a = a(o 0 I.

• ... ,

0)'

Xl

~ then a", O. If U (x) = x/aEl, £ then Ali = 1. o O Now if I E: L 2 (Q), let

then

_c---m~ e2TTiXO~,

u (x) = (2TTi)-E I' A (

~

are equivalent for all m, the estimate

the coefficient in A of DE = _a_ p Ia e

~

(5.2)

Q

I

(5.3)

Jl (f) =

r Idx. Q

where I' stands lor summation over all ~ '" 0, ~l""'~ integers. ~

n

Prool. Note first that if f E: L 2 (Q) has the Fourier series expansion Ict: e 2TTiX ' ( then c(o 0) = 0 if and only if Idx = O. Now sup­

~

. . . . . Qr

': .

pose u E: He' and u = e 217i "{ = (2TTi)EI' A

g

r

b~ e

2TT1

"{ Then Au =

(fJ b~ e 217iX {

Z' b~ A (2rrifJ

Therefore, if I is given by (5.1)

. Then the function 1-/1 (I) has mean value zero, and u' = T ii (f fl; is a solution of Au' = 1- Jl (f). Now define u = TI, where (5. 4)

Tf

=

f Then Au = [I ~:~

"t,

(fJ bg = ceo As A is elliptic, A W oJ 0 for ~ '" 0, so b~ = c~1 A (fJ. Therefore, u is uniquely determined by the relation

and Au = I, A that

"I~

(5.5)

T ii (f -

/1 (f) ) + /1 (f)

/1 (£)] + Jl (f) = I,

IITlllfI,

Q.:s:N

/1 (f) )

u o'

alld we have the estimate

11 1 -/1(£)11 0 •

Q

+

1/1(£)llluolle,

Q

;h

~N'I\I'lo.Q

!t,

(5.2). Obviously, the function given by (5.2) is indeed a solution of Au = I. Finally, as \A WI 2 c- 1 I~I r,

where we have used the estimate !luolle, Q depends only on A.

1/1 WI

.:s:

11111.

Here N'

=

N+1

+

[lull; =

(2TT)-e [I'

.:s: (2TT)-[ c

~

[I' ~

Icgl 2 IcgI

\A

WI-2 Ig\2E]'1,

Now we pass to the general elliptic operator.

operator 01 order Ein

lal

,,0

II

II' ".!

a

domain

n.

I a (x) D"" be an elliptic loci ~ Let a"" (x) be continuous il

=

e ""

E and measurable and locally bounded il lal < e. Then lor any E: n there exists a neighborhood 0 01 xC with the property that lor any IE: L 2 (0) there exists u E: He (0) satislying Au = I in O.

= (2TT)-e C 11111:.

:,... ­

THEOREM 5.2. Let A (x, D)

r~t: '

2]'4

-

l­

=

50

Local regularity of solutions of Elliptic Systems

Elliptic Boundary Value Problems

51

Proof. By a suitable change of coordinates it may be assumed that

6. Local Regularity of Solutions of Elliptic Systems

x o = 0 and Q C 11. Let Qa be the cube Ix: \xkl < a/21, and introduce the coordinate transformation

a

x = ay. Then if D'"x = Joel/ax "'I ... 1

DO(x = a-·\"'ID"'. y

If we let v (y) = u (x) = u (ay), then the equation

A (x, D) u (x)

=

!

(5.6)

In this section several results Shall be derived showing regularity properties of weak solutions of elliptic systems. The gist of these results is that weak solutions of elliptic equations can be shown to have regularity properties not explicitly required of weak solutions' of differential equations. Or, the smoother the data, the smoother the solution. In particular, we shall show that weak solutions of elliptic equations with infinitely differentiable coefficients and inhomoge­ neous terms are themselves infinitely differentiable. First we introduce some convenient terminology.

n

f (x) takes the form (after multiplying by a£) (ay) D'" v +

a

\ocI=£ '"

Letting g (y)

ax"'n,

y

!

\ocI<£

a (ay) a£-Joel D'" v

'"

=

a£ f (ay).

y

a£ f (8y), (5.6) may be written for y E: Q

=

Definition 6.1. Let A (x, D)

A' (0, D) v - Ba (Y, D) v = g (y).

(5.7)

'" to a domain 11. We shall say that A (x, D) is s-smooth in 11 if

a.

a",

c\ocI-£+s (11) for lal > £bounded for lal s: £- s.

a > 0 such that

E:

a",

is measurable and locally

for all v E: H£ (Q). Now let T be the solution operator for A I (0, D);

T has the form (5.4) and satisfies and estimate (5.5). Upon multiply­

solution of Au

ing (5. 7) by T, it will suffice to solve the equation

$hows that

IIBa

vllo.

v-TBa

(5.9)

Q

s: t Ilvlle.

Q

e,

v=Tg

for some v E: H£ (Q). Let

f

(Au, be chosen to satisfy Nt

be chosen so that (5.8) is valid; then TB

t

< 1 and let

IITBa vile. Since N'

t

< 1,

Q

s: N'

IIB a

a is a bounded linear trans­

vl\o. Q s: Nt Ilvl\e. f

a(where 1 is the identity on

He (Q)) has an inverse, and the solution of (5.9) is v

I,l!I l II

Finally, u (x)

v (a-Ix) is a solution of 4u = f for

(u, A* ¢)o. 11 '

!

I"'l~£

(-I)HD"'(a¢).

'"

The operator defined by (6.2) is called the formal adjoint of A.

Q'

III

=

f in 11. Then if ¢ €: C~ (11), integration by parts

(M o, 11 =

A*¢=

Since a the operator 1 - TB

=

a

formation of H £ (Q) into H £ (Q) satisfying

I

s, and

Now weak solutions of differential equations with nonconstant coefficients are defined in the same fashion as weak solutions of equations with constant coefficients; d. Definition 2.2. Thus, sup­ pose that A (x, D) is s-smooth in 11, s 2 and suppose u is a C£

(5.8)

1[[

!

a (x) D'" be a differ­ ~ £ '" (x) is defined for x belonging

\ocI

ential operator of order £, where a

where B is a differential operator of order not exceeding e, and Ba a has coefficients which tend to zero with (We have used the assumptions on the 8", at this stage.) Thus, for any t> 0 there exists

1\"

=

=

E: C H, upon performing the indicated differentiations in

(6.2) and using Leibnitz's rule, it is seen that A* is differential operator of order £ with continuous coefficients. In fact, the co­ efficient of D(3 ¢ in (6.2) is precisely

(1- TBa)-lTg.

Ixl < a.

'"

Q.E.D.

. ..i'.:.

~

52

Elliptic Boundary Value Problems

Local

H ~ e (- l)H ($) D",-/3 a . /3 <;..

(6.3)

0<

as a E:

'"

CI",I-f+s,

the estimate for ¢

'

the expression (6.3) belongs to

clt31-f+ s.

There­

Also, (6.3) shows that the coefficient of Df3 ¢ in A* ¢ for

(6.4)

(I, ¢)

=

e.

Then a is a weak solation in

lal s

n

IIlII o. fl II¢ 11 0 • fl'

Thus, there is a constant C such that

f- j we have by the Cauchy-Schwarz inequality that

Summing over all a,

lal ::;; e,

!(a, A*¢)\s C

Note that (6.5) is just the case j =

11¢ll o• fl'

all

¢ E: C;; (fl).

I

~I

iill

Ij

Ii:

In proving the first:regularity theorems of this section, we shall not

even need to assume that a is a weak solution of Au = f, i.e., that

(6.4) is satisfied, but only that (6.5) is satisfied. Since for sufficiently smooth coefficients, A = A**, and since A*

and A are simultaneously elliptic, in dealing with inequalities like

(6.5) we shall replace A by A*. This is more convenient notation

and it gives more general results.

We shall also derive results using a generalization of (6.5). To

motivate the considerations suppose that u E: H.J loc (fl), that

o.s; j .s; r,. and that A is i-smooth. (u. a", DO< ¢) =

Then for ¢ E: C~ (0),

(;c u, D'" ¢) = (Df3 (ao<

u), Do<-(3 ¢), where Df3 is any

e.

It will be shown below that if loc (fl); cf.

(6.6) holds and A is elliptic and j-smooth, then u E: H

J

Theorem 6.3. The key to all the following regularity results is now given in a emma. We have to work hard to establish the lemma; having the emma, all the other regularity results given in this section are ,Tproved with great ease. LEMMA 6.1. Let A (x, D) =

:",perator of order

II

(l'

we obtain

,y

(6.5)

{l'

I(u, A¢)! S const II¢lle-J. {l' ¢ E: C'; (0).

Now suppose u is a weak solution in fl of Au = f, and also that f E: L 2 (fl). Then by the Cauchy-Schwarz inequality

s

and then we obtain

IID",-(3 ¢ll • {l o

I(u, a", DO< ¢)I::;; const II¢IIE-J,

(a, A* ¢).

I(u, A * ¢)I = I({, (,6)1

53

where the constant depends only on u, a"" and supp (¢). For

Thus, A is elliptic if and only if A* is elliptic. Now we make a definition of weak solution based upon (6.1). For convenience we wiIl frequently write ( , ) in place of ( , ) o. fl' inte~rable in {l and let

s const

.s; const I¢!f-J,

1,81 = f

is just (- l)f a,8; consequently, the principal part of A* is (- l)e AT.

A (x, D) be e-smooth and of order of Au = f if for all ¢ E: C'; (fl)

1/31 = lal - f + i, E: C'; (fl) and lal > (l _ j

I(a, a", D'" ¢)I

fore, if A is s-smooth, S2 E, then also A* is s-smooth, and an easy computation shows that A"* = (A*)* = A.

Definition 6.2. Let a and f be locally

solutions of Elliptic Systems

derivative such that {3.s; a and

'"

II

re~ul8rity of

ein the

I a {x) DO< be an elliptic H=f '"

cube Q and let

8",

E: C~ and satisfy a

ftipschitz condition /a", (x) - a", (Y)!

s K Ix - YI.

jJ.,.et E 0 be the largest constant such that , Eo IgjE.s; IA (0, g)1,

ereal.

~. Then there exists a positive number (V depending on E , n, and E, t O o lauch that if \a", (x) - a", (0)1 s (Vo' then the followin~ assertion is ','Valid.

54

sec. 6

Elliptic Boundary Value Problems

Local regularity of solutions of Elliptic Systems

55

Now since D'" B_ h v h = B_ h D'" v h •

If u € L (Q) and if for all v E: C;' 2

Av)1 ~

leu,

(6.7)

C \!v\\£_l.

(u, AB_ h v h)

Q.

then u E: H 1 (Q) and

IIu \\ 1 • Q

:>; Y (C + \\u I\o. Q ),

r

function on En' so that we have u E:

H~. Let i be fixed and consider

the function B~ u, which we write Bh u for the time being (d. Defini­ tion 3.3). Note that B u E: H~ and h

0h

=

t.,.Applying this formula in the case f = a", and g = D'" v h' (6.10) becomes (u, AB_ h v h ) = I",I~£ (u, O_h (a", D'" v h ) )

Bh u having mean

it is also shown that

i(6.11)

\\vhll£.Q ~ N\\B h u\\o.Q'

where N depends only on E o, n, and £. Since is the completion of C;' with respect to the norm \ I

HE

E:

He.

since v h E:

v the function B_

h

v h: We obtain

I(u, AB_ h vh)1 ~ C I\B_ h

vhlle-l,Q'

2

) •

°-

(u, D'" v h (x - he;) .

°-

(u, D'" v h (x - he

i

h

a )

h

a )

'"

(B h u, a", D'" v h)

H=£

2

H=£

'"

1\£. Q' = - (Oh U,

I(u, AB_ h vh)\ ~ C Ilvh\\e.Q :>; CN IIB h

-

A (x, D) v h )

ullo,Q'

2 (u, D'" v h (x - he i ) • O_h aJ; H=£

" e second equality follows from the periodicity of a",. v h' and u. ext, since A (0, D) v h

By Theorem 3.14 and by (6.8) (6.9)

2

H~£

HE' and therefore we may in (6.7) insert

h

for

= -

-

it follows that (6.7) holds not only for v E: Coo, but also for v E: H£ #. # Vh

D'" v h ).

h

B: h (fg) = f (x) B_ h g + g (x - he;) B_ h f.

value zero; the existence of v h is guaranteed by Theorem 5.1, where

NoW B_

2 (u, a B H=£ "'­

u has mean value zero. Let

He be the unique solution of A (0, D) v h

• (6.8)

(u, a", D'" (B_ h v h))

Now we need a formula analogous to Leibnitz's rule for derivatives:

for two functions f (x) and g (x)

Proof. We first prove the result under the assumption that u has mean value zero, i.e., that udx = O. Then we extend u to a periodic Q

~

"'1=£

=

where y depends only on Eo' n, £, and K.

v h E:

I

=

=

Bh u, (6.11) implies

(u, AO_ h v h ) + (B h

U,

Bh u)

=

-

(Oh u, [A (0, D) - A (x, D)] v h) I10(

2 1_0 -L

(u, D'" v h (x - he;) '0

-h a'" ).

56

Local regularity of solutions of EIIiptic Systems

Elliptic Boundary Value Problems

This completes the proof in the case that u has mean value zero. We now eliminate this assumption. This is relatively simple. In \ fact, (iJo can be left unchanged. Let u E: L 2 (Q) and let 11. (u) =

Now let

(iJ

=

I·

su~

la", (x) - a", (0)\.

r udx.

x€Q,\",\=e

III

57

We now show that u'

=

u - 11. (u), which has mean value zero,

Q

Let p = p (n, f) be the number of derivatives D'" of order \a\ = r.

Then the Cauchy-Schwarz inequality applied to (6.12) implies

satisfies all the requirements of the lemma.

Now

I(u, AO_ h v h)\ + I!Oh u\\~,Q ~ Ilo h ullo,Q (iJp \lvh\lr,Q

(u', Av) = (u, Av) - 11. (u) jAVClx.

Q

+ P Ilul[o,Q K I\Vhllf,Q'

~;The quantity

I (We have used!o_h

a",\ .s; K,

a consequence of the Lipschitz condition

assumed on a",.) Combining this inequality and (6.9), (6.8), we obtain Ilo h ull ~,Q ~ CN \IOh ul\ o,Q + (iJpN

(6.13)

IIOh

ull ~,Q .

Avdx is a sume of terms of the form

Q

~ 'i:lal = e. For each such a there is

~1f31 = e-

ra

Q

+ pKN \lull o•Q

J

'"

=

=

\

\Io h

u\l o•Q·

=

Ilo h ull o •Q ~ CN +(1/2)\joh ullo,Q + pKN lIull o• Qi

oc

,

lim h->o

r a '"

Q

lim h->O

0 1 Df3 vdx h

1 r 0-h a

Q

I r B",

\\Oh ujIO,Q.s; 2CN + 2pKN Ilullo,Q'

Q

1

=

max (2N, 2pKN),

D'" vdxl

~K

• Df3 vdx,

r ID f3 vi

dx

Q

~ K [{ ID{3 vl 2 dx]~ Q

\Io~ ul\o.Q.s; y'l (C + l[ullo,Q.)' i = 1, ... ,n.

.s; K Ilvlle_l.Q'

°

This result holds for all positive h; as u E: H (Q), Theorem 3.15 implies that u E: H 1 (Q) and Ilu\11,Q.s;Yl (C+ Ilu[l o•Q)·

I

oc

ince B", and v are periodic. Thus, the Lipschitz continuity of "plies

thus,

Then, if y'

f3 DID ,

IIO h u\lo,Q'

1/2pN. Then (iJ.s; (iJo' so that after dividing both

sides of (6.13) by

=

r aD, Df3 vdx Q

=

We choose (iJo

an index i such that D'"

C;

1. Thus, for v E:

D'" vdx

r a", D'" vdx for

Q

IrQ

Avdxl ~pK Ilvlle-l,Q'

B",

58

Elliptic Boundary Value problems

Using this inequality in

III (u)\::;: Ilullo.Q'

(6.14),

together with the inequality

we find

sec. 6

Local regularity of solutions of Elliptic Systems

(6. 15)

!(u, A¢)\ ~,C

Then u E: H 1

,

I(u',

Av)!::;:

I(u,

Av)\ + pK

Ilull o.QIlvI1e-l.Q·

(6.16)

Ilull o.Q] Ilvlle-l.

(C + pK

proof. We make several reductions to simpler cases. First, we may assume A coincides with its principal part A I. For if B = A -A I, lben

Ilull o.Q+ Ilull o.Q)·

Ilull l .Q= Ilu' + 11 (u)ll l • Q: ;: \\u'lkQ ::;: l\u'l\l.Q + l\ull o.Q' it follows that

Since

+

I(u,

=

B¢)I

11¢lle-l.D + Ilullo.D C l 11¢\Ie-l.D'

e,

(6.15), II u II o.D'

and M. Thus, an estimate like

,Therefore, if (6.16) were proved in the case in which A has no lower ~er terms, we would have

L a H::;:e 0(

\Iu\ll.D' ~ y

Ilullo.D + IIUllo,D)'

Afid the result would hold also in the general case. iAlso, since the result is of a local nature, it is sufficient to obtain estimate (6.16) with D' replaced by a neighborhood 0 of a fixed

.e

A

i[~:

');P, as long as 0 depends only on the quantities E, n,

e,

K and M. Iod then by a coordinate transformation it is obviously sufficient to ;'1),

10(\ = e:

~at the case x O = 0 and D = Q.

o be the number whose existence is guaranteed by Lemma 6.1

fnd let 8 be a fixed number so small that laO( (x) - aO( (0)1::;: CU o for

\iitLet laO( (x) - aO( (y)1 .s; K Ix - YI;

and let aO( be bounded and measurable for

(C + C l

(x) DO( be uniformly

I~l £.::; IA (x, ~I, x E: D, ~ real.

CU

'~l ~ 8,

and 0 < 8 < 1/4; 8 depends only on K and cu o: Let' be a fixed real function in C~ (Q) such that' (x) 1 for Ixl ..s; 8/2,

la\.::::e:

=

~(x) = 0 for IXI); 8, and 0.5: '.5: l.

laO( (x)1 ~ M.

Let u E: L

I(u,

'~lds with A replaced by A' and C replaced by C + C 1

elliptic in D, and let E be the largest constant such that

Let aO( be Lipschitz continuous for

A¢)I +

where C 1 depends only on n,

We can easily, generalize this result to obtain REGULARITY THEOREMS (Theorems 6.2-6.7).

E

¢)I ::;: I(u, .:::: C

max [y l ' Y1 (pK + 1) + 1]. Q.E.D. (Whew)

THEOREM 6.2. Let A(x, D) =

A'

III (u)1111I1l.Q

lI u \ll.Q ~ Y (C + Ilull o•Q )' where y

(D), and for every D' cc D

where y depends only on E, n, E, K, M, the diameter of D', and the distance from W to an.

Q'

Since u ' has mean value zero and since we have already proved the lemma for such u', it follows that u ' E: H 1 (Q) and

Ilu'll l •Q~Yl

11¢lie-l.D'

\Iull l .D, ~ Y(C + Ilullo.D)'

Then applying (6.7), it follows that for all v E: C;' !(u', Av)! ~ [C + pK

loc

59

; Then for any

(n) and suppose that for all 'P ,/.. E:. Coo (n)

2 0

'#J'",=:

,:i i.,~ ,."

"

'v,

v 'v

€:C';. E: C~(Q), and upon applying

(6.15) with

60

Elliptic Boundary Value Problems !(u, A ('v)

sec. 6

II s C ll'vlle-I.o S CC 2 Ilv11e-I,o'

Local regularity of solutions of Elliptic Systems

61

Since this holds for all v E: C';'. Lemma 6.1 implies that <:u E: HI (Q) and

where C 2 depends only on 'in accordance with Leibnitz's rule. By Leibnitz's rule again A ('v) operator of order less than !(<:u, Av)1 =

s

I(u,

=

e.

Therefore.

l\

+ \(u, B I

As

,=

1 for

Ixl s 0/2,

+ C3

Ilullo,Q+II<:ullo.o]'

we have u E: HI (SS/2) and

<:Av) I

!(u, A (<:v)

Ilulll,sO/2'::;: Y [CC 2 + (C 3 + 1) [Iull o•o]'

vll

(6.17) .sCC 2 =

lI'ulll,o Sy [CC 2

'Av + B I v, where B I is a differential

This result can be generalized to obtain stronger estimates on u if inequality (6.15) is strengthened. For simplicity we shall not be quite so precise in exhibiting the dependence of yon the problem.

l\vll£_I.Q + IluII O• Qc 3 Ilvlle_I.Q

[CC 2 +

and C3 depends only on

c 3 Ilull o•o] \Ivlle-I.O' <: and

Q.E.D.

THEOREM 6.3. Let A (x, D) be a j-smooth elliptic operator of Let u E: H 2 (m satisfy the condition order £ in 0, where 1 S j .::;:

e.

M.

I(u, A¢)I s C 11¢lle-i,D'

(6.18)

We could now apply the lemma, in view of (6.17), except that A

does not have periodic coefficients. This situation is remedied by

redefining A outside the sphere So = Ix: Ix\ < oJ. One way this can

Then u E: H.I

loc

(m and for

all

¢ E: C~ (m.

every 0' CC 0

be done is to let a", (x),

a: (x) = I...

As

a",

([20

B",

(0).

S,

\x\-l Ixl

2:

1] x),

20,

where

0 < Ixl < 28

x E: (J.

a: is constant on JQ, we may extend n

a~

by periodicity to a

l~ B

a# (xl Dot is elliptic and the

<: '" 0 for Ixl 2

I

A# v)1 S [CC 2 + C 3

n.

Ilullo,o] Ilvllc-I.O·

A o ¢)I

Then for ¢ € C~

.: ;: I(u,

A¢)I +

I(u,

(m B¢)j

O. (6.17) may

s C 11¢lle-i,n + Ilullo.D c i 11¢lIe-i.D'

I

I('u,

= A - A o'

leu,

H-C '"

same constant E suffices for A #. Since be written

only on A, 0', and

Il\)r, suppose that the result has been established in such cases. and

'"

~

Ilullo.n),

e.

Lipschitz condition with the same constant K that suffices for a",' Also the operator A # (x, Dl =

y depends

(C +

Proof. The proof is by induction. We have already proved the re­ sult in the case j = 1 (Theorem 6.2). Assume the result holds with Jteplaced by j - L where 2 .::;: j s l.et A (x, D) = ~ a (x) D"'. It can be assumed that A = A ' c' 0 £-i<[ocise '" o

function on E . ObViously, the extended B# E: C~# and satisfies a

I

Ilulli.D''::;: y

(6.19)

Ixl S

So that u e:: Hi loc (m and Ilulli,D' S y (C + C I Where C I depends only on A.

r

~ .•.

~

Ilullo,n + Ilullo,n),

11\111

62

II

sec. 6

Elliptic Boundary Value Problems

1 Thus, we assume A = A o' Since IIc:I>l le-/,n s IIc:1>lle-(/-I).n, (6. 8) together with the inductive hypothesis implies that u E: H

II

/_\°.0 .

III

\

Since

u E: W 1. l~c, ~£

(6.20)

(D I

\

U,

we have for

Ac:I»

= -

Local regularity of solutions of Elliptic Systems

"here C 2 depends only on A and 0.". Combining (6.21), (6.22). and

(6. 23),

c:I> E: Coo° .

(0,)

I(D I u, all c:I> E:

(u, D I Ac:I».

Ac:I»1 ~ (C + C 2

(n").

C~

Ilulli-l.n") Ilc:I>ll e-U-l).nn,

By the inductive hypothesis that the result holds for

jreplaced by j- 1, we obtain that D; u E: Hi~olc

1\

Now

63

(nil);

(6.19) with j

rtPlaced by j - 1 and u replaced by D; u implies

1\

DI Ac:I> = D1 I

a",

D'" c:I> = I

'"

a",

D"" D; c:I> + I D;

'" AD;

=

a", .

D"" c:I>

""

c:I> + B; c:I>,

ccn nceo..

If

c:I> E: C~ (nn).

\(u, AD I c:I»\

Ilulli.n' S; y

(6.18) implies

s C I\D; c:l>lle-J.n" s C Ilc:I>ll e-i .nn. +1

(u. B. c:I» = I

I

h+ I:s;Hse

. D'" c:I»

(u, D. a I

e-J+ I~H~e

I I

I

Ilull/-l.n")·

result now follows from the inductive hypothesis, since

'1Ve ,

Definition 6.3. Let A. (x, D) =

,

i,·', ,

c:I»

(-I )1/3\ (D/3 (uD; a"" ), D"'-/3 c:I»,

e-j+1~H~e

Il uII J- 1 .n")·

(C + (C 2 + 1)

~ators of orders m.I in

'"

(uD 1 a",,' D""

liD; ullo.nn)

Q.E.D. now wish to extend these results to systems of equations.

I

"1

g,;, 0

i '~..,":

I

0., i

=

I

l"'l~m;

a l (x) D'" be differential

""

1.... ,N. This system of operators is

over-determined elliptic system at a point xo E: 0. if there is no

,

\'

+ C 2 + 1)

+

11"11 /- 1 .0." S; Y (C + Ilullo.n)'

Also \1

Ilulli-l.nn

1

l'be (6.22)

y (C + C 2

(Recall that j L 2.) As 0." is arbitrary, we see that (using Theorem 2.4) u E: H. 10 C (0,) and applying (6.24) for i = 1, ... ,n,

\(D/ u, Ac:I»\ ~ \(u, AD/c:I»1 + \(u, B; c:I»l·

Now suppose 0.'

S;

s y (C

and (6.20) implies

(6.21)

liD; ulli-l.n'

(~.24)

such that

-J:\

·.""/,,:"' i;·.·!.·.~.,·. H-m t

a~ (xo) ~

=

O. i

=

t .. ·.N.

.1_,:;.

1

\1

where for each a we choose a single index /3 such that \/3\ = la\ + j - 1 and a - /3 2 O. Since u E: H i-I (0.") and A is j-smooth.

\

_e

!\ \

II I Iii

II

\1

I

I!!

'I

the Cauchy-Schwarz inequality implies (6.23)

I'

,I I

\(u, B 1 c:I»\ ~ C 2

I\u\\j_l.nn \\c:I>\\e-i+l.n"·

!i.e system is an over-determined elliptic system in 0. if it is an over­

d8,termined elliptic system at each point of n. -An

example of an over-determined elliptic system is the pairs of

ope. rators

.ii"._

~

.

0 '"

D 21+ +D2n-l - Dn2, Dn .

64

Elliptic Boundary Value Problems

Local regularity of solutions of Elliptic Systems

65

THEOREM 6.4. Let A 1 (x, D), ... , AN (x, D) he an over-determined

n,

elliptic system in AI (x, D)

where A/ (x, D) is of order m l , and each operator

is s-smooth in

n.

Let u E: L 2

(m,

fJ € L 2

(m,

(u, A o ¢)o s

' p

i ­ I, ...N,

N

I

==

(u,

AI

and let u satisfy

N

¢)

(u, A J

== (fl' ¢), i == 1, ... ,N,

I

DC

A~ (0, D) ,1

(m

and

m-m

neighborhood of a point x 0 €

• P

n; we also take x 0 ==

Ix: Ixl < pl. ==

A o (x, D)

==!

°and let

I

Then A 0 is an operator of order 2m and for real ,; fA~ (0, Ii

I

III 1

11

Iii

I Ii I, t '

I

'Ii

, il

t'>=.!

jA; (0,

(W 1';1

2 (m-m I)

°

[C 1

I

l g

IIflllo.sp

1

11¢1I 2 m- ml ,sp

/Ifillo.s] 11¢11 2 m-i ,Sp P

P

p

N

Q.E.D.

f- 0

1-1

the coefficients of A~ (x. D) are continuous. Thus. A~ (x, f) f- 0 for real'; f- 0 and for small lxi, so that A o is elliptic in the sphere Sp' for some sufficiently small p. For ¢ E: C'; (Sp) the function m- m;

p

IIA1.(0,D),1m-ml¢llos p ' . p

Ilull/n'.$;y[C1! IIf/llos +I/ullos]' • 1-1' P , P is result can be strengthened as follows.

since the given system is over-determined elliptic at O. Now if s == 0 the assertions of the theorem are trivial; thus we may assume s 2 1, in which case the coefficients of A: ( x, D) are continuous, and hence

A: (0, D) ,1

¢)o,s

fortiori, j-smooth, Theorem 6.3 can be invoked. yielding that H 10c (S ) and for n' ccS

i "1

N

j

A 0 is elliptic of order 2m, and since A 0 is also s-smooth, and,

1

II

Ii

:0;:

m-m. A/ (x, D) A; (0, D) ,1

'

C1 1~1

:0;:

D~+ ... +D~, and set

N

~ Ilf/llos

;-1

n.

Let,1 be the Laplacian operator: ,1

A; (0, D) ,1

/ is an operator of order no greater than

/(U,Ao¢)os !:;:C

Proof. Let m == min (m , ... ,m ), m == max (m 1 , ... ,mN ). Since the N o 1 theorem is of a local nature, we need Qbtain the results only in a Sp ==

P

m-m

N

Ilull/,n ' :;: y [;~I 1If;ll o.n + Ilullo.n]'

n',

/ ¢}o s

+ 2 (m-m/) == 2m ­ m/,

n' cc n

where y depends only on A;,

m-m

I-I

for all ¢ € c~ (n). Let j == min (s, ml, ... ,mN ). Then u € HI and for

(0, D) ,1

•

==! (f/, (6.25)

(x, D) A;

/-1

I

THEOREM 6.5. Let A 1 (x, D), ... A N (x, D) be an over-determined

~lliptic system in n,

where AI(x, D) is of order m/, and each operator I (x, D) is s-smooth in n(s 1). Let u €: satisfy (6.25) for ,~tl ¢ €: where f l €: Hi~C(m. If j == mines, m + kW .. ,m + k ),

C;;-,(m, ,,,"Pen u€: H;o C(n), ,~;

2

and if

qoC(m

n' c~ n" cc n,

;;~:

¢ is again a test function on Sp' whence by (6.25)

.

.....

lI ull /.n,:o;: y [1~1 N

Ifll k

/.

n" + Ilull o•n"]'

1

N

N

66

Elliptic Boundary Value Problems

where y depends only on Ai'

ot,

sec. 6

and 0".

(u,

Local regularity of solutions of Elliptic Systems

Ai ¢) == (fi'

67

¢).

k

Proof. Let f l foe == (- 1) I D'" f,J for lal == k l , and let A.J,OC (x, D) == AI (x, D) TJ"'. Then the operators AI,,,, for i == I, ... ,N and lal == k l form an over-determined elliptic system in 11. For if zero, and if x 0 E:: 11, then for some i A, (x o. I

~ k

~

'As Ajhas coefficients in COO(O) and the system A;, .... A~ is over.

determined elliptic, and as f l E: Coo (0), we may take sand

is real non­

:!J:1,.. ·kN in

f- 0; some component

.Q.E.D.

Since (6.25) holds. we have for any test function ¢ in 11 and for lal == k i

," Because of its importance we single out the following A PRIORI

ISTIMATE.

THEOREM 6.7. Let Al (x, D), ... ,A N (x, D) be an over-deter­

(u, A" • '" ¢) == (u, A,I D'" ¢)

tiJined elliptic system in 11, where Ai (x, D) is ml-smooth and of

,9Cder mj' Set m o == min (mi, ... ,mN ). Let 11' CC 11. Then for all

.. E: Coo (11')

== (fi' D'" ¢)

"

l)H

J

:'tor all j == 1, 2, 3,---. Thus, u E: Coo by Corollary 2 of Theorem 3.9.

(say ~r) of ~ is not zero, and so AI (x O. ~ ~r I f- O.

== (-

Theorem 6.5 to be any positive integers. Thus, u E: H loc (0)

0

(D'" f l , ¢)

Ilu 11 mo '

== (fl."" ¢).

N

11''::;: y

(I~I IIA I ull o.11'

+

Il ull o.11']'

i

where we have used the fact that D'" ¢ is again a test function. Thus, u is a weak solution of the system A,1,0( u == f,I,CX . Since AI Joe is of order 1

I "

mi

+ k l , all the assertions of the theorem are immediate con-

~.be1'e y depends ~}t

~~

'",smooth. Thus (u, Aj ¢) == (Ai

III

I,

II

i! '! I I

i

I Ii' I

:1

*

U,

¢) for all ¢ C C; (0). There­

~re Theorem 6.4 yields the estimate desired, since in that theorem '.. .•.•.'.'*- mo' Q.E.D.

.1;'1, THEOREM 6.6. Let Al (x, D), ... ,A N (x, D) be an over-deter­ ·.·I~. '\C.. orollary. Let A (x, D) be an m-smooth elliptic operator of order m i and let Of CC 11. Then for all u E: Coo (11')

mined elliptic system in a, and suppose that each operator has in­ 0 finitely differentiable coefficients. Let u E: L loc (11) be a weak ,i· 2 solution of Ai u == f l, i = I, ... ,N, where fi E: coo (0). Then u E: coo (0), Il u ll m .11' ~ y (li Au ll o.11' + Il u ll o.11')' "WIf~~: if u is redefined on a set of measure zero. 're y depends only on A, 11', and 11. Remark. A differential operator A (x, D) with infinitely differentiable coefficients is said to be hypoelliptic if every weak solution u of conclude this section Hans Lewy's example of a linear differ­ Au == f for f E: Coo (0) must also be infinitely differentiable. Theorem ,.Ual equation having no solution will be presented. If the data in 6.6 implies that elliptic operators with infinitely differentiable co­ equation were analytic, then the Cauchy-Kowalewski theorem efficients are hypoelliptic. pld imply that an analytic solution existed. Therefore, some of date in the example must be non-analytic; moreover, all the data Proof. By definition of weak solution, we have for all test function' ! the example will be infinitely differentiable. ¢ on 11 sequences of Theorem 6.4. Q.E.D. The following important corollary is now immediate.

"~Ii

I

I

rl'rool. Since Ai is mj-smooth, A j has an adjoint Ai which is also

n

.~:!

II

only on A,. 11', and 11.

~o

t '~

..

J

68

sec. 6

Elliptic Boundary Value Problems

Local regularity of solutions of Elliptic Systems

rr

The example is the equation

II

(6.29)

i (x + ix ) ~ + 1 .au 2 3 ax 2 ax

(6.26a)

I

1

2

o -p

+ 1 i ~ = f (x ) 2 ax 1 '

(217 u [- i 0

yt e l () ~

3

1III

Assume that u (xl' x 2 ' x 3 ) is a weak solution of this equation in a

(u r-i (x 2 + ix 3 ) D l ¢ -

Y2D 2 ¢ -

III

(x

(6.27)

I

I'"

D

,J..

2'+'

o

= 2x

iM

2(Jr'

r Iu (r, s)1

D

,J..

3'+'

= 2x

(j!f!

3ar'

while the Jacobian of the coordinate transformation (6.27) is

2

dsdr':;; 217p

J

217 ( o

C'; (R)

do not necessarily

O.

'=

s, ()) de.

lu (r, s,

p P

0

())!2 d(),

r (

-p

217

lu (r, s,

8)1 2

d()dsdr < 00,

0

."~d

from Fubini's theorem it follows that the integral in (6.30) is iV,bsolutely convergent for almost all points (r, s) E: R, and that (~V E: L 2 (R). Also, (6.31) implies :~;ir·

~y

rf

'%1,(6.32)

(

p

(

p

r

­ 1 I U (r, s) 1 2 dsdr < 00.

o-p

J~~ Integrating first with respect to "j~ ..•

I

f (s) ¢ (r, s) dsdr.

so that

R

where ¢ (r, s) is any infinitely differentiable function vanishing for lsi 2 p or r 2 p, where p is a suitable positive number depending on the neighborhood where u is defined. Since (6.28)

r

yt (217 e'. e u (r,

IU (r, s)1 2 .:;; 217r

(6.31)

SIn

and we shall consider only those test functions having the special form

I

d()dsdr

Note that

¢=¢(r, s)=¢(x;+x;, Xl)'

I'

"=

(),

= =

U (r, s)

, (6.30)

< y~ c~s l yr e, x3

aR where

vanish on the portion of Now let

= s,

~

o -p

such functions. Note that the elements of

Y2iD 3 ¢] dx,,= (f¢dx

for all infinitely differentiable ¢ vanishing outside a sufficiently

small neighborhood of the origin. Now we introduce the coordinate

transformation

]

The equality (6.29) holds for all infinitely differentiable functions ¢ (r, s) defined for r 2 0 and having compact support in the set R = I(r, s): 0 ~ r < p, - P < s < pI; let C'; (R) denote the class ot all

neighborhood of the origin. That is, assume that u is square inte­ grable in a neighborhood of the origin and that (6. 26b)

yt e l () ~ (Jr

r

(p

= 217

where f is a real-valued, infinitely differentiable function of x l '

-

as

e in (6.29) and using (6.30),

~.

a (Xl' x 2 ' x 3 ) 1 a(s, r, 8) '" '2 '

,·,(6.33)

( U (r, s) [- i R

(6.26b) combined with (6.27) and (6.28) implies

--

69

iM - iM] as

,.."all ¢ ( Coo (R). " 0

...l.

_

(Jr

dsdr = 217 ( f (s) ¢dsdr, R

[111 I

I

70

Elliptic Boundary Value Problems

sec. 7

Now let L = I(r, s): - p < r ~ 0, - p < s < pl. We shall extend U (r, s) to the region L U 'R, forgetting that r previously was allowed to have only non-negative values. The extension is given by the formula

U (r, s) Now for

t/J

= -

=

r U ( -r, s) [ -

i

-< Since the operator ~ + i ~ is elliptic, the regularity result of ar

,!

j.

(Recall that { (s) is assumed to be infinitely that U is known to be smooth, (6.32) implies itt'that U (0, s) ':= 0, -p < s < p. l~l<'." ~i Let {(s) have the special form {(s) = (217)-1 g' (s), where g is a ;:rreal-valued infinitely differentiable function. If V (r, s) = U (r, s) '~, + ig (s), then V E: Coo (L U R) and (6.36) implies

-1fI( ­ r,

s), r;<:

0; then ¢ E: C'; (R).

s) ­ ~ (- r, s)] dsdr

i!sP (- r,

ar

as

(6.37) -

as

>Theorem 6.6 implies that U E: COO (L U R), if U is suitably modified

:Won a set of measure zero.

The relation (6.33) implies that

L

71

\~~differentiable.) Now

U ( ­ r, s), r;<: O.

C'; (L) let ¢ (r, s) €

G~ding's Inequality

=

277

r f (s) ¢ (-r, s) dsdr.

av

+i

ar

av

==

as

o.

L

!,.But (6.37) is just the Cauchy-Riemann equation for the real and t'imaginary parts of V. Therefore, V is analytic in L U R. Since ~,;,u (0, s) == 0, it follows that V (0, s) ':= ig (s), and thus g (s) is :lkanalytic, "-p < s < p. Thus, if g is any non-analytic, infinitely ,\differentiable, real-valued function, the equation (6.26a) has no weak f,;'solution in L 2' .

Therefore,

- r U (r, s) [ + i as fl:I!.

(r, s) ­

L

fl:I!. ar

=

(r, s)] dsdr

-2 17

r f (s) t/J (r, s) dsdr. L

'~)t·

:~t:~ t:
.:fA:, "I;';,

7. Gording's Inequality

k'

Conjugating both sides, it follows that since f is real (6.34)

r V (r, L

s) [ - i ~ - ~ ] dsdr as ar

=

277

r f (s) t/J (r,

':'15 The purpose of this chapter is to prove an important theorem of

s) dsdr.

L

f::.G~rding which

'i.·:.·.r.8;.......i.~; ..

will be used in the global existence theory in section

From the standpoint of motivation, this theorem should appear with

i rrthe existence theory. Therefore, the reader who is meeting this mate­

Combining (6.33) and (6.34), we obtain the relation (6.35)

r

LUR

U(r,s)[-i~-QP]dsdr=277r as

ar

f(s)¢dsdr,

LUR

which holds for all ¢ € C'; (L U R). But (6.35) just expresses the fact that U is a weak solution in L U R of the equation (6.36)

au + i au ar as

= 277f (s).

:. l'al for the first time might well proceed to section 8, assume Theo­ "':"'m 7.6 when it is needed there, and return to the proof at aome con­ 'enient moment in the future. Before giving the statement and proof of Garding's inequality, it ill be convenient to give the proof of an interpolation theorem simi­ i!Uir to, but simpler than, Theorem 3.3. Also, Poincare's inequality ill be given. LEMMA 7.1. There is a constant Yo depending only on n and such that {or any

1)

I¢IJ~

t

---

g

n

.::>:

Al. _

f

> 0 and any

Yo (f

m

j

-

A..

'I-'

I¢I;, En ~

E: C m (E ) 0

n

+ f- 1¢1~.E j

n

),

j = O, ... ,m.

n

G"arding's Inequality

Elliptic Boundary Value Problems

Corollary 1. There is a constant Yo = Yo (n, m) suc~ that for

Proof. We shall use the notation 'f for the Fourier transform of

< ( ~ 1 and for ¢

¢:

1> (.;) = (21Tr n/z r E

¢ (x) e-iX'f, dx.

~

(27T)-n/z

=

k

E:

C;;

(En)

11¢11~-1.En ~ Yo « I¢I~.E + (1-m I¢\~.E ).

n

n

Integration by parts shows that for'/" E: C 01 (E n ) 'I' D ¢ (.;)

73

r

'En

n

Lemma 7.1 is really a convexity theorem since the range on \bounded above. Indeed, we have Corollary 2. as follows:

D ¢ (x) e-iX'f, dx

f

is not

:.i!'.~.•'.'. Corollary 2. There is a constant y = y (n, m) such that for all .'!W' 1 1 ·~~,y .~:"'·"·.A".. E: m (E n ) 0

k

c

= (21T)-n/z

I ¢ E

k

n

~1

J~:

la I =

j ~

ro,

\D'"

¢\ o.E 2

{t (.;).

n

r e'" i¢ (.;)1 E r

2

t:2'"

S

r

En

~

:i~t'}

. ,~t;

A

'i,;t.:,

r

1f,\2>(-1

\¢'1 2 df, + (m-J

l

. ab

!t~,

\¢ (.;)\ 2 df,

e'" \~ (.;)1

2

df,

r

If,\2>(- 1

~·, j

:'I:>r a-

1

c",. ',.• .•.

=

a

' •. I . :

.

(-J

~

n

I

df,

+

~

n

it'hen the two terms on the right side of (7.1) are equal and (7.2)

~1esults. Q.E.D.

In fact, Corollary 2 implies Lemma 7.1: this is trivial if j = 0 or

/i;,l = m; otherwise, use Young's inequality

n

\f,12~f-1

~

2 /m 1,/..1- 2 /m _ 1,/..1 't' O,E 't' m,E

f -

Ji:,:

1\

=

],/..[<m-J)/m . - 0, ... ' ,m ' J 'I' O.E n

Proof. Let

';~Jh:

Then Parseval's identity gives 1II1

n

~;~(

q'

(i.;)'"

!,/..IJ/m 'l'm.E

',.:.;'

Therefore, if ¢ E: C~ (En) and =

1,/..1 'I' J.E ~ Y1 n

j;("7' 2) ~:: .

t:I (SJ'

't: ,....

= ISk¢

0(.;)

.,~j:;:

(x) if, e-iX'f, dx

1f,\2m

1¢1 2 df,.

l5{,

[al'" + {3-1

a-1

+ {3-1

\b\{3

1, a, {3 > 0, taking

=

2J /m 1¢1 m.E

b

(J<m-J)/m,

n

=

2(m- J)/m 1¢1 O.E n

(-/(m-J)/m

'

/

'fi}~.

a = mjj, {3 = m/ (m - j).

~

KWe shall say that 11 has bounded width ~ d if and only if there is a •...'.. ne I such that each line par.allel to 1 intersects 11 in a set whose

il"

I1III

By another application of Parseval's identity,

).. ameter is no greater than ID'" ¢I~.E ~ (-I t¢\~.En +y(m- J 1¢\~.En'1;}tiEQUALITY. Ii

n

,

LEMMA 7.3. If 11 has bounded width ~ d, then 1¢I .11 ~ yd m -/ J for all ¢ E: (11), 0 j m _ 1, where y is a constant de-

;~~l,n.11

_______________.k.

whence the lemma follows. Q.E.D. ii"

C~

dint only on m and n.

11111

d. The following lemma is the POINCARE'

~ ~

74

sec. 7

Elliptic Boundary Value Problems

n

tween x O and x O + q. By defining ¢ to vanish outside

n,

=

Then f (0) f (t)

¢

for 0 ~ j::::m -1. Q.E.D. Although the Poincare' inequality, as it stands, is completely suffi­ cient for our purposes (indeed, we shall need the inequality only in " the case that n is itself bounded), for the sake of completeness we 1. present the following generalization:

+ t

Iq\-l

q).

0, so that

=

=

(X O

J

t

1¢l j • n :::: yd 1¢IJ+1.n

we can

assume ¢ f: C~ (En)' Let f (t)

LEMMA 7.4. Assume that there are positive constants 8 and d I.. such that for every x f: n there is a point y for which Ix - yl ~ d and i. {z: 12 - YI < 81 n n = 0. Then there is a constant c depending only

fl (r) dr.

°

on nand d18, such that

By the Cauchy-Schwarz inequality, .

If (t)1 2 ~ t

(7.3)

t

J

f' (r)1

°

2

dr ~ d

1¢lj.n <;:. (dc)m- j 1¢lm.n,

r

00

If'

(r)1

2

dr. (Proof. For any n-tuple a = (a1, ... ,a n ) of integers, let !n-'l,akd

00

If (t)1 2 dt

d

=

Now express

J 0

-00

I¢I ~.n

If (t)1 2 dt ~ d 2

r

00

If' (t)\2 dt.

-00

as an iterated integral with one of the integra­

tions taken in the direction of I. From the inequality above it follows that

1¢1~.n:::: d

2

<xk

:::: n-'I,(a k + l)d,

Summing over all i, we obtain

\¢I~.n:::: 2d 1¢1~.n· 2

i

i

I

.~.mlll

Proceeding in this manner,

=

Ix:

k = l, ..• ,nl. Then En = ~ Ocr.' We shall

n.

Assume that ¢ = 0 outside

n.

Assume that x' E:

Ocr. n n.

Ocr. n n, and let y

the corresponding point guaranteed by the hypothesis. Since for

1¢1;.n·

\Di¢I~.n::::d2IDj¢I;.n:

Ocr.

prove something similar to the last estimate with n replaced by !'Then, by summing over all a, the lemma will follow for

Applying this inequality to D i ¢,

II

0 ~ j :::: m - 1,

-00

Hence

r

75

°

Proof. Let 11 b~ a line parallel to I, and assume that x and x 0 + q are points of I' n such that l' n is contained in the segment be­

an

G;;rding's Inequality

Oa

76

Elliptic Boundary Value Problems

x E: Q"" Ix - xII < d, S = lz: ly - z\ < 01.

the triangle inequality gives

Ix - yl < 2d.

Let

Now integrate 1c,b12 over Q", - S; the integral can be expressed in polar coordinates about y as

r

r r2d \c,b!2 rn-

\c,b1 2 dx':::;

~

0",-5

1

sf

rn-

1

.:::;

r2d o

2d

I

drda,

1

2 dr .

ar 1

S 2d (2d)n-l

.:::; (2dr 01- n

n

r

2d

o

I

fM

ar

2

r

n

1

which constitute

Q",

!c,b/2 r n -

1

i~:;~uality over all a, we obtain

Ip/,;

-1

~;:

1c,b1~,E

s(2d)n+l o l- n N lc,bli,E' n

n

dr l'

1

dr.:::; (2d)n+ 1 01- n

2d

r 0

I

fM ~

1

2

r n - 1 dr.

Since ac,b/ar is a particular directional deri vative of c,b, it must be bounded in magnitude by the gradient of cP; that is,

IfM 12 S ! ID i ar

B (cP) =

1c,b1~.Q

=

J

f a {3 (x) D'" c,b D{3 c,b dx.

!{3 Sm

i-I

'"

~

I'" Ism n '"

c,b12 •

e shall use the notation

It therefore follows from the preceding inequalities that

I:

~~,

1c,b12 dx

~

fa {3 (X)D'" c,b D{3 c,b dx

l"'l=m n '" 1f3 =m

..t~__ ~;

Q",-5

B' (c,b) =

f

;fot the principal part of B.

i11ii

j!ll. _ _~

drda

dx

I

Q{3

depends only on n. Summing the above in­

Thus,

2d

rn- 1

I ID. c,b (xW

where ac,b/ar1 is the derivative of c,b in the radial direction from y.

r o

c,b12

i-I

~i.

dr 1

12

f

O.::;I,,-YIS2d

ID i

.::; (2d)n+l 8 1 - n 1c,bli,Q",'

'~{Q{3 1

i-I

:~~ the center of Q", is less than 4d. The number N of disjoint cubes 0\,',

1

r2d I ilp.o ar 1

n

~

,~~here ct is the union of all cubes Q{3 such that the distance from

1

r ­

2d

0

= (2d)n+l ol-n

0

fM

(2d)n+l 8 1- n f

~

where a is the spherical measure and ~ the surface of the unit sphere r = 1. By (7.3), we have for 8.:::; r ~ 2d

1c,b[2

n

G~rding's Inequality

_

78

G~rdin~'s Inequality

Elliptic Boundary Value Problems

Definition 7.1. The quadratic form B (¢) is uniformly strongly

n if there and all x ~ n

elliptic in

real t

.'R

(7.4)

I

"'l=m

exists a positive constant Eo such that for all

~

.'RB

t-'

C; (0) is the class of (0) having compact support in n.

One bit of notation must be introduced: functions in C

THEOREM 7.6. (GARDING'S INEQUALITY). Let n be any open set, and let B (¢) be a uniformly stron~ly elliptic quadratic form on n with ellipticity constant Eo' Assume that 1°

2°

a"'fJ (x) is uniformly continuous on

n, lal

a"'fJ (x) is bounded and measurable,

lal

+

=

If3I

(¢) ~

yE 0

"II" · ,t

(¢) ~ Yo Eo 11¢11;'.n -

-

x~1!

!

laHfJl::;2m

=

If3I

in this case. By

m we have (integrating on En)

=

"~

'" Jl;+fJ I¢ (t)1 2 dt.

't

:,'J'hus, fRB (¢)

~

==

fR

==

J I¢ (tw

a"'fJ

H==lfJl=m

Jl;+fJ I~ (t)1 2

d(

IfJl .s; 2m. fR

!

a"'fJ l;+fJ dt

H=lfJl=m

. ,y the ellipticity assumption on B (d. (7.4) ),

flm,"i;"

Ao \\¢I\~.n

.'R

Eo' the modulus of continuity of a "'fJ (x) for

lal

n = En

JI)'" ¢ DfJ ¢ dx ~ JD'" ¢ DfJ ¢ dt

= m;

Here Yo depends only on nand m; and Ao depends only on n, m,

sUQ

IfJl :S 2m.

1¢1;'.n.

[,Parseval's identity, for

for all ¢ ( C;;' (0).

I

+

Proof. It is obviously sufficient to take

Then there are constants Yo > 0 and '\ ~ 0 such that

fRB

lal

;;lri

Eo Itl 2m •

The lar~est constant Eo for which (7.4) is valid is the ellipticity constant of B. m

for

LEMMA 7.7. Assume that B = B' and that B has constant co­ efficients. Then for all cp ( C;;' (0)

a",R. (x) ('+fJ

lt31=m

(x)1

,sup laafJ

:'ll:~n

79

la I = If3I = m,

!

H=lfJl=m

a"'fJ 1;+ fJ 2 yE 0

!

H=m

e"'.

erefore,

and

la"'fJ (x)!.

.'RB (¢b yEo

J I¢ (t)1 2

!

I"'[=m

e'" dt.

The proof will be through a (terminating) sequence of lemmas. In all of these lemmas it will be assumed at least that the hypothesis ,)nally, by Parseval's identity the last inequality is just ~fli ' of Theorem 7.6 is satisfied by the form B. We shall in the remainder of this section use the notation y for a~{\: fRB (¢) ~ E 1¢1 2 Q.E.D. positive generic constant depending only on n, m; and the notation '~~" y 0 m,E n • K for a positive generic constant depending only on n, m, Eo' the ~

modulu, of conlinu;ly of

a.fJ (xl £0< 1"1" \fJl

~ m, and

1..

_

80 ill

!

Elliptic Boundary Value Problems

n

Corollary. If B = B' and B has constant coefficients, and if satisfies the condition of Lemma 7.4, then there is a positive con­ stant c depending only on n, m, d, and 0, such that for all ¢ €: C~ (m

l

(¢) ~

cEo

11¢11';',n.

Thus, (7.5) implies that 1

,

P.B(¢) L (yEo - K()!¢I,;, - KV +

; {Now choose

i~:~;

Proof. From Lemma 7.4

(=

~'

'I

I,

81

*~:

·.I,:,:Ji"•• '

P.B

G~rding's Inequality

sec. 7

l)11¢11~_1'

yE /2K. Then

P.B(¢) L

Y2yE o l¢l,;, -

KII¢II~_l

=

Y2y E ol I¢I I';'

-

"~t':-

II¢Wm, n = )-0 ~ I¢I ],~ n~ j-a ~

II ,

,~

·!¢12m, n·

(dc)2(m-n

;~~

Thus, the corollary follows from Lemma 7.7. Q.E.D. LEMMA 7.8. Assume that B' has constant coefficients,. Then

~~:,

P.B

(¢L:::. y Eo 11¢\I';',n -

K

(¢) =

B'

(¢)

C

'~~'

I

';~l

~

+

H+lf:3I~2m-1

lal

+

D'"

¢ DfJ ¢ dx.

1f31 ~ 2m -

1

!.r

P.B (¢)~P.B' (¢)-K

111

1

-

KII¢II~·

!¢I,;, -

11¢ll m - 1 \¢I m

-

K

11¢11,;,-1'

the latter inequality being a consequence of Lemma 7.7. For any numbers a, b, (, with ( >0,

pr~ncipal

.'I" :~, '.:;'constant f,' .,' . LEMMA 7.9.that Assume that B = B p such ",

I.

Then there exists a pooitive

,,','

'\',

, .

P.B(¢) ~ yE

o

II¢Wm,n

'or all ¢ €: C~(m such that the diameter of supp (m is less than p. f1'he constant p depends only on n, m, Eo' and the modulus of continuity ~-'f the coefficients aaf3'

11¢ll m - 1 1¢l m-K 11¢11';'-1 K

G~rding's

This lemma completes the proof of inequality in the case "Ithat the part of B has constant coefficients. For the general ';,{'<:ase, we fust prove the theorem locally. il\i,r ,

Thus,

1 1'1

~yEoll¢ll;,

P.B(¢) 2

\~Q.E.D.

J a"'fJ (x) n

1£ a"'f3 (x) D'" ~ DfJ ¢ dxl ~ K 1¢IH,n 1¢llfJl,n' ,

~ yEo

+TJ1-mll¢II~).

; h:;

By the Cauchy-Schwarz inequality, for

C7.5)

Y2yEoll¢II,;,-K(TJII¢II;,

P.B(¢)L

I:' If we set TJ = yE 0/4K, then

11¢11~.n·

Proof. We have B

+ Y2yE o)II¢II';'_l'

I;CBy Corollary 1 of Lemma 7.1, since ¢ can be considered to be in ~iC~CEn)' we have for 0 < TJ ~ 1

;1:

for all ¢ €: Cg' (m,

(K

..

"Proof. Since the coefficients are uniformly continuous, for every ,JPositive ( there exists a positive number p such that laafJ(x) _

~;aafJ(Y)1 < ( if Ix -

yl < p, x, y :€ n. If ¢ :€ cg'cm and the diameter of ii"supp (¢) is less than p, let x O ~ supp (¢) and let

,c.;)

1

I'll ,'ill1'1

III

I,

2 labl

~ ( lal 2 + (-1

Ib1

2•

"

,

[i"

B" o(¢)

= I

~

lul=lf3j=m

aufJ(x

O )

Jsupp(¢)

U

D ¢Df3¢dx.

IIIIII

I

82

Then by the Cauchy-Schwarz inequality

'1111

::RB(p) =::RB

L:RB

!II

"

o(¢) +::R

J

1

lal=I,BI=m supp(¢)

G~rding's Inequality

sec. 7

Elliptic Boundary Value Problems

(7.9) :RB(¢) L a [aa,B(x) - aa,B(xO)) . D ¢Df3¢dx.

83

yEoll¢II~.n - K(II¢II;',n - Kt- II I¢II;'-I.supp(¢)'

;~;~; By Lemma 7.3

11¢llm-l.supp(¢) ::; ypll¢llm,supp(¢)

o(¢) - ( I IDa¢l onI Df3¢1 0 n x 14=!t31=m ' .

=

ypj I¢II m ,o,

so that (7.9) becomes ?:RB x

a(¢) - K.tll¢II~,o·

:RB(¢) L (yE 0 - K( -

Applying Lemma 7.8 to the form B x (7.6) becomes :RB(¢) 2 (yEo - K()II
°

which has constant coefficients,

Kt- 1 p 2)11¢11;'.o.

Now simply choose ( = yE o/2K and then p2::; tyE o/4K. Q.E.D. The follow ing lemma is similar to Theorem 1. 9 in that it stafes the existence of a slightly modified partition of unity.

'

- KII¢II~,n'

LEMMA 7.10. Let F be a compact subset of En and let v Feu 0., where each 0/ is open. Then there exist functions (. ~ .~

Now choose (

(7.7)

=

yE o/2K, so that

:RB(¢) L

j'#

Choosing

(J

I il

il

1 (/x) 2

1 and

=1 for x ~ F.

Proof. Choose 0:. 0:' such that 0; cc 0: cc 0;, i .~ I, .. . ,v, and , v jFC ,U 0:'. Let (: ~ C:;'(E), i = 0, 1, ... ,v, such that: 0 s; (: s; 1 for ,- j

v

all i·' ('(x) 0

(V2yE o - K y 2p 2m)(\cbll;'.n·

= -

v

Ion i~UI 0"· supp (,0 C /_Uj 0', and for 1 -< i -< V , J' i'

(' =

i­

on OJ'. supp ('., cO ,

.. Then for 1 ::; i ::; v let

(/x)

Proof. Assume ¢ ~ C~(m and supp (¢) has diameter less than p. where p is the constant guaranteed for the form B' by Lemma 7.9. Then, by the lemma,

1

{

lal+If3I::;2m-l'su pp(¢)

=

v Yo (:(x) [}1 (/(x)2 + (1- (~(x) )2]- 2

,I',Since the quantity in brackets never vanishes, ( "

1',;1- (~(x)

=

0 for

x~

F,

f

(x)2

=

1 for x

;-1

~ F.

~ C:;'(O);

and since

Q.E.D.

An easy modification of the above proof gives a (3(x)Da¢Df3¢dX

a

,

LEMMA 7.11. Let d> 0 and for each n-tuple a

~; integers let

=

(a

1

, ... ,a)

~'

(7.8)

2

1

sufficiently small, the result follows. Q.E.D.

:RB(¢) = :RB'(¢) +

I

,II

I

:s;

Z

i-I

Corollary. Lemma 7.9 remains valid if we omit the assumption thatB=B'.

III

l/

I

C:;,(O /) such that 0::; (i

V2yE oll
Finally, since the diameter of supp (¢) is less than p, Lemma 7.3 implies 11¢ll o .s yp mll¢lIm' and then (7.7) becomes :RB(¢) L

1

YEoll
Using the inequality 2ab::;

ta 2

+ (-lb 2 for a, b 2 0, (7.8) becomes

I

J t~

.: l

Q", =

Ix:

d(a k

-

1) < x k < d(a k + 1), k

Then theee exists a function (

< C~( Q,)

=

1.... ,nl.

such that 0

~ ( < 1 and

of

84

sec. 7

Elliptic Boundary Value Problems

GJrding's Inequality

85

R(¢) =

1 ~(x - a) 2 == 1.

'" That is, if ~"'(x) == ~(x - a), then ~'" ~ 1(",(x)2

=

C';;'(Q,J

1 I : 1a

and

00

m -m

IfJ

1.

.~

t

1

1

1 ysa

1 8sf3

y,Ja

o-jf3

f aaf3 (a) 8 11 y (~)Da-y( . DY¢ .

' .

°

D ¢dx.

'" Proof. Let ~' ~ C~(Q 0) be chosen such that (x) == 1 for x ~ 3/4Q . Then set o

Thus, by (7.10)

IR(¢)I s K

(7.13)

(x) = ('(x)[l~(x - a)2)-'I,

III

'"

III I

As in the proof of Lemma 7.10, the functions (x - a) satisfy the requirements of the lemma. Q.E.D.

We are now ready for the

II

,'t;

1-1

~ f

II1

IR(¢)I

=

1

KI!¢ll m.11II¢llm-1.11·

f ~

lal=!f3I=m 11 '-1

=

2"

f

11

IDY¢/ IDO¢ldxj

sK

1

f

/yl+!ol.s;2m-1

11

[DY¢! IDO¢ldx.

:l:By the Cauchy-Schwarz inequality

IR(¢)I {(7.14)

B'(¢)

IDY¢/ IDO¢ldx

Q,

[using this relation in (7.13),

Now

I

Thus

t

'-1

ID"'(I s K, lal sm.

:KB(¢b :KB'(¢) -

/00

IIU aQ., a set of measure zero.

Now let ¢ ~ c~(m. Then by (7.8) (7.11)

IDO¢ldx.

Now:the cubes Q. overlap in such a fashion that any fixed point is 2" distinct cubes, except for points on

I,

(7.10)

1 ~ f IDY¢! lyl+1 8Is2m-1 '-1 Q,

~!.cpniained in exa~tly

Proof of G~rding's inequality. We shall apply Lemma 7.11, taking the cubes Q", of side d, where d is chosen such that the diameter of Q", is less than the number p of Lemma 7.9; thus ynd < p. Enumerate the cubes Q", and the corresponding functions (", in some order: Ql' Q2"'" and ~1. ~2'"'' Since the functions ~, are just translates of (1' we have the estimates

II

a

s. K

1

lyl+18jS2m-1

IIDY¢llo.11IIDo¢llo.11

.s; KII¢lI m.11II¢ll m- 1.11'

(x)2 aaf3 (x)D ¢Df3¢dx

1-\1'hus, (7.12) implies

1\

I:

II 1

1II

1

la/=If3j=m

~ f

,-1 11

11

1'

11

'r

(7.12)

III

] ~I

II' I11II

Df3-li~

where by Leibnitz's rule

aaf3Da«(¢) Df3«(,¢)dx + R(ep),

,(7.15)

J ;!4~

:KB(¢b.

~ :KB'(~,¢)-KII¢llm.11II¢lIm_1.11' ,-1

86

G~rding's Inequality

EIliptic Boundary Value Problems

:RB'(~A») ::? yEo II ~A)II;'.n·

lal'::::m 1f3I~m

Hence, (7.15) implies

(7.16)

:RB(ef>)2. yE

0i_!I lIC:i ef>II;',n -

KIIef>lIm.n 1/ef>II m-l.n .

1°

If for alI ef> €: C:;"(n)

(7.17)

:RB(ef» 2 collef>II:',n -

shows that ~

1-1

n

,­

~ 0=

1

1

\c
J IDC<'ief> \ 2dx n

(. then for almost alI x €: p

':;(7.18)

1

H:S:m

! J, i-I

n

1

J

n

21 DC<ef>12dx + R /ef» I

\DC<ef>\ 2dx + R 1 (ef»

= 11ef>11;',n + RI(ef»,

2°

'(7.19)

(7.14):

i___

n and for alI real e 1 ea.

aa{3(x)(l+f3 2. Co

If fJr

lal=m

an';p €: C';;'(n)

IllV/;) 12 collef>II;',n -

Aollef>II~.n,

n and for alI real e

:;l 'L~"~

I

1

a f3 aaf3(x)e + , 2 Co

lal=If3I=m

1

e 2a.

lal=m

::w moreover n is connected and a af3(x) €: CO(n)

jR 1 (ef»1 ~KIIef>I\m.n 11ef>llm-l.n·

for

''lhen there exists a real number () such that for alI x

~'teal ~,

Combining these results with (7.16),

e

.21) :RB(ef»

1

(then for almost all x ~

;(7.20) where R 1 (ef» is a quadratic form satisfying an estimate similar to

:R

Aollef>II~.n.

lal=l$/=m ~..

H~m

n

Let co> 0 and Ao ~ 0 be given constants.

The same type argument as was used to derive (7.15) from (7.12)

1 11'1ef>11;,

J aa{3(x)Daef> Df3ef> dx.

1

B(ef» =

87

~ yEol\ef>\I;',n -KIII1 m-l.n·

As in the proof of Lemma 7.8, this implies the required estimate in Theorem 7.6. Q.E.D. We shall conclude this section with a proof that Garding's inequal­ ity is very nearly equivalent to the ellipticity of the quadratic form

B.

:R (e l () 1 '\ lal=If3I=m

aaf3(x )e a +f3

)2 c

lal = 1f31 = m, ~ n and for alI

1 ea,

lal=m

'here c is some positive constant independent of x and e. . Proof. Let eland e 2 be any non-zero real vectors and let t/J 1 '

~~y, 2 €: "

C';(n) be such that supp (t/J I) and supp (t/J 2) are disjoint. Set ef>(x) = t/Jl(x)e ITe · X + t/J2(X)e iTe · X •

THEOREM 7.12. Let aaf3(x) be bounded measurable functions in 811 open set

n for lal ~ m, 1f31 ~ m,

and let

(We shall apply either (7.17) or (7.19) to the function ef> and let

T ->

~.

88

sec. 7

Elliptic Boundary Value Problems

G~rding's Inequality

So in computing B(¢), \\¢11';',,2' and 11¢11~.0, it will be necessary to

keep track only on those terms containing the factor T 2m . Thus, since supp (l/J1) nsupp (l/J2) = 0, Leibnitz's rule implies

I ~ J a fJ(X)(iT~i)OC l/J .(x) . lal=lf3I=m i-lOa }

B(ep) =

2

= r 2m I

c·)a+ (,,,J

I

i-I

fJ

lal=lfJl=m

II¢Wm,~£ n=T

+ ...

}

J aafJ(x)Il/J/x)I2 dx + ... , 0

where the other terms are of lower order in (7.23).

(iT~i)fJ l/J .(x) dx

T

that 2m. Likewise,

2

2m

I (~i)2ocJ 1l/J·(xWdx+ ... ; 11¢11~0=0+'" OJ •

i-I

Assuming (7.19), if we use this special function ¢, the estimates (7.22) and (7.23) imply, after letting T --> 00, that (7.24)

I

t

(~i)a+fJ J

I

}

lal=lfJl=m

2

I

(~i)20<

/-1

J

0

1l/J/xW dx.

l/J /x) = {

Ix - xii < p.

,

i

0,

Ix - xii 2 p,

where P.} is a non-negative number. If x 1 and x 2 are in the Lebesque sets of all the functions aafJ(x), lal = IfJl = m, then applying (7.24) to the special functions l/J.} and letting P --> 0, we obtain (7.25)

2

Ii-I 1

p

2 /

I

(~i)2OC.

i loc\=m

To prove (7.20) let x 1 be any point in the Lebesque set of all the

functions aafJ(x), lal = IfJl = m, let PI = 1 and P2 = O. Then (7.25)

becomes

a fJ(x 1)(e)a+fJ l.:2 c I (e)a+fJ

I I lal=lfJl=m a .. 0 lall=m '

which is just (7.20). Like~ise, (7.18) can be proved by exactly the ,:, same type argument. ' is continuous for jal = IfJ\ = m and , Finally, assume thafa~(x) , a assume (7.19). Then (7.S) hold~ for all xl, x 2 ~ 0, Xl';' x 2; and the continuity of a afJ also implie~ -that (7.25) holds even if xl = X 2. _Thus, if we let PI = 1, P 2 =

I

vp

for P .:2 0, and if we define

a (3(x)(l+fJ,

[H(x 1, e) + pH(x 2,

m.

-n

p2

laj=lf3I=m a

This inequality has been derived for functions l/J l' l/J 2 ~ C~( But since C';;'(O) is dense in L/O), (7.24) is valid under the assumptions that l/J l ' l/J 2 ~ L 2(0), supp (l/J 1) n supp (l/J2) = 0. Now let xl and x 2 be distinct points of 0, and let for j = 1, 2

P

/-1

H(x, () =

aafJ(x)Il/J/xW dxl

2

I

0 L Co

p

~ Co

89

lal=~I=m aafJ(xj)(~j)a+fJl

e)IL c~(le\ 2m + pl~212m), e,

an inequality which holds for all xl, X 2 ~ 0, all real vectors ~1, and all numbers p ~ O. Let Z be the set of numbers of the form H(x, () for x ~ 0, ~ real. ,Since 0 is connected and H is continuous, Z is a connected subset ¥of the complex plane. The set Z contains, together with any non­ !. zero complex number z, all the numbers of the form rz, 0 < r < .00. ._ Therefore, we need only show that the angle between the line from 0 '~to H(x 1. e) ana the line from 0 to H(x 2• ~2) is less than 17 - 0 for ,;-'some 0 > 0, 0 independent of xl, x 2, ~1. ~2. Obviously, we may aSSume lei = 1~21 = 1. Then (7.26) becomes !H(x 1

e) + pH(x 2• e)1

L c~(l + p).

Since the coefficients aafJ are bounded, we have !H(x,

()I s K for

90

Elliptic Boundary Value Problems

lei = 1.

eJ) = r .e i8j;

Now let H(x J ,

18 1

!r1e-

+ pr2e

182

}

then

\ L c~(l + p).

Let p = r/r 2 ; then

Ie

18

1

+e

18

21

'( - 1 +r-1) 2 '1K Lcorl 2 L Co .

But this means that 8 1

-

Q.E.D.

8 2 cannot be too near an odd mutiple of

'fT.

­

8. Global Existence In this section we shall assume that A is a strongly elliptic opera­ tor of even order = 2m which has been normalized so that

e

(- l)m~A'(x, ~

>0

f~ 0

for

(d. Definition 4.1). Consider the problem of finding a function u such that Au = f u =90

(8.1)

in on

-< :: = cPl on ......... ,

a

m- l

u

an m ­

l

--- =

n,

The reformulation can be motivated in the following way. Suppose at the boundary and the solution u are sufficiently smooth. Then e condition u = cPo on an automatically prescribes all derivatives If u in the directions tangent to an. Likewise, the remaining bound­ ,ry conditions automatically prescribe all derivatives of u on an in 'hich enter at most m - 1 differentiations in the direction normal to Therefore, under sufficient smoothness conditions, the Dirichlet undary conditions of u automatically prescribe at an all derivatives "u, \al ~ m- 1. Thus, instead of all normal derivatives of order than m, we could prescribe for u all derivatives of order less an m on an. . However, we obviously C~I16.ot prescribe arbitrarily on an all de­ vatives of u of order less' than m, since these derivatives are not / 11~'.uldependent. One way out of this difficulty is to list a number of 'r~ompatibility conditl?ns these derivatives must satisfy. However, an ;~.asier procedure is sImply lQ_pc>stulate the existence of a function 0" (x) which is in em - 1 and whose derivatives at an of order less 'i~than m are precisely the prescribed derivatives for the solution of the ,~rbirichlet problem. Thus, the Dirichlet boundary conditions for u now (~.sume the form: ~'1 11t D"'u = D"'g on \aISm-1. ':jl' ,

m)

an,

~(isfying systems ,~j'

such as (8.1). One of the more important techniques

;iwQS first used by Riemann to prove the existence of a solution of X'lj

J~:·the Dirichlet problem for Laplace's equation. This method makes use

on

an,

where alan indicates differentiation in the direction of the exterior normal to This is the Dirichlet problem for the elliptic operator A in n. The existence theory developed in this section is for the Dirichlet problem. However, the formulation of the problem given above is fraught with difficulties. For instance, considerations involving normal differentiations at the boundary are quite complicated, and, indeed, we would like to consider domains whose boundaries may fail to have tangent planes. Therefore, we shall reformulate the Dirichlet boundary conditions in (8.1).

an.

91

$1' There are many ways of proving the existence of functions u sat­

an, an.

cP m - 1

Global Existence

sec. 8

'iof the Dirichlet principle: a solution u of Laplace's equation on i~inimizes the Dirichlet integral,

.I,.:~ : • 'j

"t~:

,,'

f}:

n

j-l

n

(~)2 dx, ax 1

ong all smooth functions u which satisfy the given boundary con­

;:i
0

genemUzotinn, the loll owing p,"g
Jprove the existence of solution of (8.1) . 1~! In previous sections we have already discussed the meaning of the ;\statement that a function u is a weak solution of an equation Au = f. ,/{In section 5 we proved the existence of weak solutions of elliptic ~i~uations, and in section 6 we established several regularity results,

92

allowing us to give conditions on the smoothness of A and f which automatically insure that weak solutions are sufficiently regular to be classical smooth solutions. For boundary-value problems the procedure is similar. We shall first formulate a weak sense in which a function has certain boundary values, and shall then in sections 9 and 10 give conditions on the data which allow the conclusion that the weak solution of the boundary-value problem is indeed a classi­ cal solution. We shall now give several formulations of the weak boundary con­ ditions. The discussion will be motivated by considering the situa­ tion in which the functions treated and the boundary are sufficiently smooth. Suppose u is a smooth solution of Au = f. Let v be a smooth function, and assume that n has a smooth boundary. By Green's for­ mula we have

(8.2)

2m-1

(A*\!, u) - (v, Au)

=!

i-O

f

an

aiu N 2 -l-iv, m ani

da.

This formula is non-trivial; it will be derived and discussed in sec­ tion 10. Here N k is a linear differential operator of order k. If D"'v = 0 on au for 0 S; lal ~ m - 1, then v is a function with zero Dirichlet data. For such a function, and for u satisfying (8.1), the relation (8.2) becomes m-I

(8.3)

(A*v,u)-(v,f)=!

i- O

J

an N 2

_1_·v.
m

= (v,

f),

n.

so that u is a weak solution of Au = f in For the boundary condi­ tions, let g be any smooth function such that aig/ani =
(A*v, g) - (v, Ag)

=

~

j-O

f

an N 2m -

I -.v· I

'where now v is assumed to have zero Dirichlet data. This subtracted from (8.3) gives (A*v, u - g) - (v, Au - Ag) =0•

. (8.4)

Comparing this relation with (8.2) with u replaced by u - g, it follows that

(8.5)

mil f i- 0

an

N

2 -I-iV' aj(U -

g)

da = 0

manj

for all sufficiently smootl{ v having zero Dirichlet data. But it is reasonable to expectthat the functions N 2m-I-iv for 0 ~ j .::;: m - 1 can be essentially a,rbitrary for such v, so that (8.5) implies ai(u - g)/ ani = 0, 05;,1 s-m-l} That is, aiu/ani =
I

Conversely, suppose that u satisfies (8.3) for every function v having 2m continuous derivatives up to the boundary and zero Didchlet data. Then u satisfies (8.1) in the following weak sense. If v ~ C;(m, then (A*v, u)

93

Global Existence

Elliptic Boundary Value Problems


for all smooth v having zero Dirichlet data. This is evidently a sat­ isfactory definition, since no boundary integrals appear in this formula­ tion. Indeed, this is almost our ultimate definition. It still needs modi­ fication so that something analogous to Dirichlet's integral for Laplace's equation appears. We now proceed to the definition that will be used here. If both

, lj;) -

(
Alj;)

= O.

This result is obtained by transferring all the differentiations on lj; in

the integral (
transfer only some of the differentiations to obtain different formulas;

since


94

integrals contain only terms of the form Daep . DfJt/J with lal + IfJl ~ 2m ­ 1, it is seen that all the boundary integrals vanish. In particu­ lar, suppose we integrate by parts the expression (ep, At/J) in some manner so that no derivative for either ep or t/J of order greater than m results. Thus, we obtain some formula (8.6)

(ep, At/J) =

I J cafJ(x)Daep DfJt/J dx. lalC:;;m n IfJl~m

Now we motivate our final choice for definition of weak solution. AsSume all the functions and are, sufficiently smooth, and let g be a smooth function having the Dirichlet data aJg/an i = epJ' O:s: j:S: m - 1. ::If u is a solution of (8.1), then u - g has zero Dirichlet data. Thus, ,itu ep has zero Dirichlet data, (8.8) implies

an

.,.tI:

(ep, A(u - g) )

=

B[ep, u - g];

lor setting U o = u - g, using Au

.i~:;:~

(There is no reason to expect that there is a unique set of functions c {3(x) with this property; we shall give an example below to show a that there are in general many distinct possibilities.) We shall set aafJ(x) == cafJ(x); the bilinear form

95

Global Existence

Elliptic Boundary Value Problems

~i:

(ep, 1- Ag)

=

B[ep,

~~\~

=

I,

u~Y.

(

l.

Thus, we shall say (wproximately) that u is a generalized solution of ~i;' (8.8) if u = u 0 + g,(\Vhere u 0 is a function having zero Dirichlet data [1; and satisfying \. ;1~1'

B[ep, t/J]

(8.7)

L

=

(Daep, aafJDfJt/J)

\al:S:m

=

=

B[ep, t/J],

L

(-UH(ep,

~:::tions

.~;;

ep

DG.(BafJDfJt/J) ).

lal:S:m

';'[~:!

;~\(8.10)

an.)

1. If A=

I

(_l)la!

I

DaaafJDfJ,

in (8.9), then the formal adjoint of A is given by

(8.11) =

~ C~(n).

lal:S:m

IfJlsm

Thus, since even the functions ep in C';;'(n) are dense in L 2 (n), we have

At/J

B[ep, u o] = (ep, f) - B[ep, g], all ep

',:(Here we have used the relation (8.8) with t/J = g; the relation holds 'i since all derivatives of ep are zero at After a few definitions, we shall make the statement (8.10) precise. I :.:First, we give some examples.

IfJl~m

(8.9)

00

~ C o(n), we require only that

:' -\\,

all smooth ep, t/J having zero Dirichlet data. Note that if in (8.6) we transfer all the differentiations of ep by in­ tegrating by parts, we have formally

(ep, At/J)

B[ep, u o] = (ep, I) - (ep, Ag),

~all ep having zero Dirichlet data. Or, if we look only at the test func­

~~:,'

is called a Dirichlet bilinear form corresponding to the operator A. The relation (8.6) can be expressed

(ep, At/J)

~,'

;'1',.

1",\,

IfJl~m

(8.8)

~')

(_1)\0:1 Da(aafJDfJt/J).

lal:S:m

IfJl:S:m

A* =

I

(_l)la!

DaafJa DfJ.

lal:S:m

\fJl~m

Thus, if aafJ = afJa ' then A

=

A*, a relation which is expressed by say­

96

97

Global Existence

sec. 8

Elliptic Boundary Value Problems

B 1 [1>, ¢] = (~1>, ~¢);

ing that A is formally self-adjoint. Note that in the case aafJ = afJa' the Dirichlet form B[1>, ¢] is Hermitian symmetric; that is,

2

B 2[1>, ¢] = ([D 12 - D 22]1>, [D 1 - D /]¢) + 4 (D 1D 21>, DID 2¢)' B[1>. ¢] = B[¢, 1>].

The B 2 is a Dirichlet form corresponding to ~ 2 follows from the identity

Moreover, if A = A*, then some Dirichlet form for A is symmetric in the sense that aafJ-= afJa' For, suppose A has some Dirichlet form B 1[1>. ¢] =

I

(D 2+D 2)2_(D 2_D 2)2+4(D D)2

12

a (D 1>, cafJ DfJ¢).

lal:S:m

IfJl:S;Jl1

Here A is assumed to be any linear differential operator, not neces­

sarily of the form (8.9).

Then A* has the Dirichlet form

-12

12'

Note that from Bland B 2 one pbtains infinitely many distinct Dirichlet forms: B = tB l + (1- t)B 2, tJand arbitrary real number. Now we give a rigorous d~hnition of weak solution of (8.1). a

Definition

8.1.

/'

H/(fi) is the completion of C oo o (n) under the norm (m

·1\ Ilm.n.

\"" a

(8.12)

B 2[1>.¢]=

I

Clearly, H ma (n) can be identified with the cl osure in H m (n) of

(Da1>,cfJaDfJ¢),

C';~n); thus, Hm (n) is a closed subspace of Hm (n). The functions

lal.:s;m IfJl:S;m

as shown by a simpl e computation. But if A = A*, then B 2 is also a Dirichlet form for A. Thus, '!2{B 1 + B 2 1is also a Dirichlet form for A, and '!2{B 1 [1>. ¢] + B 2[1>. ¢ll =

I

I

B[v. u] =

a

(Dav , aafJDfJU)o,n'

lal~m

(D 1>, aaf3Df3¢],

IfJl:S:m

lal:S:m

IfJl:S;m

and we assume only that the coefficients aafJ(x) are bounded and . •. measurable in

where

U

in H m (n) should be thought of as having zero Dirichlet data (D"'u = 0 on an, lal·S; m ­ 1) in some vague sense. We shall now even weaken the concept of differential operator. We shall assume that a bilinear form B is given:

n,

and that the leading coefficients aafJ(x),

lal

=

IfJl

=

m,

~.a~e unifo.rmly continuous in n. With such a form is associated a formal l/2(cafJ + cfJa ).

aafJ =

Thus, aafJ

=

afJa ' as was desired.

2. The biharmonic operator in two-dimensional space is ~2 = (D 2 +D22)2. 1

For

~2

there are the two distinct Dirichlet forms:

"~.f.

dIfferential operator Au =

I

(_l)l a l Da(aafJDf3U),

lalsm IfJl:S:m in accordance with (8.9). It must be emphasized that A is just a symbol, and may not be a differential operator at all, since the coefficients aafJ are not assumed to be differentiable. Note that if we start with a genuine differential operator A, and if B is a Dirichlet form associated with

98

Elliptic Boundary Value Problems

. sec. 8

A, then the fonnal operator associated with B is just the differential operator A itself. Finally, note that if A is a differential operator, then its principal part is

A I = {_l)m

};

Elliptic Partial Differential Equations, Annali della scuola Normale Superior di Pisa, v. 13 (1959 pp. 134-135). THEOREM 8.1. Let B[u, v] be a bilinear form on a Hilbert space H with norm II II. If there are positive constants c 1 and c 2 such that

aaf3Da+f3.

lal=m 1f3I=m

99

Global Existence

\B[u,

v]1 ~ cillullllvll,

IB[u, u]:~ c211u112,

Thus,

(-l)m~A'(x, ~

=

~

};

aa{3(x)(a+ f3 .

lal=m 1f3I=m

for all u, v ~ H, and if F(x) is/a bounded linear functional on H, then . there exist unique v, w ~ H Juch that //

F(x) = B[x, v] = BLw/x]'

all

x ~ H.

Hence, the assumption of strong ellipticity of A is equivalent to the assumption of strong ellipticity of the quadratic form B[¢, ¢] (cL Definition 7.1). We can now define THE GENERALIZED DIRICHLET PROBLEM: given g ~ Hm(n) such that and f~ L 2 (n), find a function u ~ H m

(m

1)

o u-g~Hm(n).

2)

B[¢, u]

=

(¢, f)

for all

¢

~ C';(n).

Alternatively, by setting U o = u -;g this can be stated in the equiva­ lent form, "find a function U o ~ Hm(n) such that B[¢, u o] = (¢, f)­ B[¢, g)." Such a function u will be called a generalized solution of Au = f with the same Dirichlet data as g. It will be convenient to abbreviate "generalized Direchlet problem" to GDP. The motivation for this definition of generalized solution is given in the discussion leading to (8.10). Our first task is to show that in certain cases the GDP has a solu­ tion. The regularity theory in section 6 will then show that this solution is a solution in the classical sense, if enough assumptions are made on the regularity of A, f, g, and The proof of existence makes use of the following LAX-MILGRAM theorem, which is derived and at the same time is a generalization of the Riesz-Fre'chet representation theorem (see L. Nirenberg, On

an.

University of

V'~~h;r()"f(m

li6rary

100

sec. 8

Elliptic Boundary Value Problems

IlukJ - Uk i Ilm-1 ,un ~ Iluk/ - ¢ki Ilm-1 n +II¢k / - ¢k J Ilm-I,n '

(8.17)

101

Global Existence B[¢, Tf]

=

(¢, f),

all

¢

o

€ Hm(O)·

o

+11¢k J -

Uk

J

Ilm-I,n

k 71
it follows that

IU k I converges

0

k.,, k.1

as

Now T maps L 2 (0) into Hm(O), so in (8.17) we may take ¢ '" Tf;

therefore,

coIITfll~..n s IB[Tf, Tf]1

=:

I(Tf, f)1

~

IITf\lo.n Wllo,n

s;

II Tfll m ,n pt(\ o.n .

--+ 00,

o

in Hm-I(O). Q.E.D.

i

We shall call the form

I

B"{u, v] '" B[v, u]

Hence,

(/

\

the adjoint of B. The formal differential operator associated with B* is given in accordance with (8.14) by

IITfllm,n s c~llIfl\o.n ' o

(_l)la\

I

A*v '"

so that T is a bounded linear transformation of L 2 (0) into H m (0).

Da(af3aDf3v).

2 Now (8.15) implies also that IB*[u, u]12 c ollul1 m,ll.£o n. Therefore, for each h ~ L 2 (0) there exists ~ unique v '" T I h € H m (0) such that B*[t/J, T lh] '" (t/J, h) for all t/J € Hm(O). Or, by the definition of B*,

lal:=;m

1f3I~m

In case both A and A* are differential operators, then A* is the formal adjoint of A in the usual sense; d. (8.11). Now assume that for some positive constant co' (8.15)

IB[u,

u]1 2 col \ull~.n,

all

U ~

o

Hm (0).

By theorem 8.2, the GDP has a unique solution. In particular, if we take g '" O. corresponding to zero Dirichlet data, there is for each o f ~ L /0) a unique element u ~ Hm (0) such that (8.16)

B[¢, u] '" (¢, i))

all

¢ € C;;'(O).

Note that since both sides of (8.16) are continuous functions of ¢ o for ¢ in Hm(O) , we can even assert that (8.16) holds for all ¢ € Hm(O). Now we introduce the notation u '" Ti. Thus, (8.16) becomes

(8.18)

B[T I h,

t/J] '" (h,

t/J),

all

t/J

o

~ H m (0). o

Moreover, T I is a bounded linear transformation of L 2 (0) into Hm(n). Now we can obviously think of T and T I as being linear transforma­ tions mapping L 2 (0) into L 2 (0). With this indentification we have THEOREM 8.4. In L 2 (0), T I (h, Tf),,,, (T Ih, f), Proof. By (8.17) with

all

by (8.18) with

t/J '"

Tf,

B[T lh, Tf] '" (h, Tf).

T*. That is,

f, h € L 2 (0).

¢ = T Ih,

B[T lh, Tf] '" (T lh, f)j

'"

102

Elliptic Boundary Value Problems

sec. 9 o

Comparing these expressions, the result is immediate. Q.E.D. The previous results were derived under the assumption that B satisfies (8.15). Now if A is a differential operator which is strongly and uniformly elliptic, then we have seen that the quadratic form B[c,b, c,b] is uniformly strongly elliptic. Therefore, under the assump­ tions that aaf3 is bounded and measurable, and aaf3 is uniformly con­

u f H m and f f L 2' (8.20)

tinuous in n for lal = 1131 = m, we have at least the validity of o Garding's inequality; d. Theorem 7.6. Moreover, a consequence of Theorem 8.2 and the corollary to Lemma 7.9 is that in sufficiently small subsets of n the GDP for Au = f is always uniquelv solvable. More generally, we have THEOREM 8.5. Assume G~rdin~'s inequality holds: for some > 0 and Ao ~ 0, and for all c,b :€ C~Cn}

constants Co

:RB[c,b, c,b] L collc,bll;',n -

I

,

,i<

4\'.

Aollc,bll~.n'

Then for any A 2 A the GDP for Au + Au = f has a unique solution. o Moreover, if n is bounded, then A satisfies the Fredholm alternative; o that is, if N(A) is the null space of A (the set of functions u ~ H m such that Au = 0 in the ~eneralized sence) and if N(A*) is the null space of A*, then NCA) and NCA*) are finite demensional and have the same dimension. Also, the GDP for Au = f and u havin~ zero Dirichlet data has a solution if and only if f is ortho~onal (in L Cn))

103

Global ReRularity

B[c,b, u]

= (c,b, f),

all

o

c,b~Hm(n)~u-AoTou=f.

In particular, for u f Iim , u f N(A) <='9 B[c,b, u] = 0, all c,b ~ Iim (n)
identity mapping of HmCf}) into Ho(n~ = L 2 Cn}. This identity mapping

is compact, by Theorem 8~3: Thus; To is a compact operator on L 2 (n). and we can apply the Riesz-Schauder theory of compact operators. Thus, NC1 - AoT 0) and N(l - AoT~) have the same finite dimension. Also, Au = f is solvable for u f Ii (m <='9 B[c,b, u] = (c,b, f), all c,b ~ o m Hm(n) u - AoT OU = f, by (8.20). By the Riesz-Schauder theory, the range of 1 - AoT 0 is precisely the orthogonal complement of NO - Ao~)' which, as we have seen~ is N(A*). Thus, Au = f is o solvable for u ~ HmCm if and only if f is orthogonal to N(A*). Q.E.D.

2

to NCA*).

Proof. Let B + A be the bilinear form B[v, u] + A [v, u]o,n' Then for A 2 Ao' :RCB + A)[c,b, c,b] = :RB[c,b, c,b] + Allc,bll~.n ~ collc,b\I~.n, so that Theorem 8.2 applies to B + A. It remains to verify the Fredholm alternative. Let To be the operator associated with B + Ao according to the above analysis; that is, To is a bounded linear transformation of o L 2 Cn) into Hm(n}, satisfying (8.19)

(B + Ao)[c,b. T of] = (c,b, o

t)) all

o

c,b ~ HmCn)

(cL 8.17). Now if u ~ H m Cn), then u is a generalized solution of 0 Au = f precisely if B[c,b, u] = (c,b, f), all c,b f H Cn}. This is true if m 0 and only if (B + Ao)[c,b, u] = (c,b, f + Aou), all c,b f HmCn). By (8.19), this is equivalent to the assertion that u = T 0([ + \u). That is, for

9. Global Regularity of Solutions of Strongly Elliptic Equations In section 6 we discussed local regularity results for elliptic systems. In the present section we are again confronted with the problem of regularity. This time, however, we shall obtain regularity properties up to the boundary of the domain; moreover, we shall in the process obtain some more local regularity theorems. It should be noted that we have already given results which show that the solution of a generalized Dirichlet problem for an elliptic equation is regular in n. For suppose u :€ H (n) satisfies B[c,b, u] = Cc,b, f) for all c,b :€ C~(n}, where B is a Dirichiet form corresponding to an elliptic operator A. Suppose that A has coefficients which are smooth enough to guarantee that B[c,b, u] = (A*¢, u). Then we have the result that u is actually a weak solution of the equation Au = f.

104

Thus, the local regularity results of section 6, especially Theorem 6.3, give conditions on A and I which allow the conclusion that u is as regular in 0 as we please. We shall not dwell on this point, however, since the regularity theorems of this section imply results both in the interior and at the boundary of O. ° . The method consists of using Garding's inequality to show that if a solution u already has a certain regularity, say u ( H°m (0), then u actually has more reguIarity, say u (H m +].(0). As in section 6, the principal lemmas (Lemmas 9.2 and 9.3) require a considerable amount of work; the more precise regularity theorems then follow with relative ease. We will consider only the Dirichlet problem with zero Dirichlet data. Asa first result we have

ao

LEMMA 9.1. Let Xo ( and assume that the coordinates can be rotated so that some neighborhood U 01 Xo has the representation

U=

Iy:

l(y') - h o < Yn < l(y') + h o'

I(y'),

Prool. Let ¢k (C';(O). For fixed

• < rol. Iy'l

y'

such that

Iy'l < r 0'

(1',)_h

!

¢k(Y',/(y')-h)=

,

Dn¢kdYn'

I(1' )

ao.

since ¢ k vanishes on

[¢k(Y', 1(y1)-hW

By the Cauchy-Schwarz inequality,

~h /(~'~/lb:¢kI2 dYn' f(1'/)-h

;

,

so that, on integrating '(Vith respect to y', for r

< r 0'

I.~ I,

J

l¢k(Y', l(y') - hWdi2

where Gh

h£ ID n¢kI 2dy, h

=

Iy:

l(y') -h
< l(y'),

Iy'j
Thus,

hI

J J o

l¢k(Y', l(y') - hW dy'dh

11"1
° (0) and assume that u is defined and continuous in U Let u ( HI Then u(Xo) = O.

S

nil. ~

J

h

o

J

where 0

< hI < h o'

\D n¢k[2dy

Gh

lhhi GJ \D n ¢kI 2dy, h

Let ¢k

-+ U

I

° (0); we see that in HI

hI

h7 1 Jo r

lu(Y', l(y') - h)\2dy'dh

\1"I
an

~

IY'I

~ ro

Y'

and for

O
11"1
Iy'l < rol,

where y, = (YI""'Yn-I)' r o is a positive constant, I is a continuous

lunction lor [y'l < r 0' XO = (0, ... ,0, 1(0)), and

un ao = Iy: Yn =

105

Global Regularity

sec. 9

ElIiptic Boundary Value Problems

lhh1f G

ID nu\2dy h

I

lhh1luli·

106

sec. 9

Elliptic Boundary Value Problems

Since u is continuous, on letting h 1

J lu(Y', \y'J
f(y')

)1 2dy'

=

~

107

Global Regularity

support in Ix: Ixl < RI. Note that (need not vanish on the flat part of the boundary of the hemisphere. Now we are ready to prove the fundamental regularity lemma. The ° essential tool will be Garding's inequality.

0 we obtain

0,

LEMMA 9.2. Assume so that u(y', f(y')) = 0 for Iy'l
au

1°

that the bilinear form B[v, u]

=

an, au,

B[c,b, u] for all

=

i

( \ I

has bounded (\aa{3l&M) measurable coefficients in G R , and that, for lal = m" a ~ satisfies a uniform Lipschitz condition: "at-'

laa{3(x) - aa{3(y)[ ~ K[x - yl,

x, y ~ G R ;

that the associated quadratic form is uniformly strongly el­ liptic: for ~ real and x ~ n

then

~ .I IB[c,b,

~tifJD{3u) o.G R

(Dav,

1{31~m

(c,b, f)

c,b ~ C;'(n),

I lal~m

lal=rn

u]1 ::; C11c,bllo.n = C[lc,bllm-m.n,

aa{3(x)~a+{3 L EI~12m;

\{3!=rn

where C is some number depending on n, m, n, u. Even this inequality is more than is needed. Indeed, we shall assume only that

IB[c,b,

u]1 :s: C11c,bllm-J.n

for some j: I::; j ::; m; and we shall show that if u ~ H°m (n), this in­ equality implies that u ~ H m + J.(n). This fact will first be proved for a neighborhood of a part of the boundary which is flat. Since the principal lemma can be stated for either a sphere or a hemi­ sphere, corresponding to interior or boundary estimates, respectively, the following notation will be convenient. By G = GR' we shall denote either the sphere Ix: Ixl < RI or the hemisphere Ix: Ixl < R, x n > 01; whenever a distinction between the two is necessary, we shall indicate which should be considered. We shall write G' = G R , and Gil = GR", assuming that R' < Rand R" = %(R' + R). The symbol (will denote a real function which is infinitely differentiable on En and has its

2°

3°

that u ~ Hm (G R) and that (u .€ (G R ) for all (~c';
i(

IB[c,b,

u]1 ::; C11c,bllm-l'

Then D.u ~ Hm (G R I) and there is a constant y, depending only on m, , n, E, M, K, R, and R', such that

IID/ullm.GR. :s:y(C+ Ilullm.GR),

108

Elliptic Boundary Value Problems

sec. 9

(1)

i

=

1,

,n, if GR is a sphere, or

(2)

i

=

1,

,n - 1, if GR is a hemisphere.

Proof. An argument, familiar by now, shows that without loss of generality we may assume that aaf3 = 0 for lal < m; for if hypothesis 3° of the lemma holds, then it certainly holds if aaf3 is replaced by zero for

IB[¢,

u]1

lal < m, =

By this convention we also have Df3vh fore, for any ¢ (C';(G) (9.1)

B[¢, v h] =

since from 3° and the inequality,

I ~ (Va¢, lal=m 1f3ISm

~I ~ la\=m

Baf3Df3u)o.G +

~

a (D ¢, aaf3 Df3u )o.GI

laJ<m 1f3I~m

(Va¢, aaf3 Df3U ) -

C111¢llm-l.Gllull m.G,

lal=m

1f3lsm

Let k be a generic constant depending onlyion m, n, M, K, E, R, and R' (k will also depend on (, but ( is fixed for fixed Rand R'). In (9.1) we shall use Leibnitz's rule; we theri\ obtain terms of the form 0hDf3-Y( . DYu. The proof of Theorem ~/.i3 imp1i~ that for [yl S m - 1

(9.2)

110 h (Vf3-Y(.

DYu )

O,G

R

~ \\Df3~Y~--;/DYulll .G =

it follows that, for any ¢ ~ C';(G)

aaf3 Df3U )I ~

= 0hDf3«(U) in GR' There­

~ (D a¢, aa f30 h Df3«(u)) 0 , G'

1f3I~m

I ~ (D a¢, lal=m

109

Global Regularity

(C + C11Iullm,G)II¢llm-l.G'

IIDf3-Y( .

R

+h

DYuJI 1 ' GR

Skllullm,G'

1f3I~m where C 1 depends on m, n, and M. The remainder of the proof consists of an estimation of difference quotients. Let (be a function as described above, such that ( == 1 on G' and ( == 0 outside Gil. Throughout the proof the function (will be fixed. For 0 < h < R - Ril, let v h = o~«(u) = 0h«(U); if G is a sphere [hemisphere], i can take on any of the values 1, ... ,n [1, ... ,n - 1]. By 2° (u ~ H° (G R ). Since i 1= n in the case of the hemisphere, it is not m ° difficult to see that vh .. 0h «(u) ( Hm (G R)' Moreover, since we may assume ( == 0 for Ixl ~ R" (and x n > 0 in the case of the hemisphere), we may consider (u to be defined (== 0) for such x, and then we obtain a version of Leibnitz's rule:

Da«(u)

=

~ %)Df3(. Da-f3 u,

I.B[¢,

vh]1

S l~ \a\=m 1f3I~m

Now the first term on the right can be estimated as follows:

I ~ la[=m

(Va¢, aaf30h(t)Jf3v.))1

\f3J~m

=I

~ (Va¢, 0h(aaf3(Df3u))

la\=m 1f3I~m

f3sa

valid for x ~ Gr' any r > O. Here D -f3u(x) can be defined to be anything (say zero) outside of GR' and the formula remains valid. a

(Va¢, aaf3 0h «(Df3U))f + kll¢llm,Gllullm,G'

-

~ (D a¢, 0haa.f3 . (x + he J )Df3u(x + hel)1

la\=m 1f3I~m

110

~I

a (D ¢, 8 h(aaf3(;Df3u )!

I

lal=m

1f3IS:m

111

Global Regularity

Elliptic Boundary Value Problems

+kll¢llm,Gllull m.G.

Since this inequality holds for every ¢ ~ C~(G), by continuity it holds . 0 'Lior all ¢ ~ H m (G). In particular, it holds for ¢ = v h' Thus,

'L

\B[v h , vh]\.s: k\\vhllm,G(C +

Ilullm,G)'

where we have made use of the Lipschitz condition on aaf3' Thus, o

By Garding's inequality, Theorem 7.6, IB[¢, vh]l.s:

I

a (D ¢, 0h(aaf3,Df3U ))1

I

lal=m 1f3IS:m

= I -I lal=m

+kll¢llm.Gllullm.G

IlvhI1 2m•G.s: y~lE-I[9tB[vh' vh]+Aollvhll~,G] ~ y~IE-l[kllvh\lm,G(C +

('O_h Da¢, aaf3Df3u ) I +kll¢llm.Gllullm,G.

~

1f3I~m

I

la\=m 1f3I.s:m

(Da('O_h¢)' aaf3Df3u ) I +

+ Ilullm.G + IlvhIJo.G)

/;where we have utilized (9.2) with f3 tfOividing by Ilvhllm.G.

kll¢llm.Gllullm.G

Iloh('u)ll m•G = Illvhllm.G.s: k(C +

y = 0 and m replaced by 1.

Ilull m.G).

.'u

u]1 +kll¢ll m•GIlull m•G·

The functioo 'O-h¢ is a test function on G; here we use again the fact that for the case of a hemisphere, i oj n. Therefore, by hypothesis 3 0 of the lemma,

u]1 s: CI\,O-h¢llm-I.G'

IB[¢, vh]j.s: ClI'o-h¢llm-l,G +

k\\¢llm.Gllullm.G

Ilullm.G)\I¢llm.G'

rSince , == 1 on G', the conclusion of the lemma follows. Q.E.D.

'[~ Since this lemma can be applied to any sphere contained in a domain

'In, it implies interior regularity. More than this, it provides regularity

I properties up to a flat boundary. This is true even for the normal de­

rrivative D n u, although the proof above certainly does not show this.

: For m = 1, the situation is quite simple. If u satisfies the equation lB[¢, u] = (¢, 1)0.0 for all ¢ :€ Co 00(0), then for any sphere S cO we have IB[¢, u]s I.s: 11 11\ 0,5 11¢III-l.S for all ¢ :€ C~(S), and the lemma can be applied in S. It follows that u ~ H~OC(O) and satisfies Au = I 'Weakly in 0, if A is the elliptic operator with Dirichlet form B. If GReO is a hemisphere with the flat part on its boundary contained in and if the coordinate system is rotated so that the flat part of . iJG R lies in the plane x n = 0, then it follows from the lemma that, for t.,,, R' < R, D.ilI € H1(G R ,), i = 1, ... ,n - 1, whence D.D.u ~ HO(G R ,), J }

an,

Again we have used a modification of Theorem 3.13; d. (9.2).

:€ H m (G') and that

IIDlullm.G'.s: k(C + Ilullm.G).

iY.

Thus

= k(C +

=

By Theorems 3.15 and 3.16, this implies that D ,

= \B['O_h¢'

IB['o_,,¢,

Aollvhll~,G]

.s: kllvhllm,G(C + Ilull m.G+ Ilul'L:a),

We have used the vanishing of 'outside of G" to transfer 0h from the right side of L) to the left. Using Leibnitz's rule and the proof of Theorem 3.13 again, we see that IB[¢, vh]l.s: \

kllvhllm,G(C

Ilull m•G) f

sec. 9

Elliptic Boandary Valae Problems i = 1, ..• ,n - 1, j = 1, ... ,n. Since B is uniformly strongly elliptic, the coefficient a nn is bounded away from zero; indeed,:Ka no 2 E. Hence, assuming A is really a differential operator, the equation Aa = f can

113

Global Regularity

1°

that a :€ L 2 (I R ), that a has weak derivalives D:u :€ L 2 (I R i = 1, ... ,n - 1; and

2°

that there is a constant C and an integer m 2 j such that

),

be written 0-1

D 2 a = a- 1 (t _ n

nn

1

(9.3)

n

n

1 aJ.D.D.a - 1 b.D.a - ca).

i-I j _ 1

J

1

]

i-I

J

I(D:¢,

a)1 .$ C[ 1¢ll m - J,I

R

J

Since all of the terms on the right are in H o( G R I), it follows that D~a :€ HO(G ,) and that a€: H 2 (G R ,). Moreover, by the estimates ob­ R tained in Lemma 9.2 for the derivatives D I.D J.a, i Ie- n,

for all ¢ :€ C~(~). Then for every R' < R, a :€ H/~I) and n-1

Ilullj'~1 ~y(C+ J~1 IID:ullo.~ + Ilulloi~}'

IIDnalll,GR'.$ y(C + Ilalll,G)'

\

where y = y(n, m, R, R'). Thus,

IlaI12,GR'.$ y(C + Ilalll,G)'

an

Even in case is not flat, but is sufficiently smooth, the same re­ sult holds. This will be seen later; d. Theorem 9;7. Note further that if n ~ 3, then 2 > n/2, so that Sobolev's inequality implies that, after modification on a set of measure zero, a :€ Coal). Therefore, Lemma 9.1 implies that a == 0 on Thus, for second-order elliptic equations Aa = f, with only the assumption that f :€ L 2m) and weak assumptions on the smoothness of A, it fo11o'1!S that any solution u in H/U) is in c<:(ID and vanishes on Moreover, the regularity results of section 6 allow us easily to give smoothness conditions on 2 A and f which show that a is even in C (U) and is thus a genuine solution of the entire Dirichlet problem. We shall later in this sec­ tion obtain regularity results that allow the same conclusions for any

an.

au.

n and any m. If m > 1, this method will not work. Instead, we shall use a lemma which guarantees the existence of weak derivatives of a function known to have certain pure derivatives and known to satisfy a certain inequality: LEMMA 9.3. For any r> 0, let 1, = Ix: Ixl < r, x n > 01. Let R be a positive namber and let j be a positive integer. Assume

_________ 4

Proof. The proof is by transforming the boundary problem into an interior problem by a type of reflection, and then using the local re­ sults of Theorem 6.3. First, observe that inequality (9.3) holds for any function ¢ :€ C~(~) since J(¢ -+ ¢ in Hm (~), by Theorems 1.7 and 1.8. Moreover, (9.3) holds for any function ¢ :€ C m (IR ) such that for some {; > 0, ¢(x) = 0 for R - 8 < Ixl < R and D~¢(x) = 0 for xn = 0, S = O, ... ,m (note that it . then follows that D""¢ = 0 for xn = 0, lal ~ m): to see this, let

¢(x - (en)

x( I

for

R

-8' xn ~ (,

¢(x) =

o

for all other

x;

then ¢ix) (C~(~) and ¢( -+ ¢ in Hm(~) as Now we extend u to En : let

o

(9.4)

for

(-+

O.

[xl::::R,xn>O,

a(x', x n ) =

2m + I

1

k-I

Aku(x', - k- 1 x) n

for

x < 0, n

where x' = (xl' ... 'Xn _ I ); here the \'s are constants which are chosen so that a is sufficiently differentiable (in the weak sense) for xn = O.

_

114

2m+l

f

In order for the function and its tangential derivatives (that is, de­ rivatives D'" with a n = 0) to match nicely, it is sufficient to assume

k-l

that =

I

2m+l

A D"[u(x' _

I

k

k-l

n'

r

l

f x )]1

n

'

x )\ n

k-l

x n -0

n " n -0

I k-l

D"'¢· udx =

k

D"'¢(x) . u(x', -

f "n >0

r

l

x )dx n

D"'¢(x', - kx n )

•

u(x)dx,

f

D"'¢o' udx,

I R

SR

2m+l

= D"u(x'

kA

I

D"'¢ • udx +

<0 n

where we have substituted - kX n for xn ' Recall that an = 0.) Thus,

For the normal derivatives, we must have

D"u(x', xn)1" n-0 n

k "

2m+ l

f

"n>o

Ak = 1.

k-l

f

A

I

D"'¢ • udx +

"n>o

2m+l

115

Global Regularity

sec. 9

Elliptic Boundary Value Probl ems

where

Ak (- k)-";

2m+l

¢o(x) = ¢(x) +

kAk¢(X', - kx n )·

I

k-l

i.e., 2m+l

I

(9.5)

k-l

By (9.5) with s = - 1, ¢o(x', 0) = 0; however, cPo is not necessarily a test function on ~, since its support may intersect I x: x n = 01. To circumvent this problem, let peA) be a infinitely differentiable function on the real line with peA) = 0 for A.:>: 1, peA) = 1 for A2 2. Let

,\ (_ k)-S = 1. k

Thus, in order to have derivatives up to order 2m - 1, it is sufficient to assume that (9.5) holds for s = 0, ... ,2m - 1. This imposes only 2m conditions on the 2m + 1 quantities Ak • k = 1, ... ,2m + 1. Because we will need (9.5) to hold for s = - 1 later, we take IAkl to be the (unique) set of real numbers satisfying (9.5) for s = - 1, 0, 1, ... ,2m -1. (The determinant of the coefficients in the system of equations (9.5) is a Vandermonde determinant for the quantities - k- l • k = 1, ... ,2m + 1, and thus is not zero.) Note that the argumen.t - r l x n in (9.4) is chosen so that the value of u at any point in SR = Ix: Ixl < RI depends on only the values of u at a finite number of points in I R • Thus, u ~ L 2 (SR)

s(x) =

Then, D,s( = 0 if i .j,. n, so that

f

f

f SR

"n>o

D"'¢ • udx +

f

D"'«(,(¢o)' udx.

S(D"'¢o' udx

= (-

l)H

f

(,(¢OD"'udx.

~

Thus,

To see this, let ¢ € C,;(S R) and consider

f

f ~

~

Moreover, if lajs 2m - 1, if an = 0, and if D"'u exists weakly in ~, then D"'u exists in S R and it is the extension (by (9.4)) of D"'u on I R •

D"'¢. udx:=

=

Since (,(¢ 0 is a test function on ~, the existence of D"'u on I

\Iu11o,SR':>: Y!lu11o.IR·

SR

(,p"'¢o' udx

~

and

f

p«(-ll xn l)·

D"'¢ • udx

"n
~~

D"'¢' udx

=

f ~

(,p"'¢o' udx +

f ~

(1 - (,()D"'¢o . udx

R

yields

116

Elliptic Boundary Value Problems = (-

(1 - (,c)D"'¢o • udx.

(,c¢oD"'u dx + f

l)H f

~

~

2m + 1

..;.:~

=f x

As c tends to zero, the second term on the right tends to zero, and the first term tends to be obvious limit, so that

f

D"'¢.udx=(-l)Hf

= (-

n

>0

D 2 mt/J • Ii dx +

I

n

A k

k-I

e -2m f

x

>0

D2m [t/J(x' ,-kx )]. u(x)dx, n

n

n

¢oD"'udx

IR

SR

117

Global Regul arity

sec. 9

where we have substituted -kx n for x n ' Let, for xn > 0,

2m+ 1 (¢(x) + I k;\¢(x', -kxn))D"'u(x)dx

1) I'" I f

I

t/J*(x)

k-I

=

2m + 1 Ak(-k) 1- 2mt/J(x' ,-kx n ). I

t/J(x', x ) -

k-I

n

R

=

(-1) H [f x

+

n

n

<0

k- 1

Let t/J

s yllD~ullo '~R ~ ,

10:: C:;O(SR)'

(D 2mt/J, u) n

I

f

D 2m t/J . Iidx +

x

n

D2mt/J. Iidx + f

x <0

n

1C

2m + 1 Ak f I k-t

D~mt/J • udx + n

>0

x <0

x n

=

O. s

=

\(D2mt/J*, u)o ~

"

D 2m t/J ·lidx n

g~R

n

1=

I(D;:'¢,u)o I '

R

I

s CII¢llm-J .I R ~ CIIt/J*11 2m - J.I R •

n

(D 2mt/J) (x', -kx ) • u(x) dx n

-1, ... ,2m - 1, we

0, ... ,2m.

holds for ¢:

D~mt/J(x). Ii(x', - k-1x )dx

>0

=

Note that also t/J* = 0 for R - 0 s Ixl s R, xn > 0, some 0 > O. Let ¢ = Dmt/J*; then. by the remark made at the beginning of the proof, (9.3)

n

2m+1 Akk f I k-I

O'''''R

n

n

n

=f

(D 2m t/J*. u) ~ ;

n

n

=f

=

DSt/J*= 0 for x n n

t'

x >0

O,SR

have

1, ... ,n - 1.

=

Then =

O,SR

i

n

moreover, by the assumption that (9.5) holds for s

where we have substituted - k-1x n for xn . This proves that the ex­ tension of D"'u is, indeed, the a-th derivative of the extension of u in SR' Moreover, from (9.4) it follows that for a = je f [ID~u\\o , SR

(D 2mt/J, u)

(9.7)

2m + 1 AkD"'u(x', - k-1xn)dx], ¢(x) I

f x

>0

Then

¢(x)D"'u(x)dx

Thus, from (9.7),

n

(9.8)

I(D~mt/J,

U)0,SR

1

S

C11t/J*112m-J.I

R

~ yCI\t/J112m-J,S R

118

Elliptic Boundary Value Problems

I

for any t/J c;:: C':;'(S R)' For i f n, we can use the fact that u has the weak derivatives D: in SR' together with the estimate (9.6), to obtain (9.9)

1(D~mt/J, U)o ,

5 J

R

1= 1(D;m-1t/J,

DjJu)O

5

' R

119

Global Regularity

Clearly, o~('u) c;:: Hm(G) for sufficiently small Ihl. Also, o~' ~ l for sufficiently small Ihl, so that o~'(x) • u(x + he ) ~ o H (G) for sufficiently small \h \' by the assumption on the behavior

C':;'(Ix: Ix\ < R\) m

I

0

of u. Therefore, 'o~u c;:: Hm(G) for small Ihl· For suitable R' < R, supp (~ n G c G'. By the proof of Theorem 3.13, we have for small

::; 11t/J11 2m - i ,5 R IID:u l1 o,5 R ~yIID:ullo , I

sec. 9

Ihl

Ilo~('u)!\m,G = \lo~('u)llm,G"

\\t/J112m- J,5 R ' RI

::; II'ull m + 1 ,G Now let A be the elliptic operator of order 2m given by A=

::; Y\I'\lm+1,GIIl{~lm+l,G'

n

I D2m; j_

1

\

I

by Leibnitz's rule, Then (9.10) implies from (9.8) and (9.9) we see that (9.11)

n-1

!(At/J,

U)O,5

R

I ~ y(C + i~1 \ID:ullo,I R ) 11t/J112m-J,5 R

for all t/J c;:: C,:;,(S R). Thus, the interior regularity theorm (Theorem 6.3) can be applied: for R' < R we have u c;:: H R I) so that, in particular, uC;::H/I R ,), and

/S

n-1

IluIIJ,I R , ~ l\ull i ,5 R '::; y(C + 1~1 IID:u11o,I R

+

\1<:-o~u\\m.G~ Y111 ullm+1,G o

for small lhl, where Y1 is independent of h. Since H m (G) is a Hilbert

o space, any closed sphere Iv: \\vll m ~ AI in H m (G) is weakly sequenti­ 0 ally compact. By (9.11), 'OhiU is uniformly bounded in H m (G), and so o a subsequence 'o~ u converges weakly in H (G). In particular, for m

k

any ¢ c;:: C':;'(G)

Ilullo,I R )'

lim

f

k400 G

'o~ u· ¢dx

=

k

lim k-+oo

f

G

UO~h ('¢)dx k

Q.E.D. =-

One additional lemma is needed before proceeding again to the bound­ ary regularity question. LEMMA 9.4. Let G = Ix: Ixl < R, xn > 01. if u c;:: H +1(G) and if o om x 'u c;:: Hm(G) for all' c;:: C':;'
'(x)O~u(x) = o~('u)(x) - o~o(x) .

f n. Defining u

u(x + he l ).

f

uD I('¢)dx

G

=

f

DIu' '¢dx.

G

o

Hence,

Q.E.D.

0

'o~ u converges weakly in H (G) to 'DIu, and so (,D IU ~ Hm(G k

m

Now we can extend Lemma 9.2. We shall show that u and all of its derivatives of order ~ j are smooth up to the flat part of the boundary of G. The theorem is stated precisely in Lemma 9.6. We will use the notation C~(m for the class of functions which have bounded, continuous derivatives of all orders ~ k on U.

120

Elliptic Boundary Value Problems

sec. 9

Furthermore, for ¢~ C';(G"I),

Definition 9.1. A Dirichlet bilinear form B[¢, ljI]

I

=

a (D ¢, aaf3Df3lj1)o.n

m-J
1f3I~m is right j-smooth in able in nand aaf3

~

1f3I~m

n if the coefficients aaf3 are bounded and measur­ dO:!+J-m

(m for laJ

LEMMA 9.5. Assume that j

~

+j -

m and

2°

° that u ~ Hm(G), that (u ~ Hm(G) for all (~C,;(\x: and

3°

that there is a positive number C such that

¢

~

I

Ixl < R!);

•

m-J<[al~m

1f3\.s;m I m-J
(Da¢, D j a • af3

Df3u)o.GIIl

1f3\~m

(9.12)

I

=-B[D/¢u]-

(Da¢,Dlaaf3·Df3u)o,G."

m-J
II u IIm+ J, G' ~ y( C + II u II m•G)'

the inductive hypothesis gives u ~ Hm+J-1(GIII). For j = 1, DiU ~ Hm+J_1(GIi), i = 1, ... ,n - 1, by Lemma 9.2. To establish a similar result for j > 1, we observe that u ~ H + 1 (Gill); hence, by Lemma 9.4 o m ' i)J IU ~ H m (Gill), i = 1, ... ,n - 1, for any (~C';(\x: IXI < Rill!).

/-­

(DaD i¢' aaf3 Df3U )O.G III

I

C';(G).

u]1 ~ CII¢llm-J,G.s; CII¢llm-(j-l).G'

.d'·iJ f3 u)o Gill

a..}J

1f3I.s;m

CII¢ll m- J•G

Proof. The proof is by induction on j. Observe that the lemma is trivial for j = O. Thus, it suffices to establish an induction step. As in the proof of Lemma 9.2, we assume, without loss of generality, that aaf3 = 0 for lal + j - m .s; O. Assume that 1.s; j ~ m and that the lemma holds if j is replaced by j - 1; we will show that it holds for j. Let Rill = Y2(R + R"), and Gill = GRill. Since

(Da¢,Dja

m-j
Then, for any R' < R, u ~ H m +.(GI) and there is a constant y = } y(m, n, B, R, R') such that

[B[¢,

(Da¢,D I.(aa f3Df3U))o • Gill

m-J<\al.s;m 1f3I~m

that B is uniformly strongly elliptic and right j-smooth in G;

for all

~

m> O.

1°

IB[¢, u]\.s;

a (D ¢,aaf3 Df3D jU)o.G nt

I

B[¢, DIu] =

.

lal~m

121

Global Regularity

1f3lsm I

i.

Consider any term in the sum on the right; integrate by parts lal ­ m + j - 1 times (which is possible, since, by the inductive hypothesis, u ~ Hm+J-1(GIII», transfering lal - m + j - 1 differentiations from ¢ to the other side of ( , )0 G III. By the Cauchy-Schwarz inequality,

.

a I(D ¢, D i aaf3 •

Df3u)o,G",I.s; yll¢llm-J+l.GI\ullm+J-l.G;

note that we have used the fact that the derivatives of D iaaf3 up to order Ja\ - m+ j - 1 are bounded. since B is right j-smooth. Thus, by hypothesis 3° and (9.12), IB[¢,

Mhz

Dp]\ ~ CIID j ¢lI m_J.G + YII¢llm-J+l.GI,llullm+J-l.a'" III

--.----­

122

ElIiptic Boundary Value Problems

sec. 9

a _ (D -

IIuII m+J- 1 • G,n )II¢IIm- J+1 • GIII.

S y( C +

123

Global Regularity e

/¢, aafJDfJDP)o.G'"

....~;.

Here y is a generic constant depending only on m, n, B, R, R'. There­ fore, D /u satisfies the conditions of the lemma with j replaced by j - 1, and G replaced by G'''. By the inductive hypothesis it follows that D;u f Hm+J-1(G") and that

(9.13)

- 1 • G,,::; y(C + Ilullm+J-l.GIII IID;ull m+ J S

+

iEstimate the right side by transferring lal - m + j - 1 differentiations a e ifrom D - /¢ to the other sides of ( , ) and applying the Cauchy-Schwarz il:inequality. Since B is right j-smooth, the derivatives of aafJ which i~f·occur will be bounded. For the first term we have ::<

IIDpllm.GIII)

y(C + Ilullm+i-l.alll)

y(C +

Ilullm.a).

e

I(D

a

-

e /¢,

I

a

I(D

a

I

Therefore, from (9.14), a

n

I

DfJu)o.G" S IB[¢,

.fJ

u]1 + yll¢ll m- i .G". . (C + Ilullm+i-l.G·III).

Let

af,m en

IfJlsm

v

For any term in the sum on the right, there is an i f, n such that / I D a ¢ = De D a- e ¢. Thus a e

a . G" = _(D = -

I

IfJlsm me

a (D ¢, aafJDfJU)o.an.

m-J
a (D ¢, a fJDfJu)o

Ilullm+i-l.GIII).

.fJ

+

(9.15)

p)o.G" I S YII¢llm-J.GllnDjut~m+i-l.GII

aafJ DfJD

¢, aafJDfJu)o.G" I::; yll¢llm-J.G"(C +

DfJu)o.a"

n

IfJlsm me

yll ¢ll m- i •G" Ilull..,; +J-l.G II;

::; yll¢llm-i.G"(C + Ilullm+i-l.GIII).

!(D,;:¢, B[¢, u] = (D';:¢,

DfJ u ) o.G" I ::;

"Thus, (9.15) implies

In the case of the hemisphere, we still must show that these relations hold for i = n. The proof is completed by using Lemma 9.3 to show that DnU f H m+r1 (G') • For r/.. f COO(G") 'f' 0 (9.14)

¢, D i aa{3

for the second, by (9.13) we have

(note that m + 1 S m + j - 1 since j > 1). In the case of the sphere, inequality (9.13) holds for i = 1,... ,n; this completes the proof in this case, since u f Hm+/G") and by (9.13) and the inductive hypothesis

Ilull m+i•a" S

/

a I(D -

a (D -

I

a

IfJlsm men.fJ

DfJu.

Then, by hypothesis 3°,

ICD,;:¢.

;¢, D;Ca fJDfJu))o a"

a.

e

=

/¢, D ;aafJ • DfJu)o.G"

v)o.G"\::;

CII¢ll m_J•G" + yll¢llm_J.G"(C + Ilullm+i-l.GIII)

S y(C+

....

Ilullm+i-l.GIII)II¢llm_i.G"'

124

From the inductive hypothesis,

B[¢, u]

f

Ilu\lm+]-l,GtlI'::;: y(C+ Ilullm,G)' Then u

V)o,G"1

~ y(C

(¢, f)o,G

~

Hm +},(Gt) and

Ilull m +'}, G':S; y(llfll o, G+ Iluli m, G)'

+ Ilullm,G)II¢llm_i,GII,

Gil,

Proof. Hypothesis 3° of Lemma 9.5 is

v satisfies the conditions of Lemma 9.3. (Note that Dju ~ Hm+r1(GII), if,.n.) Hence, Lemma 9.3 implies v ~ H/G') and so that, in

=

for every ¢ ~ C';(G).

Therefore,

!(D;:'¢,

125

Global Regularity

Elliptic Boundary Value Problems

D~v~ L 2(GII) for i';" n, since we have shown

IB[¢,

u]1

imm~diate, since

I(¢, 1)0,GI ~ IIfllo.GII~llo,G:S; IIfllo,GII¢llm-i,G

=

n-l

Ilvll·1, G'

~y(C_ + Ilull m, G+ i~ 1 IID~vllo ' Gil ~ y(c

+ Ilvll o, Gil)

. Therefore, the lemma applies and the theorem is proved. Q.E.D. As an extension, we have

+ Ilullm,G)'

THEOREM 9.7. If hypotheses 1°,2°, and 3° of Theorem 9.6 hold, and if for some k 2 0

(We have used (9.13) in this estimate.) Since, by the ellipticity of B, a .;,. 0, it follows that n

me ,me

4°

f ~ Hk(G)

5°

B is right (m + k)-smooth,

n

Dm u = (a n

me n ,me n

)-l(V -

~ a nfJ D{3u) ~ HJ(G'), IfJ\:S;m me, l3f.me n

and that

IID;:,uIIJ,G':::: y(C + Ilullm,G)'

Q.E.D.

From Lemma 9.5 we can easily get global regularity theorems for strongly elliptic equations. As a first result we have

then u ~ H 2m +k(Gt) and ~.•..

Il ul1 2m +k,G' ~ Y(lIfll k,G + Ilullm,G)'

,

Proof. The proof is by induction on k. By Theorem 9.6 the con­ clusion holds for k = O. Suppose that k 2 1 and that the theorem is true if k is replaced by k - 1. Then u ~ H 2m+k-l(G"), and (9.16)

IluI1 2m +k-l,G II .:s: y(llfllk-1,G'+ Ilullm,G)

THEOREM 9.6 Assume

1° 2°

=

that u ~ H (G), that for every ,~ COOo (Ix: Ixl < RI), 'u ~

°

H m (G)j

3°

S;

that for some j, 1 ~ j ~ m, B[¢, 0/] is a right j-smooth, uni­ formly strongly elliptic Dirichlet bilinear form in Gj . Thus, for any i

that f

~

m

L 2 (G) and that

B[D ,~ ,ri.,

Y(llfll k, G+ Ilullm,G)'

1, ... ,n - 1, and for ¢ ~ C';(G")

u] =

~

la\::::m IfJl.:s:m

(D IDa cp,

a a fJD fJ u) 0Gil ,

126

Elliptic Boundary Value Problems

I

whereE.=D.f-A.u• Clearly,E.~Hk I(G"). Moreover, 1 1 1 , ­

(Dacp, D /aa{3D{3u))o.G"

la\::;m I{3!:=;;m

[[E/ll k ­

I

== -

by

(Dacp,

B

1

(_l)l a l(cp, Da(a i {3Df3u))o G" - B[cp, Dju] lal::;;m a, 1{3I::;m -

:S: y( II Ej II k-l, G + II U 11 2 m. G " ) ::; y(IIEllk.G + Ilull m• G), where (9.16) has been used again. Thus, we have obtained the re­ quired estimates on D tU for i ,t. n. To treat the normal derivative, let A be the elliptic operator

A

~ cJa[+k(G) c

(D jCP' f)o,G" = ­ (cp, D/)o,G II;

The operator A can be written as a genuine differential operator with bounded coefficients, since aa{3 ~ CJa\+\G) C cJal(G). By (8.8) and hypothesis 3°, (cp, Au) = (cp, f), all cp ~ C';;'(G), since we know already that u ~ H 2m (G"). Thus, Au = E, and so, by Leibnitz's rule,

n

me ,me

=

(cp, D/f)o , G" - (cp, Aju)o , G"

=

(cp, Ej)o.G'"

D{3. a{3

CJal+I(G),

(_l)m a

thus,

B[cp, D I u]

(_l)l a ID a a

I

=

la\.\{3I::;m

since k L 1. Thus, A j is a differential operator of order at most 2m, having bounded coefficients. Now =

,i

IID p I1 2m +k-I,G' ~y(IIEillk-I,G" + [IDjullm,G")

(9.17)

la[:=;;m \(3I::;m

B[D /cp, u]

II1IJLJ, G)' "~

B[cp, D 1.u],

Note that the smoothness of B implies aa{3

+

According to Lemma 9.4, (JJ.u ~ H',,({;J!), so that, by the inductive 1 m __" hypothesis applied to D jU, D jU ~ H 2m +k-l (G') and

.I

j (_l)la IDaa {3D{3. a

I

IIfjII k-I,G" ::; y( IIfII k,'G

\

where

A. =

(~.16),

a{3D{3D iU)O.G'"

Let a ai {3 = D.a {3; in each term in the first sum on the right we trans1 a fer all the differentiations from cp to the other side of ( , ), obtaining

(cp, A 1.u)o • G"

IID/llk-I.G" + [IAjullk-I,G"

::; y(IIEllk.G" + IluI12m+k-I.G");

la\::;m \{3\::;m

B[DiCP, u] = ­

I .G":=;;

(Dacp, D j a a {3 • D{3u)o • G"

lalS:m 1{3I~m

I

127

Global Regularity

n

Dn2m u=E+

I [a\::;2m

b Dau, a

where the coefficients b belong to C*k(G), and b = O. Now a 2me n a n is bounded away from zero, since the Dirichlet form B is men,me

uniformly strongly elliptic on G. Since, by the results above, every

128

Elliptic Boundary Value Problems

From these remarks, it is also seen that if 12 is bounded and of class Ck with k 2: 1, then 12 has the restricted cone property. For, the Jacobian of the transformation (9.20) is one.

term on the right in (9.18) is in Hk(G') and satisfies bounds as given by (9.17), it foll ows that D~mu ~ Hk(G') and

IID~mullk,G' ~

(9.19)

i

THEOREM 9.8. Assume

1°

Since D~mu ~ Hk(G') and D/u ~ H 2m+k-l (G') for i J,. n, it follows that D"'u ~ Wk(G') for Jal = 2m. Therefore, Theorem 2.2 and Theorem 2.3 imply that u ~ H 2m+k(G'). The estimates we have derived, (9.16) and (9.19), then show that

I

I

y(jlfllk,G + IIYllm,G)'

Il ul1 2l1'1 t k.G' S Y(llfllk,G

+

Ilullm.G).

(xl'''''x i _ 1, xitl'''''x n )

Yi = Xi

Xi

> g(x'),

=

0, x' ~

B is right j-smooth in 12, then for every R,

u'l.

/ i 1=,

n 12 is mapped into ly: y/ > 0 I. In fact, we can assume that

an

s m,

If for some k L 0, 12 is of class C 2m+k, f ~ H k(O)' and B is right

(m+ k)-smooth, then u ~H2mtk(nR) and

-g(x');

n 12 is

f)o,n

Ilulimt l.n R s Y(llfll 0.12 + Ilull 0.12)'

mapped onto a hemisphere: about X O there is an open neigh­ borhood which is carried into a sphere centered at the image of X O having radius small enough so that the sphere is contained in the image of U; since is compact, a finite number of these neighbor­ hoods cover it. U

~

u ~ Hm+/(nR) and

IluI12m+k.nR S y(llfll k•n + Ilullo.n)'

n

Remark. If 12 is bounded, then the theorem holds with R replaced by This statement is also true if only is bounded or if 12 is 1ultimately cylindrical (i.e., it for sufficiently large R, 12 nix: Ixl> RI ~. consists of a finite number of truncated cylinders). ,t l'

then U

\\.

B[¢, u] = (¢,

n.

O X l' J"

(

H°m (0),

If for some j, 1 s j

an:

= x"1 -

~

for all ¢ E: C';(O).

Observe that such a region U n 12 can be mapped, by a mapping of class Ck, onto an open set with a flat part of its boundary correspond­ ing to U n simply let Yj

/

that u

~ U',

where U' is the projection of U on the hyperplane x.I and g ~ Ck(U'); and if

(9.20)

12 nix:

4° that f ~ L 2 (n), and that

an

2° U n 12 is contained in the half cylinderlx:

=

3°

~ 1, an open set 12 is said to be of class

= g(x'), x'=

thatn R

that B[¢, t/J] is a Dirichlet ;oifinear form of order m satisfy­ ing G~ding's inequality-tn 12,

Q.E.D.

about every X O ~ there is an open neighborhood U such that for some i, U n an has the representation Xi

c 2 m,

2°

Ck if

1°

that 12 (bounded or not) is of cl ass

Ixl < RI,

In order to extend the above results to arbitrary domains with smooth boundaries, we make

Definition 9.2. For k

129

Global Regularity

an

n

Proof. By the definition of the class c 2 m, every point in has a

neighborhood which can be mapped into either a sphere or a hemi­

sphere by a mapping of class C2m[C2m t k if 12 ~ C2l1'1 t k]. Only a

. finite number of such neighborhoods are needed to cover~. Theo­

.rem 9.6 and 9.7 can be applied in each of the mapped neighborhoods;

since the mappings perserve the desired properties of u, we obtain

130

Elliptic Boundary Value Problems

131

Global Regularity

(e.g., for the second part of the theorem) for each of the neighborhoods

THEOREM 9.9. Assume

U that u :€ H 2m + k(U) and

1°

that 0 is of class Coo and is bounded,

2°

that A is a uniformly strongly elliptic operator whose co­ efficients are in the class COOm),

Il uI1 2m +k,U ~ y(Jlfll k.O+ Ilullm.o)· Since a finite number of neighborhoods U cover OR' we obtain u ~ Wloe (0) and 2m+k

(9.24)

\Iul12m+k,OR ~ y(J Ifll k •O + Ilullm.o)·

3 ° that f

4°

~

COO(O), and

that u ~ H (0) is a solutio,b of the GDP for Au m / infinitely differentiahle-Virichlet data. (,

Proof. We are given f, g

(9.23) we n~ed only apply Garding's inequality (Theorem 7.6). For, since u ~ Hm(O), we may in (9.21) take ¢ = u to obtain =

(u, f) 0

.o.

°

Also, we may apply Garding's inequality to u, obtaining

B[¢, u 0]

I(u, f)o.ol

+ Aollull~.o

.$

Ilullo.ollfll o.o + Aollull~.o

~ ('12 + Ao)llull~.o + 'hl\fll~.o' Thus,

=

~

COO(f!),

~~d

u0

=

u - g satisfies

(¢, f) - B[¢, g]

for all ¢ ~ C';(O), where B is some Dirichlet form corresponding to A. This is from the definition of GDP. Since the coefficients of

A are smooth, we have B[¢, g] = (¢, Ag), so that for all ¢ ~ C';(O)

B[¢, u 0]

collull~.o ~ ~B[u, u] + Aollull~.o ~

f with

Then u can be modified on a ~et--f)fmeasure zero so that u ~ COO(fn.

T~us, indeed u ~ H 2m +k(OR)' To obtain

B[u, u]

=

=

(¢, f - Ag).

Now the second part of Theorem 9.8 applies, and k can be any posi­ tive integer. Thus, U o ~ H 2m +k (0) for every llositive k. By Theo­ rem 3.9, u 0 is equivalent to a function in Ci(O) for every positive j (note that 0 has the restricted cone property). Since u = u 0 + g, the result follows. Q.E.D. The results of this section can be used as a priori bounds. We shall . first need LEMMA 9.10. if 0 is bounded and in class cm, if u ~ cm(f!) and 0 if D u = 0 for lal ~ m - 1, then u ~ Hm(O). DC

Ilullm.o 5: c~~('h + Ao)'l,llull o.o + c~'l,r'l,llfllo.o. Substituting this in (9.24) yields (9.23). Likewise, (9.22) is proved. Q.E.D. As a corollary we have

an;

Prool. First we prove the theorem locally. Let XO ~ XO has a

neighborhood U such that 0 n U can be mapped onto the hemisphere

lR = Ix: Ixl < R, xn > O} by a mapping of class cm(11 n U), There­

fore, we can consider the restriction of u to n U as a function in

Cm(l R)' Assume that supp(u) C Ix: Ix I < R'}, R' < R, and let

n

_=

132

Elliptic Boundary Value Problems

for

uCx - (en)

u/x) =

o

x - (en ~

,,,]

_.,._.,_.,..

--_.

_-'=.::::~

w_:::_.,

"':'._....

~~

"._;;;;;;s==.::..~.

-

-

=... -_. _., ----_.,.

_._-,...._-......

lR'

-~:;:,-_-'-

133

Global Regularity

IIA'ullk,n ~ IIAullk.n + ylluI1 2m +k-1' k ~ O.

otherwise. o

I

-"::-.- ---

:!iTbus, by (9.26) applied to A' instead of A, we have 0

For « R - R', u( ~ C~C1R) CHmC1 R). Clearly, u( -+ u in HmC1R ) as O. Hence u ~ HmC1 R). To prove the result globally, use a special partition of unity as in o the proof of Garding's inequality. Q.E.D.

IluI12m+k.n~yCIIAull k,n+ IluI12m+k-l.n+ Ijullo,n)·

(-+

~

i

f,

/

hBy the interpolation inequality, Theqrem 3.4, for any (> 0 there is a itconstant C( such that i

~

I

THEOREM 9.11. If A

lI u I1 2m +k,n ~ Y(IIAulllc.n+ (I lulhm+k.n + C(llullo,n)'

a"p oc

=l H~2m

is B uniformly strongly elliptic operator on n, if for k ~ 0, n is bounded and in class C2m+k, if a", ~ ckcn), and if u ~ C2m+kCO) satisfies D"'u = 0 on

(9.25)

an for

lal ~ m - 1, then

Il u I1 2m +k,n ~ yCIIAullk,n + Ilullo,n)'

PiOof. First, assume that A = A' has constant coefficients. By o Lemma 9.10, u ~ HmCO). Moreover, f == Au r;: HkCO). Since, for ¢ ~ C';CO),

B[¢, u]

=

C¢,

AU)o,n =

C¢, f)o,n,

where B is any Dirichlet bilinear form for A, it follows from Theorem 9.8 that (9.26)

IluI12m+k,n~yCIIAullk.n + Ilullo,n),

so that the theorem holds in this case.

Now assume that A I has constant coefficients. Then

A'u

=

Au -

! H~2m-l

so that

BocDocu,

Choose ( so small that y( ~

Y2. Then

lI u I1 2m +k.n ~ 2yCII Au ll k,n + C(llullo,n) i' and (9.25) holds in this case. I"~ The extension to the case in which A I need not have constant co­ l" 0 i-'refficients is similar to the proof of Garding's inequality. For, con­ ~.. sider the form BCu)

=

IIAull;.n

for u ~ c 2m+kCO). Since the coefficients of A are in Ckcn), this is a ~'< Dirichlet form0 of the type we have considered. We wish to prove a ~·version of Garding's inequality: BCu) 2: collull;m+k - '\ollull~,

an

for the subclass of functions satisfying: D"'u = 0 on for la\ ~ m - l. We have succeeded in establishing this in the case that A I has con­ stant coefficients. Exactly as in the proof of Lemma 7.9 and its corollary, we can establish (9.27) locally: that is, (9.27) holds if o supp(u) has sufficiently small diameter. Of course, in Garding's in­ equality we assumed that u ~ C,;CO). But the proof showed that we really needed only two facts concerning the class of functions u: (1) the validity of an interpolation inequality, and (2) the fact that for

134

Elliptic Boundary Value Problems

135

Coerciveness ,OC:

(D'" v, D

l; ~ C';(E n)' l;u is in the same class of functions as u. These are both valid in our case. Thus, the interpolation inequality of Theorem 3.4 implies (9.27) holds locally. And the partition of unity argument of pp. 84-86 then implies the estimate (9.27) for all u ~ C2m+k(fi) satisfying: D"'u = 0 on for lal S; m - 1. Q.E.D.

where x'

an

(-1)

= oc:

1

n

= _ (D'"

n u)

(xl' .... xn _ +

••••

+oc:

n-I

10. Coerc iveness

l

+e

v, D

oc:-1

n

n

This chapter contains a treatment of some boundary value problems other than the Dirichlet problem. We shall need the following special form of Green's formula, which we have already mentioned (vid. equa­ tion 8.2). For simplicity, we shall assume throughout this section that the differential operators and the bounded region are sufficiently smooth.

n

[

D'" vD

f

",-I n

udx'

n

Ix'I
(D'"

t

11

+e V,

D

oc: - 1

n n

'

--txt I
(-1)

u) - (v, D"'u) ,

"'I +"'+"'n-I

=

I

u)-

and dx' = dxl ... dx n _ l • Therefore

)

+1

n

t

"'-I

D'" vD n n

udx'.

i,

On applying this argumentiin ':.. 1 more times, we obtain ,r,1

(-l)H(D"'v, u) - (v, D"'u)

'" - I '" -k-I n :~; THEOREM 10.1. Let A be a strongly elliptic operator of order f D""+ke v • D n udx'. = (-1)1"" I+k f n in Then there exist linear differential operators Nk(x, D) for k-O Ix'\
i

n. an,

n

n - (v,

(A*v, u)o

Au)o

•

l"'I

=

.H

f-I 2

J-O

f

an

aJu

Nf_I_' v , --J do,

] an

aJ/ ani is the exterior normal derivative of order j at an. Proof. First, consider the case that n is the hemisphere Ix:

«-l)HD"'(a",v), u) - (v, a",u)

where

R,

'" - I n

2

[xl <

xn > 0\ and

tive D'"

=

v v~mishes for Ixl : .: R', where R' < R. For a deriva­ D ,.D"'-e' with i ~ n, an integration by parts gives I

- (Div, D"'-e u) - (v, D"'u)

=

0,

0:.

1

+ •.. +0::

1 n-

'

terms:

t1

D'" +ke (a v) • D '"

IX'

n

-k-l

udx'.

n

On summing over a, we obtain

(A*v, u)-(v, Au)

'" n - I

QI".

(D'" v, D nu ) - (v, D"'u) = 0, n

where a' == a - a n en. Further partial integrations contribute boundary

~

f Ix'I
k-O

since the boundary terms vanish. Indeed, after al+...+an _ 1 partial integrations, we obtai n (-1)

I 'I

(-1) '" +k

=

2

2

l"'l.sf

k-O

f-I 2 j- 0

2 '"n::': j + 1

H.sf

(_l)I"'~I+k

I

f IX'[
(-l)H-j-1

n

f Ix'I
-k-l

l)'

D'" +ke (a v) • D '"

n

udx'

n

D"'-eJ+I)en(a",v). D~udx'.

136

Elliptic Boundary Value Problems

J

Let No

~

.v =

r.-I-}

(_l)!"'!-ID"'-(i+ l

....

)e

n

Ix'\
(a"'v.

.)

H~

where

a(O, ... ,o,e)

eDe-i-I, n

(0 •••• ,0, )

cannot vanish since A is elliptic. Also, e-I

(10.2)

(A*v, u) - (v, Au) =

I i-O

J

I H~I "'.fl e n

J

Ix'I
N; f-I-'" = n

I

Ioct I;51-"'n

(-l)HD""(d", INe-I-j)

'

associated with (-D n tnu is of order at most e-1-a n , and does not contain the pure normal derivative of order e- 1 - a n . The relation (10.3) can be written (Ne_I_lv, (-Dn)lu )' = (dINe_I_jv, D~u)'

i-I

oc

- I

(N; e-I-oc v, (-D)

oc -0 n

'

n

d", , iD"',

(¢, !/J)'

d,Ne_I_'v , Dludx'

}

:',G l,

where a' = (al' .. n_ 0). Note that an < j. Hence, the term with the oblique derivative oto~der j can be represented as a term with the normal derivative of order j plus terms in which the normal deriv­ ative is of order less than j. Moreover, the operator

Ne-I-Jv, (-Dn)Judx',

where d l and doc • 1 are smooth functions of x'. Since v vanishes for Ixll > R', an integration by parts gives

Ix'I
icxI:S:1

(-l)HD""(d"',jNe_I_I V ) ' (-D ntnudx'

C'

,

so that the theorem is proved in the case of a hemisphere and v vanish­ ing for Ixl L R'. In (10.2) the normal derivative -D n can be replaced by any non-tan­ gential derivative, whose direction is a sufficiently smooth function of x'. Such an oblique derivative DB can be represented as a linear com­ bination of D 1' ... ,D n' Thus for some d i .f 0, d J .Dis = (-D n )i +

J Ix'I
",.fle n

Then the surface xn = 0 is non characteristic for Ne-I-J' since Ne-I-J contains the term (_l)e-I a

Nor.-I-I v· (-D n )Judx'

I

+

"'n2J+ I

(10.1)

137

Coerciveness

10

B

=

J

¢(x')!/J(x')dx'.

\x'I
=

e-1, we obtain

e-I

I (Ne_I_lv, (-D n)i u)' = (de_IN ov, D:-Iu)'

J- 0

J

Ixll
I

Ne_I_ v , (-D )iudx' l

e- 2

- I

J

icxI~J I x'I
Ne-I-iv, d""pocudx I

ex :0

n

,

'"

(Ne_1 e-I-'" v, (-D n ) '

n

n u )'

n u )',

138

Elliptic Boundary Value Problems

sec. 10

£-2

where the operator Nt{-i is of order j and the surface xn = 0 is non­ .. characteristic for this op~rator. Therefore, (10.2) becomes

+ i:O (N£-l-i V, (-Dn)J u )'

= (N01v, D£-lU)' S

+ ~ (N) J-O

/ ~/

1

t- -

JV, (-D )J U)', n

where

J

=

d£_lN O'

N£-l-i

=

'

­

N£-l-J - N£-!,£-1-i' 0::;; ] ::;;

to,

J-O

L-l-i'

n

=

(N 2v, D£-2u)' + 1

S

O~ ee OJ'

~ (N&

where the surface x = 0 is non characteristic for n Therefore, (10.5) becomes

£--1

~ (No

i-O

n

NtL-l-i , 0::;; j

::;;

£- 2.

v (-D )i u )' = (N1v D£-lU)' + (N 2v D£-2 U)'

L-I-J'

nO'

l'

S

s

£-3

+ ~ (Ng J-O

L-l-i

v, (-D )JU )'. n

n-UO~eus;.

n.

v (-D )Ju )',

L-I-J'

S~ ee Si'

Let It;;) and I(il be collections of functions in COO(E) such that (I L 0, (i 2 0, supp«(.) e O~, supp«(i) e S'., and ~(_I + ~(i = 1 on I 1 J Let VI = (IV, vi = (iv. By the remarks above,

~3

i- O

(10.7)

(A*v i , u) - (vi, Au) =

~

~ (Nt~_JV'

J-O

9

I

~£

the transformation is 1. If A~ is the formal adjoint of AI' then we must have

£-1 (N£_l_IV, (-Dn)J u )' =

J

[(u, A*v)o,opn] into (Aju, v)O,H[(U, Alv)O,H]' since the Jacobian of

An induction argument then shows that

J-O

o.

The boundary regions are more difficult to handle. Consider a par­ ticular boundary neighborhood Oland let r j be the mapping which carries 0 inn onto a hemisphere H. The mapping r j carries the operators A and A * into smooth linear differential operators A j and A.,J respectively, on the hemisphere. The operator A.I is the formal adjoint of A: if u, v,;;: C';(O _ n n), then r. carries (Au, v)o 0 nn I

£-1

D~udx',

-'r~j

an eUO~,

Nt-l- J

v (-D )Ju)'

.

--­

where N£-1-J = N£-i~'i-~/ Now we continue with the proof of the theorem. If v vanishes for Ilx I L R', then the theorem holds trivially for the sphere Ix: Ixl < Rl, since the boundary terms vanish. Now consider an arbitrary bounded region n,;;: C£. Let 1011 be a finite open covering of of the type described in the remarks following Definition 9.2; i.e., there is a smooth mapping carrying 0 1 n n onto a hemisphere. Let IS.l be a finite collection of spheres J such that S.J en and n - UO.I euS J.• Let 10'I.1 and IS'.l be sets satJ isfying

Now a relation similar to (10.4) holds with N£-l-i replaced by ' 0::;; j s: £- 2. Therefore, the argument just given shows that

~ (NJ

\

0

f N£-l-iv, Ix'I
an,

£- 2.

Since N£-1,£-1-J does not contain the derivative D~-l-J, but N£-1-J does contain this derivative, the surface x n = 0 is non characteristic this surface is non charac­ for Nl- 1- i , 0 s. j ::;; £ - 2. Since d£_1 teristic for N~.

~2

J-O

I o

N~

£-1

(A*v, uL:::fv , Au) = _~

(10.6)

£-2

(10.5)

139

Coerciveness

Diu)', S

(v, Alv)O,H = (Aju, v)O,H = (u, A:'v)O,H

=~-=-~-=

-=~-=-.::=-===~~.=-"""'=

140

Elliptic Boundary Value Probl ems

sec. 10

1\

= A~. Thus, by the first part of the for all u, v <;: C';(H); hence. proof, for Ai' At, we have an identity of the form (10.6) with the oblique derivative Ds chosen in such a direction that its image under is a normal derivative on an. Thus. after mapping back into 0 1 n n, we obtain

,;1

Coerciveness

II Nf-l-J , I VI =

141

1 1(,1'1.,

#-l-Jv

~

u-1-J +...,

Il

where l 1]1 cannot vanish since the numbers :R 7J1 all have the same

sign and III = 1,i(,1 ~ O. Thus, the result is proved. Q.E.D.

As a corqflaryto the proof, we have

'I

(A*v l , u)o

I

nn -

(vI' Au)o

I

THEOREM 10.2. Let

nn

aiu d

= ~ f Ne-l-J IVI • i-O anno., . anJ 0,

where N k.1 is a linear differential operator of order k, and the surface n 0 I is non characteristic for N k, j" Recalling (10.7), we sum over all i to obtain

an

I

B[u, v] =

f-l

(Dau, aafJDfJv)o,n

lal5::m

IfJkO:m

be a bilinear form which is uniformly elliptic in n (n bounded while aafJ are sufficiently smooth functions defined in 0). Then there are linear differential operators N k(X, D), x <;: such that for u, v <;: C 2 rn(fi),

an of order k, k = O, ... ,m -

(A*v, u)-(v, Au)

=

=

f-l

I

i

I

i-a

f-l

B[u, v] = (u, Av)o

f

anna I Ne-l-·

1.

IVI • -aJu do anJ

fan (~N v ) aJu 1 e-l-J,l l ani

I

J-O

do,

A=

I

where Nf-l- J is a linear differential operator of order e- 1 -j, for is non characteristic. By Leibnitz's rule, we have which

an

J-O

f

an

aJu

- N 2 _l_Jvdo,

anJ m

lal:$m

IfJl:$m

an is non characteristic for N k'

Defil)ition 10.1. Let V be

a linear space, Rm(n) C V CHm(n).

A bilinear form B[ul. v] =

I

(Dau, aafJDfJv)o.n

lalsm

IfJl:$m

#-l-lv

(,/1 1 an E- 1 -i +....

where 1]1 does not vanish and ... represents terms in the tangential derivatives and lower order derivatives of v. Observe that the sign of :R'1 is (_l)f-l-J, if A is suitably normalized (d. (10.1)). Thus, 1

I

(_l)laIDaaafJDfJ,

and the surface

II N f-l-i,I V I = Ne-l-J v ,

=

m-l

where

since v I vanishes outside 01, Therefore, we must show that

Nf-l-J.l v i

n+ •

is coercive over V if and only if there are constants Co and A such o that co> 0 and

(10.8)

:RB[u, u] 2

collull;,n - Aollull~.n

1,

142

Elliptic Boundary Val ue Problems

for all u ~

v.

sec. 10

two assumptions of the nature of V. First. if v ~ V, then (v ~ V. Second. if v ~ V. then sufficiently small translations of (v tangent to the surface x n = 0 al!¥l~give elements of V. Thus. if both these I assumptions hold, then one can establish the boundary regularity of j u. As an e~mple. consider the case in which V = H m (n). Both the above conditions are obviously satisfied. so that u is highly regular are sufficiently regular. Let us then assume in in case f. B:antl that everything is sufficiently regular for the following manipulations. We shall show that u is actually a solution of a certain boundary-value problem. For. according to Theorem 10.2,

B[u, v] is strongly coercive over V if (10.8) holds with

.1. 0 = O. From Theorem 7.12. we see that a coercive bilinear form must be strongly elliptic. Indeed. it would suffice to assume (10.8) with :RB(u. u] replaced by IB[u, u] I.. Then Theorem 7.12 shows we could multiply B by a complex number to achieve strong ellipticity for B. Assume that B[u. v] is strongly coercive (a strongly coercive qua­ dratic form can be obtained from one that is just coercive by adding Ao(u, v)o,n)' Then. if f ~ L 2(n), by the Lax-Milgram theorem there is a unique u ~ V such that

(10.9)

B[v, u]

=

n

n

(v. f)o,n B[v, u] = (v. Au)o

for all v ~ V. For this we must assume V is closed in Hm(O). Thus, we have sol ved the problem of finding for each f ~ L 2 (n) a function u ~ V such that (10.9) is satisfied for all v ~ V. In case a V = H (0). this is just the generalized Dirichlet problem already m a discussed. In case H m (0) is a proper subspace of V, then we have no longer solved the GDP. but some other type of problem. which we can consider as an abstract boundary-value problem. The question now arises concerning the regularity of the solution u. The interior regularity is no different from the theory we presented in section 9. or. alternatively, the interior regularity theory of sec­ tion 6. Thus. in case f and B are sufficiently regular. then we can gi ve precise statements concerning the regularity of u. Then. under sufficient assumptions. u ~ H~:C(n), and since (10.9) is satisfied a

for all v ~ C obtain m-l

L

j-O

L

2m

m).

n+ •

m-l

L j-O

f

ajv

-. N 2m-l-judo

an anI

But (10.9) implies B[v, u] = (v. AU)o,n. so we

f

ajv an anj N 2m_l_judo = O.

Since the functions ajv/ anj are essentially arbitrary. this implies that on an (10.10)

N 2m-l-jU = O.

j = O, ••• ,m -

1.

Recall that N 2m-l-j is order 2m - 1 - j, and N 2m-l-jU contains the term a2m-l-jU/an2m-l-j with a non~vanishing coefficient. Thus,

fortiori for all v ~ C~(n). u is a weak solution of Au = f, where

A =

143

Coerciveness

the boundary conditions for u are specified by m differential operators on an, having orders m, m + 1•... ,2m - 1. Note that in this case the boundary conditions actually depend on the choice of the Dirichlet form B. These are so-called natural boundary conditions. The idea can be generalized. Let B 0"'" B k-l be sufficiently smooth linear differential operators on an. where k ~ m and the order of B i is less than m. Assume that the orders of B 0"" B k-l are distinct, and that an is non characteristic for these operators. In this case V shall be the closure in H m(0) of the set of functions ¢ .€ . cm(f!), such that B/¢ = 0 on an, i = D, •.. ,k - 1. If k < m. let Bk..... B _ be differen­ m l

(_l)lalvaaat3v,B.

lal':=;;m

1,BI~m

As usual. the question of boundary regularity is more complicated. We can reduce the considerations to the case of a hemisphere Ix: Ix I < R, x n > 01, by making sufficient smoothness assumptions on n. Then we need to consider functions (~C~(!x: Ixl < R\), as in the results of section 9. If one examines the proofs in section 9. it is evident that the same arguments will work in our case if we make

.

-._------­

144

Elliptic Boundary Value Problems

an,

an

tial operators on such that is non characteristic for these operators, and if 0 :::;; j :::;; m - 1, then, for some i, B I has order j. Then, by the proof of Theorem 10.2, we obtain a Green's formula m-1

B[v, u]

(v, Au)o n + I

=

1-0

.H

Coerciveness

(10.14)

B[v, u]

an

B.v •

n

J

2

_l_.uda= m

+ I (v, cP,u) + (v, du), j-

in a non-unique fashion. The form B comes from integrating by parts so that B[v, u]

(10.16)

on

an, j = k, ••• ,m -

o an,

n

n

I aIID;DI+ I aPI+a,

l.j-1

I- 1

so that the inequality

1.1- 1

J

n

-

2

alP ,uDludx.:::t E o lul 1

.n

1-1

where all isreal, and for real n

I al/~le.

',J-1

n alP,uDludx,

ID lu\2,

Eo I

I n

J

so that the condition (10.13) implies the ellipticity of this particular o form, and, hence, by (10.15) the ellipticity of B. Note that Garding's inequality is trivial for this second-order case, since the integrand in (10.16) is already bounded below by

n

(10.13)

(v, Au)

_

1.1- 1

1.

Thus, in addition to the given boundary conditions, B ou= ...=B k _ 1 U = on the function u also satisfies the natural boundary conditions (10.11). Note that if k = m, then no natural boundary conditions a appear. It is therefore seen that the critical fact needed for treatment of general boundary-value problems is the coerciveness of a quadratic form. For an example of the usefulness of this criterion, let us con­ sider the case of second-order problems (m = 1), and a strongly elliptic operator

A=-

=

n

I

(10.12)

1

is satisfied in case v, u (C';(n). Now one possible selection for B has the principal part

an,

M2m- 1 -P = 0

(D IV, blu)

'-1

O.

J

v (C"'(fi), Blv = 0 on i = O, ... ,k - 1. Thus, since Bkv, ••• ,B"'_lV can be chosen essentially arbitrarily, we obtain

(10.11)

(Dlv, alPlu) + I

1.1-1

(10.15)

",-1

an Blv. M

n

I

=

M2m _ 1 _ l uda,

an

I J

145

n

J

is non characteristic. If where M2m-1-j is an operator for which u ( V and (10.9) holds for all v ( V, then we obtain

j-O

sec. 10

J

2

f,

Eol~W·

By the procedure given in section 8 we can associate with A a Dirichlet bilinear form,

is immediate. In higher order problems it is no longer the case that the integrand in B[u, u] is positive. Since the lower order terms in (10.14) can be easily estimated in terms of lul:.n with a small co­ efficient, and lui ~.n, it follows that B is coercive over H 1 (n) if we assume only (10.13). As we have remarked, the boundary conditions are connected with the choice of B. In the second-order case, the computations are fairly simple. Let us write A in the form

146

n

A

= -

n

n

I-I

I-I

au au -au + g-+

2 a IJD p. + 2 a P I + 2 {3 P I + a,

1,1- 1

J

an

where, of course, by comparison with (10.12) we must have alj + ajl = all + aJI'

147

Coerciveness

sec. 10

Elliptic Boundary Value Problems

aT

=

0,

where T = (T 1 ,. .. ,rn) is/ iil gi~en smoothly varying unit tangent vector. Since the pr()1JI~Ill/6f choosing the quantities a l is trivial, we con­ centrate on'; the choice of all' Now the given boundary condition can be written \

a l +{3I=al • Then, if n.1 is the i formula gives (v, alDlu)

= -

th

component of the exterior normal to

an, Green's

Since A (D .(alv), u) + 1

- (v, aljDpju)

=

Ian va .un .da, 1

(D/allv), Dju) -

I

=

B[v, u] + Iafl

[- 2/ ,}. a.Jn:DJu + 2 1;.n.u]vda, 1

1

1'\, the coefficients a ..J must satisfy 1

ail

Ian vallJ-;;nlda,

1

I

for a certain bilinear form B. Thus, the associated boundary condition for u is

(10.17)

=

(10.18)

assuming a IJ.. to be real. Thus, summing these expressions gives (v, Au)

au -a + 2.g rJD.u + au = O. n J J

- 2 1,ja lj np j u + 2 1a ln l u = O.

In case a IJ.. = alJ = a JI' the expression 2 1,. J.a t..n .DJu is called the J 1 conoTmal derivative of u corresponding to the operator A; if A = 1'\ (the Laplacian), the conormal derivative becomes 2 np ju = au/an, J the normal derivative. Usually, of course, the boundary condition is given in advance. Thus, it is important to be able to pick the form B in such a manner as to obtain the given boundary condition. That is, a I..J and a l must be chosen in such a fashion that (10.17) coincides with the given boundary condition. We shall now demonstrate how this can be ac­ complished in the case that A = 1'\. Suppose then HB t the given boundary condition is

= 1,

a IJ .. +a J..J =O,

i:pj.

we shall show that we may indeed choose alj

=

0IJ + (n/ j - n/)g.

°

Here I J is the Kronecker o-function while the functions n I' T I and g which a-priori are defined only on the boundary are supposed to possess smooth extensions into Using these extensions, a IJ.. are _ smooth functions in n which obviously satisfy (10.18). Moreover, the boundary condition (10.17) becomes

n.

0= 2 1)01j + (n/ j - n/I)g]npju - 21aplu

= 2 jnp JU + g2/p j u2 In; - g2PP ju2P/1 - 2Pl n lu =

anau

Ii

+15

an

au + 0- 2 a n.u, aT / 11

since on 2n~1 = 1 and InIT.1 = O. Therefore, this boundary condition is equivalent to the given one, if the quantities a l are chosen correctly.

148

Elliptic Boundary Value Problems

sec. 10

We can also treat mixed boundary-value problems using this proce­ dure. For instance, suppose we wish to obtain a Dirichlet form giv­ ing the boundary conditions

11

B'[v, u] = 2 (D/v, a/Piu)O.n i,i- , corresponding to

au + au = OonalO, ae

AI

=_= :1

u = Oona 2 0,

a

th~

principal part A' of a strongly elliptic operator,

n /-­

2' a i i D P l' i,i-'

'"

a

an.

where I 0 and 20 are disjoint; and their union is Th~n we take for V the closure in H ,(0) of the cl ass of functions in C ~(O) \l{hich vanish vanish on 2 0. And then the Dirichlet form must be chosen in such a manner that the boundary condition on alo emerges as a natural bound­ ary condition. It should be noted that our regularity theorems hold except in neighborhoods of points in a,o n a20. Our theory fails there because at such points tangential translations of functions in V do not remain in V. And, indeed, singularities actually Occur at such points. To conclude this section, we shall briefly mention some a priori estimates. Let A be a sufficiently smoo'ch, uniformly strongly ellip­ tic operator on 0, let 0 ~ C2, and suppose that u ~ H 2 (0) n C '(0) satisfies

a

au + au = 0 on an, ae where a;ae is a smoothly varying directional derivative that is not tangent to an. From the regularity theorems, we obtain the a priori

~B'[U, u] = Jh . ~_PDiU, '.1

=

!.

i.i- ,

where a~i = Waii +

(DiU,

a;;').

11

:E a~/li = ~

i,i-'

+

Ilullo,o).

We can obtain such results for strongly elliptic second order problems which are not real, such as problems for the operator D ~ + D; + eieD;, where e is a constant, lei < 17.

aiPiu)O,O + (a/Pi u,

a~Piu)O,O'

For real ~,

11

2 ailli > O.

i,i-'

Hence, for some positive constatn co'

i

i a~lli2col~12.

,i- I

1

It follows that

estimate

Il uI1 2,0:s,; C(IIAullo,o

149

Coerciveness

~B'[U, u]

=

J

o

11

2 Diua~iDiudx i,i-' n

2C O

2

21 D i u lo•o

i- 1

= coluli.o·

D/u)o,O]

150

Elliptic Boundary Value Problems

11

Therefore, B' [u, u] is coercive over H 1(0). Hence, we obtain the regularity results and the a priori estimates. As an application, we present the following trick which will be use­ ful later. Let A be a uniformly strongly elliptic second order operat~r over 0, normalized in the usual fashion, such that A has real coeffi­ cients. For x = (x!'""'.":n)' let (x, r) be a vector in E n + 1 • For some constant 0,101 < TT, let L

=

~u =

0,

~~u = o. an This is ~()t)~ regular boundary-value problem; the form B 1 is not co­ erciv~/over H 2(0) since the space of harmonic functions in has infinit~mension. However, the second,

n

A - eiODi

B 2 [cp, l/J] on the cylindrical domain 1 = 0 x E l ' Then L is uniformly strongly elliptic on 1, as is easily verified. Suppose that u ~ H 2 (f'} n C 1(f') satisfies

au

iJl + a(x)u = 0 on

e

151

Coerciveness results of Aronszajn and Smith

al,

where is a smoothly varying vector parallel to the 0 plane, but not tangent to Then we have the a priori estimate

an.

(Wi -

=

D;]cp,

Wi -

D;]l/J) + 4(D 1 D 2 CP, D 1 D 2 l/J),

leads to a regular boundary problem. This form is coercive over H 2(0), since the polynomials and 2';1';2 have no common complex zero, except'; = O. Cf. Theorem 11.9. More generally, it can be shown that if one considers the Dirichlet form tB 1 + (1 - t)B 2 , t a real parameter, then this form is coercive over H 2(0) if and only if - 3 < t < 1. For a proof see S. Agmon, Proceedings of the International Symposium on Linear Spaces, Pergamon Press, Jerusalem 1961, pp. 1-13.

';i - .;;

Il ull2,l ~ C(IILull 0,1 + Ilull o,r)· 11. Coerciveness Results of Aronszajn and Smith We note that if B[v, u] is a Dirichlet bilinear form corresponding to A, then

B t[v, u]

=

f

00

10 B[v(x, t), u(x, t)] dt + e (D tV, D tU)o,l

-00

is a Dirichlet form corresponding to L. Finally. we present a brief remark concerning the biharmonic opera­ tor in two variables. As mentioned on p. 96, there are the following two distinct Dirichlet forms corresponding to ~ 2. The first, B 1 [cp, l/J] = (~cp, ~l/J), comes from the Green's formula

(v,

~ 2U ) = (~v, ~u) + fan

(v

:li- : ~li

and gives the associated boundary conditions

)da,

In this section we shall give some very precise conditions for a special type of quadratic form to be coercive over Hm(O). Before ob­ taining these results, we need two auxiliaries: some results of Calderon and Zygmund, and the Sobolev representation formula.

Definition 11.1. A Calderon-Zygmund kernel is function on E n - 10\ satisfying the conditions: 1°

k is (positively) homogeneous of degree -n; that is, k(ax)

2°

a measurabl e

= a -nk(x),

a

> 0,

x

f, 0;

and

for lal = 1 k(a) is a bounded measurable function on the unit sphere I, and k has mean value zero; that is,

f I

k(a)da = O.

152

I

II!

Elliptic Boundary Value Problems

Coerciveness results of Aronszajn and Smith

11

In general, a function h(x) is said to be (positively) homogeneous of degree r if h(ax) = a"h(x) for a > 0 and x ~ O. Note that if h is differ­ entiable, then D jh is homogeneous of degree r - 1. The following result exhibits a wide class of Calderd'n-Zygmund . kemels. LEMMA 11.1. Let h ~ C1(E n - 1Oj) and let h be homogeneous of degree 1 - n. Then for i = 1, ... ,n, D jh is a Calderon-Zygmund kernel.

Proof. We have remarked that D jh is homogeneous of degree -no Thus, we need only exhibit that D jh has mean value zero. Let p(t) be a non-negative function of a real variable such that p t;: C(E 1)' supp (p) c [1, 2], and (" Eili2 dt = 1. Now an .integration by parts o t shows that

LEMMA 11.2. Let k be a Calderon-Zygmund kernel, let 0 < i < R,

i,lJtId let k(x), ki,R(X) =

E

E

n

2

R.

-\

T~en if ki,R is the Fourier transform of k£,R' there exists a constant

c,independent of i and R, such that for all

e~ En

/\

\ki , R(g)\ '::;;c. Moreover, for all

e~ E

....

£-+0

Ixl

n

< Ixl < R,

n

there exists

1:\

k(g) == limk£ R(S/.

h(x)p'(lx\) !L dx,

Djh(x).P(lx\)dx=-f

i

Ixl .: ; i, I,xl

0,

."

f

153

R-+oo

'

.J\

where p'(t) = dp/dt. Next we introduce polar coordinates x = ra, r = Ix I, in each of these integrals to obtain (D jh) (a)

00

f

f

o I

rn

00

f

p(r)rn-1dadr=-f

h( )

---.!!..-p'(r)a.rn-1dadr,

I r n-

0

1

1

which simplifies to

(11.1)

foo eJr) dr f o

r

I

(D.h)(a)da= - foo p'(r)dr f h(a)a/da. 0

1

I

Remark. The notation k is used here since this function behaves

somewhat like a classical Fourier transform, even though k itself

does not have a Fourier transform in the usual sense. In the distri­

i\ bution sense, k is actually the Fourier transform of k, though we shall not need this fact. As a corollary of this lemma we have LEMMA 11.3. Let k be a Calderon-Zygmund kernel and let

f ~ L/E n ). Let k£,R *f be the convolution

(k c R*f)(x) = f 1;;.

k£ R(X - y)f(y)dy.

B' n

Since foo r-1p(r)dr = 1 and since foo p'(r)dr = 0, (11.1) becomes

o

f

0

(D /h)(a)da = O.

I

Q.E.D. We shall need also the following basic lemma of Calderon and Zygmund, which we state without proof. For an elementary proof see A. Zygmund, on singular integrals, Rendiconti di Matematica 16 (1957), pp. 479-481.

Then, kf,R *f converges in L 2 (E n ) as £ -+ 0, R -+ 00. If k*f denotes the limit in L 2 (E), then the following estimate holds:

Ilk*fll oE .s::(21T)n/2c~lflloE ' ,

n

• n

Where c is the constant appearing in Lemma 11.2.

Proof. Using a familiar fact about the Fourier transform of a con­

volution,

154

Coerciveness results of Aronszajn and Smith

Elliptic Boundary Value Problems

(11.4)

*_

kf,R f - (217)

nil'kf,Rf.A

Moreover, for a certain constant c depending only on h,

I 1.1

i

IID,(h*f)lle,E ~ cllfll e • En • n

Thus, Parseval's relation shows II kf,R *f - kf',R,*fll o2 ' En

=

ll[" ·"]/\ 2 (217 ) n kf,R - kf',R' fllo,E

= (217)n f

A

E

"

Remark~/Although the last estimate shows that really D i(h*f) ~ .L 2(E n ),istill we must write h*f ~ H~OC(En)' since h*f itself may not

n

Ikf,R(x) - kf',R'(x)\

2

.

n

1\

• If(xWdx. Using the previous lemma, the integrand in the last expression is bounded by the integrable function 4c211(x)12, and the integrand tends to zero a.e. as f, f' .... 0, R, R' .... 00. By Lebesgue's dominated con­ vergence theorem, the integral tends to zero, so that by the complete­ ness of L 2 (E), k f R *f converges to a function k*f in L/E). By (11.4) and Lemma 11.2, •

,O<x)\ ~ 2

b~)nL/E).

~vf, f

En

Ih(x - y}1 If(Y)ldy ~ max Ih(a)\ sup If(Y)I. \al-l y~En

- f

(h*f)(x)D /¢(x)dx

En

=

lim -

fR

O

oo

If <€ lx-yl
f

~

h(a)a,da.

Let f be a bounded measurabltf function having compact support in En' Then h*f ~ H~oC (En)' and h*f has weak derivatives given by the formula

D i(h*f)

=

ki*f + c /­

Ix _ yll-ndy. supp(f)

h(x - y)f(Y)Di¢(x)dydx.

By Green's formula, we have for fixed y (11.7) -

f h(x f
y)Di¢(x)dx

= -

f

IX-yl-R

- f

x/_y, h(x - y)¢(x)~ da x y/-x, h(x - y)¢(x)- da

\X-yl-f +

C/ =

f

Let ¢ ~ C';;'(E n). Then

(217)n C 2\'ftx) I2.

THEOREM 11.4. Let h ~ c1(E n -lOn, let h be homogeneous of degree 1 - n, and let k/ be the Calderon-Zygmund kernel given by k i = D J.h. Let

We have

Since f is bounded and has compact support, it follows that h*f exists

as an absolutely convergent integral, and is uniformly bounded.

(11.6)

If we integrate this expression over En and use Parseval's relation, (11.3) follows. Q.E.D. Now we state a result we shall frequently use in the remainder of this section.

(11.5)

155

f D ih(x f
f

x

y)¢(x)dx.

Suppose that fey) == 0 for Iyl L A, and suppose ¢(x) == 0 for Ixl L B. Then if Iyl < A and R > A + B; it follows that for Ix - yj = R we have

Ixl > R - A > B, so that ¢(x) = O. Thus, if y ~ supp (I) and R> A + B,

then the integral in (11. 7) taken over the sphere Ix - yl = R vanishes.

Also, if in the integral over the sphere Ix - y\ = f we set x = y + w,

then we have

•

156

Elliptic Boundary Value Problems - f

y/-x i h(x ­ y)¢(x)-- da

Ix-yl-(

(

h ( ) _a_-/..(y + w)a

f

=

I

x

(

n-I

~

I

THEOREM 11.5. Let C be an open cone of height h and open­ ing with vertex at the origin. That is, for some unit vector t', the axis of the cone, C = {.;: 1';1 < h, .; • ';0> 1';1 cos el. Let I be the surface of lhe unit sphere, and let A be the portion of I subtended by C. Let ~ coo(I), supp (¢) c A, and suppose

(n- 1da

e

I

= f h(a)a.¢(y + w)da I I

IIII

11

cl¢(Y)

=

157

Coerciveness results of Aronszajn and Smith

sec. 11

/

I

¢(a)da = (_l)m

(m - 1)! '

+ f h(a)al[¢(y + w) ­ ¢(y)]da,

I

for

by (11.5), The last integral tends to zero with (uniformly for y ~ En' since DI¢ is bounded. Thus, from (11. 7) we have for R sufficiently large and for y ~ supp (f) - f

«lx-yl
h(x ­ y)DI¢(x)dx

f

=

E

~

00

R

(h*f)(x)D/¢(x)dx E

=

lim f ( .... 0

n

is defined as in

R~oo

=

lim f

( 0 E R oo n

= lim f

( 0 E R oo n

- f

¢(x)dxf

E' n

I

(_l)m

(m- 1)!

/ 0

~ u(ta)dt.

dt m

Multiply this relation by ¢(a) alld integrate over A to obtain n

(11.8)

u(O)

h

=

f ¢(a)da f t m -

A

1

dm -

dt m

0

u(ta)dt.

f(x)¢(x)dx.

E

Now

n

2

=

tm - I

f(y)¢(y)dy

E

(k; R*f)(x)¢(x)dx + cJ •

u(O)

f(y)¢(y)dy n

k; R(x ­ y)f(y)dy + cJ

(11. 9)

(E n ) to kl*f. Thus, we have

(h*f)(x)DI¢(x)dx = f [(ki*f)(x) + c.f(x)]¢(x)dx. E E ' n

E

n

•By Lemma 11.3, k/ R*f converges in L (.

k; R(x ­ y)¢(x)dx + cJ

E'

n

L D"'u(y)dy.

Iyl Iyln

m! ¢(L)

f

'"

f(y)dyf E

I

locI-m c a!

Proof. Let a be any unit vector such that a . ';0> cos and con­ sider the function of a single real variable, u(ta), 0 ~ t ~ h. Expand this function in its Taylor series about the point t = h. Then, since u(ta) vanishes for t 2: h - 8,

(11.2). Substituting this is (11.6), - f

=

cm(c), where u == 0

8, some 8 > O. Then

e,

•

where lim J(y) = 0, uniformly for all y.Here k/

Ixl : : h u(O)

k; R(X - y)¢(x)dx + cl¢(Y) + J({Y),

n

I. Let u ~

where da is the surface element on

~ u(ta) = m dt

Q.E.D.

I

I'" I -m

m! a"'(D"'u)(ta). a!

This formula can be proved by induction on m. For, if it is valid for a given value of m and we differentiate once more, we obtain

n

We now pass to the proof of the SOBOLEV REPRESENTATION FORMULA:

(11.10)

L

dm+1

dtm+l u(ta) =

I

locI-m

I

n

m. a'" I a!

I-I

a (D"'+elu)(ta), /

158

Coerciveness results of Aronszajn and Smith

Elliptic Boundary Value Problems

so that, for any {3, !{3! = m + 1, the coefficient in (11.10) of af3D{3u is

m! = m! I i-I {31, ... ({3i - l)! ... {3n' i-I ({3 - e i )!

i

m!

{3! = (m

I

H-m

f ( A

L

{3j

(tl:92 Col lull;' ~ (C + Ao)llu\ 1;'-1

/-1

+ 1)!

(3!

then L i~4inite-dimensional. Pro()t! Since B is coercive over L, (11.1) and Definition 10.1 imply

•

Thus, (11.9) is established by induction, as it is obvious for the case m= O. Substituting (11.9) into (11.8),

u(O) =

~

\~B[u, u]\ i: Cl\ul\;'_I.n,

i

n

L of Hm(n). If for all u

159

t m- I m! ¢(a)a"'(D"'u)(ta)dtda.

a!

Setting y = ta-

for

u ~ L.

Alth-.9 u gh L CHm(n), we shall consider L as a subspace of Hm_I(n); let L be the closure of L in the space Hm_I(n). It follows from (11.12) that L cH m (n) and that (11.12) holds for u ~ L. Let lu k } C _ L, llukllm-I i: 1. By (11.12), the norms Ilukllm are bounded. Thus, Rellich's theorem, Theorem 3.8, implies that a sub~equence of lu k} converges in Hm_I(n). Therefore, the unit ball in L is compact. But if the unit ball in a normed linear space is compact, the space must have finite dimension. Thus L is finite-dimensional. Q.E.D. LEMMA 11.7. Let n be a bounded domain having the segment property. Let B[u, u] be a coercive quadratic form over a subspace V of H (n), and suppose ~B[u, u] 2 0 for u ~ V. Then the set m

u(O) =

I

H-m

f

l

All

m! ¢(L) !y!m-l(L)"'(D"'u)(y)dtda.

a! Iyl Iyl

Since the formula for volume integration in polar coordinates is dy = tn-Idtda, we obtain

u(O) =

I

H-m

f m! c a!

L = lu: u ~ V, ~B[u, u] = O} is linear and of finite dimension. Proof. By Lemma 11.6. we need only show that L is linear. For

u, v ~ V, let

¢(L) y"'lyl-nD"'u(y)dy.

l.vl

Q.E.D. The preliminaries having been completed, we shall now discuss some important coerciveness results due originally to Aronszajn; we shall gIVe proofs of K. T. ~mith. First, two lemmas are needed. LEMMA 11.6. Let n be a bounded domain having the segment property. Let B[u, u] be a coercive quadratic form over a subspace

B 1 [u, v] = Y2B[u, v] + Y2B[v, u]. Then B I [u, u[ = ~B[u, U]2 0 and B I [u, v] = B I [v, u]. Thus B I is a Hermitian symmetric, non-negative definite, bilinear form on V, and the Cauchy-Schwarz inequality must be valid for B I :

IBI[u, v]1 ~ BI[u, u]

';.; 2

BI[v, v]

'I, 1

= l~B[u, u] • ~B[v, v]}v..

160

El1i ptic Boundary Value Problems

Coerciveness results of Aronszajn and Smith

161

:: Amp (z)e Az ' x

Thus, if u, v €: L, then B I [u, v 1:: 0, and

k

:: 0,

B1[u+v, u+v]::B1[u, u]+2~Bl[U, v] + B1[v, v]

:: 0, or, ~B[u + v, u + v] = 0, so that u + v€: L. Obviously, L is closed under scalar multiplication. Thus, L is linear. Q.E.D. We now turn to the results of Aronszajn and Smith. The first result ° is almost trivial, using Garding's inequality and the previous lemma.

so that B[lfA, -~Al:: O. Obviously, ~B[v, v12 0 for any v. Since the set of f~~tions uA of the above form is not of finite dimension, when A tak~§hn all complex values, Lemma 11.7 implies that B cannot be

coercive over Hm (0). Q.E.D. , NQw~apartia! converse of the second part of this theorem will be given. Tne-6nly additional assumption is a requirement on the do­ main. The proof is taken from K. T. Smith, Inequalities for formally positive integra-differential forms, Bul1. Amer. Math. Soc. 67(1961), THEOREM 11.8. Let PI (t), ... ,P N«() be homogeneous polyno­ 368-370. mials of degree m, having constant coefficients. For u, v€: Hm{n) let THEOREM 11.9. Let P /(), ...,PN(() be homogeneous polyno­ n __ mials of degree m, having constant coefficients, and let P l«()""'PN«() (11.13) B[u, v]:: I f P k(D)uPk(D)vdx, have no common non-zero complex ze roo Let U be a bounded open k-l U set having the restricted cone property. Then the quadratic form in (11.13) is coercive over Hm(O); that is, there exists a constant C where U is a bounded domain. such that for all u €: Hm (0) 1° A necessary and sufficient condition that B be coercive ° over lIm(U) is that there is no common non-zero real zero N of P )«(), ••. ,PN «(). (11.14) Ilu\\m,U-S: C[k:1 IIPk(D)ullo,u + Ilull o,u1. 2° If U has the segment property, a necessary condition that B be coercive over Hm (n) is that there is no common non­ zero complex zero of P /(), .•• ,PN «(). n

Proof. Note that the non-vanishing of

I

jP k«()1 2 for all non-

k-)

zero g is equivalent to the ellipticity of the form B[u, u]. Thus, in 1° the sufficiency is an immediate result of G~rding's inequality, and ° the necessity follows from the converse of Garding's inequality, Theorem 7.12. Thus, 1° is merely a restatement of previously de­ rived results. To prove 2°, suppose z = (z), ...,z ) is a common non-zero complex zero of P I""'PN • Let uA(x) = eAz.~; u €: H m (n) since U is bounded. Now P k(D)uA :: P k(Az)e Az . x

Proof. According to Hilbert's Nullstellensatz, if P«() is any poly­ nomial which vanishes at all common complex zeros of PI «(), ... ,PN «(), then some power p of P is a linear combination of P l'''''PN , in the sense that there exist polynomials A 1 «(), .•• ,~«(), such that pP = N

I

AkPk' Since the only common zero of the polynomials P 1,,,,,PN k-l is zero, the requirement on P is simply that P(O) :: O. Notice that if the above formula holds for pP, then a similar expression is obviously valid for pP', any p' > p. For a proof of the Nullstellensatz, cf. Van der Waerden, Modern Algebra, II, Ungar Pub. Co., pp. 5-6. Now we apply this result to the polynomial p«() :: f. We then ob­ I tain for some sufficiently large p and for some polynomials A~>«()

162 (11.15)

Coerciveness results of Aronszajn and Smith

EIIiptic Boundary Value Problems

~p

N

1

A~)(~P k(~' j

l

=

k-l

=

1, ... ,N.

Therefore, if m' 2 pn and if lal = m', then a 2 p for some j, so that J (11.15) implies that for some polynomials A~k(~

where cPoc(y) (L COO(E n -10l), cPoc(y) is homogeneous of degree m' - n, ,and supp [cPoc(Y/lyl)] C A/, where AI is the portion of the unit sphere ,I subtended by C I' Now utilizing the formula (11.18) for DOC, the representatiolL{11.19) becomes

,/ (

(11.16)

C' =

N

A~k(~Pk(~'

l

k-l

lal

,.

m', 1<

We can certainly assume m' 2 m. (This is actually implied by (11. 16), though we need not use this fact.) Now if Aak(~ is the part of the polynomial A~k(~ containing terms of degree precisely m' - m, then the homogeneity of P k(~ implies N

(11.17)

e'= l

k-l

Aock(~Pk(~'

/ "­

"

-

/

(11.20) j u(x) =

=

lal =m'.

/

,

N

l

l

H-m'

k -1

Doc = l

Aock(D)PkeD),

k-l

lal

ac.

ac..

(11.21)

l

H-m' ia!

tfi k(Y)

=

l

H-m'

J c/

cPoc(y)Docu(x + y)dy,

k-l

C/

(-l)m'-mA ock (D)cP)y).

Then tfi k is homogeneous of degree m'- n ­ (m' - m) = m ­ n; in par­ ticular tfi k is integrable over C I' so the integration by parts leading to (11.21) is fully justified, and we have

n

n

u(x) =

,

l (_l)m -m J Aock(D)cP)y), Pk(D)u(x + y)dy.

l

=

Let

n

n.

(11.19)

u(x)

H-m'

~ m',

where A ock is a homogeneous polynomial of degree m' - m. Obviously, it is sufficient in proving (11.14) to assume u (L COO(E), by the approximation theorem, Theorem 2.1. Let 10/1 be a finite open covering of and let C/ be the cones guaranteed by assumption that has the restricted cone property. If h is the minimum height of the cones C J , we may refine the covering 1011, if necessary, to insure that the diameter of 0/ is less than h. We have for any x (L n 0 1 that x + CI c Now suppose first that u has its support in 0/. Since the diameter of OJ is less than the height of C j , it follows that if x (L 0/ the cone x + C J has the spherical part of its boundary outside 0/, and therefore outside the support of u. Thus, we may apply the Sobolev representa­ tion formula (Theorem 11.5) to obtain for each fixed x (L no/

n

cPoc(y)Aock(D)Pk(D)u(x + y)dy.

C/

\

N

(11.18)

J

[We ,,\iislnoj!:1~~grate by parts in (11.20), moving all the differentiations ; in Aock(D) so that they apply to cPoc' Note that there are no boundary contributions, since u(x + y) vanishes for y near the spherical part of and cP-(y) vanishes for y near the lateral part of Thus, since I I Aock(D) has constant coefficients and is homogeneous of degree m' - m, (11. 20) becomes

If we interpret (11,17) as a formula for derivatives, it follows that N

163

N

,,(11.22)

u(x) =

l k-l

~;,.

J

C

tfi k(Y)P k(D)u(x + y)dy, x

(L

n no /'

I

Also tfik(L COO(E n -101) and supp [tfik(Y/ly\)] cAl' We now need to apply Theorem 11.4. To do so, it is necessary that the formula (11.22) be modified to contain integrals over all of En' not just ove r C I' To accomplish this, let

164

Elliptic Boundary Value Problems

y ~

P k(D)u(y),

Coerciveness results of Aronszajn and Smith

n,

165

CiThus, (11.24) implies

,.. . k(y) =

0,

y

n.

~

liD; Df3ullo.n Also let C~ be the infinite cone which subtends the protion A of I.

J Since supp (u) cOJ' u(x + y) vanishes for y ~ C~ - CJ' and the inte­

gration in (11.22) may be taken over all of C~ with no change in the

value of the integrals. Next, the integration may be performed over

En' Isince t/Jk(Y) vanishes outside C:. Thus, (11.22) becomes

,/'~-

~ k~l

N

JE

N

c I I IPk(D)ul lo.n· k-l

Since this holds for all f3, 1131 = m, and all i = 1, ... ,N, we obtain

Note that we must restrict x to be in n n OJ for the validity of this formula. Now replace x + y by y in this formula to obtain N

(11.23)

u(x)

=

I

O.E n

N

t/Jk(Y)Wk(X + y)dy, x~nnoJ' n

W k)11

~ I cllwkll o' En k-l

=

u(x)= I k-l

liD J(t/J f3k*

k-l

JE

t/J~(x - y)wk(y)dy, x ~ OJ

n n,

N

lulm,n~ c 1 k~l IIPk(D)ullo,n· ! By the interpolation result of Theorem 3.4, we have

n

N

where t/J~(y) = t/Jk(- y). Since t/J~ ~ C""(E n - {O\) and t/J~ is homoge­ neous of degree m - n, we may differentiate formally the relation

(11.23) a total of m - 1 times to obtain for 1131 = m - 1

(11.24)

Dl3 u (x)=

I

k-l

JE

t/J(:) (x-y)wk(y)dy, n

I-'k

x~OJ nn,

where t/J f3k = Df3t/J~ is homogeneous of degree m - n - (m - 1) = - n + 1 and is in C""(E - {O I). This formula is valid because t/J(:) (y) is in­ n I-'k tegrable near y =0. By Theorem 11.4 it now follows that each of the functions t/J f3:

W k(x)

=

JE

t/J f3k (x - y)wk (y)dy has weak derivatives of first n

order which satisfy

IID J(t/Jf3:wk)O.E n s

(11.25)

11ullm,nsc2[k~1 IIPk(D)ul lo.n+ Ilullo.n],

which is just (11.14) for the case in which supp (u) cOJ' Thus, we have established (11.14) locally. The procedure for ob­ taining the global result is precisely the same as for obtaining o ­ Garding's inequality from the local result in Lemma 7.9; d. pp. 81­ 83. Therefore, (11.14) is established. Q.E.D. We now extend this result to show that the existence of certain weak derivatives of u implies the existence of all weak derivatives ,up to a certain order. THEOREM 11.10. Let P1(tJ, ...,PN(tJ be homogeneous polyno­ mials of degree m, having constant coefficients, and having no com­ mon non-zero complex zero. Let n be a bounded open set havin~ the restricted cone property. Let u ~ L/n) and let P k(D)u exist weakly and belong to L/n), k = 1, ...,N. Then u ~ Hm(n), and

cllwkllo.En' N

Ilullm,n sC [ k~1 IIPk(D)ullo,n + Ilullo,n J•

166

I

Elliptic Boundary Value Problems

Coerciveness results of Aronszajn and Smith

Proof. Obviously, once we have shown that u ~ H m(0), the previous n. Thus, we need only show theorem implies the estimate on Ilull m.u that u 10: Hm (n). It is easily seen that u ~ H10c(D,). For, extend u to m

vanish outside n, and let ue = JeU' where Je is the mollifier introduced in Definition 1.7. Then ue ~ C~(E) and, by Theorem 2.4, u .... u in e L;OC(n), P k(D)ue .... P k(D)u in L;OC(n), k = 1, ... ,N. If S is a sphere such that S cc n, then (11.14) implies N

Iluell m•s .s; C[k~l I[P k(D)uell 0.5 + Iluell 0.5)' Therefore, it follows that u e .... u in L 2 (S) and Iluellm.sis bounded as e .... O. By Theorem 3.12 it follows that u ~ Hm(S). Since S is any sphere whose closure is contained in n, it follows that u ~ H~o Now let 10~1 be a finite open covering of and let IC II be the associated cones such that x + C i cO for x ~ 0 no';, By approximat­ ing each open set O~ by finite unions of spheres contained in O~, we may actually assume that there is a finite open covering \0 i I of ao such that each set 0i is a sphere whose diameter is less than the Let 0/ be the union of all the cones x + C.I for x £ 0 no I.. height of C.. I Obviously, n.I itself has the restricted cone property. By decreasing the size of the sets 0 i' if necessary, we can also assume that for any fixed unit vector g in the cone C /' the translated set O.I . e = eg + OJ satisfies O.1. e CC 0, if e is sufficiently small. Let ue(x) = u&c + e.;) for x ~ 0,. Since x + eg ~ n,I . e for x ~ 0/, and since I we have shown already that u ~ H m10 c(O) c H m (n,I . e)' it follow> : that u e 10: Hm (0.). Moreover, we have the estimate I

an

cu:n.

Therefore, when f .... 0, u f .... u in L 2 (0 i) and

s: C[k~1

IIPk(D)uello.Oi + Iluello.o/],

which follows from (11.14) since 0i has the restricted cone property. But then it follows, since u e is just a translation of u, that

Applying Theorem 3.12 again it follows that u ~ Hm (0.) c Hm (0 no.). J I Since this holds for each i, it follows that D""u ~ L/n) for lal.s; m. As mentionedOt'\ page 10, any function in W~oc(O) whose D""u belong to L 2 (n), \Cf( .s; m, is in Wm(n). By Theorem 2.2, it follows that u ~ Hm (O!- Q.E.D. Thi§,te'8ult will prove extremely useful in section 13. In that appli­ cat~()n, it will prove essential that no assumption is necessary on the \ smoOthness of ¢no This emphasizes the importance of Smith's proof, since Aron~~~ln's original proof required a smooth boundary. Specifi­ cally, we shall need the foIl owing corollary of the theorem. Corollary. Let 0 be a bounded open set having the restricted cone property and let u ~ L/n). Suppose that D~u exists weakly and belongs to L/O), k = 1, ... ,n. Then u ~ Hm(n) and for some constant C depending only on 0 and m n

Ilullm.n

s: C[k~1 IID~ullo.o

+ Ilui 10.0)'

For the case of variable coefficients we have

1°

for each XO ~ 0 the polynomials non-zero real zero; and

2°

for each XO ~ the polynomials mon non-zero complex zero.

an

P~(XO,

D) possess no common

P~(XO,

D) posses no com­

Then there exists a constant C such that for all u ~ Hm (0) N

N

Ilufllm.Oi .s; C[k~1 liP k(D)ull 0.0 + [lull 0,0)'

is bounded.

I

N

Iluellm.Oi

II ufll m.n.

167

(11.26)

Ilullm.O~ C[k~1 \\PkullomO + Ilullo.o]·

168

Elliptic Boundary Value Problems

Proof. First we will prove the theorem for a neighborhood of the

an.

169

Coerciveness results of Aronszajn and Smith

+ NC1C Z sup

boundary. Let XO :€ Then the polynomials P~(XO, fl, k = 1, ... ,N, have no common non-zero complex zeros. Let be a small neighbor­ hood of xo; since n has the restricted cone property, we may aSSume that U = n n has this property. By Theorem 11.9 there is a con­ stant C 1 such that for u :€ Hm(U)

°

H-m

15:k5:N

°

N

Ilull m• u ~ Cl(k~l IIP~(xO, D)ullo.u + [Iullo,u)'

!Pk.",(x) - Pk.",(XO) I Ilullm,u

",(u

+ NC 1C 2 Ilull m- 1 + C11lullo,u' ~y/fti.e interpolation inequality, there is for €

>

~at

.

°

a constant C €' such

NC1Czllullm_1,u':::;: € l lullm,u + C€ l lullo,u'

Now P k(x, D)u

=

~

Furthermore, since, for la\ = taken small enough that

P k.",(x)D"'u

H~m

= P~(XO, D)u

!

+

[p k,,,,(x) - P k.",(xO))D"'u

H-m

m,

pk,,,,(x) is continuous on

n, V can be

NC1C Z sup Ip k ",(x) - Pk ",(xO) \ < .€

rt:u' ,

H-m

lSk5:N

+

~

H'::;m-l

P k.",(x)D"'u.

Hence, for some constant C z ,

Thus, N

Ilullm,u 5: C 1 k~l IIPk(x, D)ull~,u + 2€/Iull m,u + (C 1 + C€ ) llullo,u'

IIP~(x~ D)ull o. u .::; liP k(X, D)ull o,u Choose € < 1/4. Then we have + Cz[sup rt:u

IPk.",(x) - Pk.",(xO)lllull m• u

H-m + Ilull m- 1• u ]·

Thus, N

Ilull m• u 5: C 1 k:l IIPk(x, D)ullo.u

N

Ilullm,u S C[}l liP k(x, D)ullo,u + Ilullo,u]' where C = 2(C 1 + C€). Now let 0l""'OV be a finite covering of such sets OJ thus, for u :€ H m (n n 0/)'

an by

N

Ilul1m,nno/5: C[k~l

(11.27)

v LetO o = VOl' I- 1

11 P k (x,

D)ullo,nno + Ilullo,mo ]' l /

170

Elliptic Boundary Value Problems

Let U ccn' ccn, where U is chosen such that n - 0 0

B(u)

N

=

1

k-l

cu.

Let

N k-l

IP~(x, (;)\2 f, 0,

x€:: n'.

ItI = 1

t

1 IP~(x, (;)1

2

k-l

2 E o ltl

2m

•

Furthermore, the coefficients of B are uniformly continuous on U. ° By Garding's inequality there are constants Yo> 0 and 1.. 0 .); 0 such that for u :€ : H°m (V) :RB(u) 2

Yollull~.u - Aollull~.u;

in other words,

Ilull~,u ~

D)ullo.u +

Ilullo.ul.

On comb}n~ng this inequality with (11.27), it is seen that we have actually proved a local version of (11.26). As in the proof of G~rding's inequality, the global version now follows from a familiar I a~ent utilizing a partition of unity. Q.E.D. . The original result of Aronszajn can be expressed as follows. Sup­ \ pose,.o !s pf class Cm and consider the condition 3°

Hence, for x€:: U and ItI = 1 this quantity is bounded away from zero. It follows that there is a constant Eo such that for real and x :€ : V, N

N

Ilull m • u S; C[k~IIIPk(X,

2

IIP;(x, D)ull o.u'

Then B(u) is a uniformly strongly elliptic quadratic form on U. For, since P~ (x, (;), ... ,P~(x, (;) possess no common non-zero real zero t for x€:: 0,

I

171

Coerciveness results of Aronszajnand Smith

N

y;l k~l IIP~(x, D)ull~.u + Aoyoillull~.u'

By the Cauchy-Schwarz inequality, it follows that there is a constant C such that

an

an

for each x°!O: let n be the unit normal vector to at xo. For each real tangent vector tat Xo the polynomials in T given by P~(xo•. t + Tn) possess no common non-zero complex zero.

n

Remark. The estimates guaranteed by the preceding theorems can be extended to more general classes of norms: for instance, L p norms • and norms involving the Holder continuity of derivatives of u. We conclude this section with an application of Sobolev's repre­ sentation formula and the Calderd'n-Zygmund theorem already men­ tioned. The theorem will not be needed in the following but is of con­ siderable interest in itself. THEOREM 11.12. (CALDERON'S EXTENSION THEOREM). Let o be a bounded domain having the restricted cone property. Then there exists a bounded linear transformation {[, of Hm (0) into Hm (E n ) such that for every u :€ : Hm (0) the restriction of ([,U to n coincides with u. Proof. We prove the theorem first for u :€ : Cm(O)

N

Ilull m • u S; C[k~l

IIP~(x,

D)ullo.u + lIullo.ul.

As above, the interpolation theorem yields

n

The theorem of Aronszajn states that if is of class C m and is bounded, then the conditions 1° and 3° are necessary and sufficient for the validity of (11.26). We shall not give the proof of this result.

an

n Hm (0).

Let 10, I~_ I be a finite open covering of with associated cones IC ,I such that x + C, cO for x€::O no,. As in the proof of Theorem 3.10 we add to this covering an open set 0 0 ceO such that 10il~-o

is an open covering of let Co be a sufficiently small cone such

that x + Co cO for x ~ 0 0 , We can arrange this covering such that

n;

172

the diameter of 0 1 is less than the height of C/' Next let 0; cc 0/ be such that IO~1~_0 is still a covering of Let I'II~-o be a partition v

of unity subordinate to 10;1. such that I == 1 on O. Then u =

v /-0 /~o u i ' where u/ = is in cm(f!) and supp(u/) C 0/. To extend u to

n.

U'i

'I

En it is clearly sufficient to extend each u/' By the Sobolev representation formula we have for x (11.28)

u/x)

=

Coerciveness results of Aronszajn and Smith

sec. 11

EIIiptic Boundary Value Problems

v

(11.33)

vex)

0

no/

I I rp"'I(y)D"'uI(x + y)dy,

I"I-m c/ where rp",/ :€ COO(E n -101), rp",/ is homogeneous of degree m - n, and supp [rp",/(Y/lyJ)] C \ ' the portion of I subtended by C I • As in the proof of Theorem 11.9, we shall arr.!.nge things so the integration in (11.28) is' taken over all of En' First, let

=

I v/(x),

x ( En'

/- 0

By the yery construction, vex) = u(x) on O. Now.Jis in the proof of Theorem 11.9, and by Leibnitz's rule, i

~

173

D{3v (x)

(U~34)

/

=

I

({3)D{3-Yt/J I(X) • I DYrp~/(x - y) • wai(y)dy, 0

~(3 Y

IT\1l\.~ni - 1. Here DYrp~/ is homogeneous of degree m - n - Iyl. We now need\to estimate the L 2 norm of D{3v j • In this computation K will stand for a generic constant depending only on 0, the functions

rp",/ and

t/J /'

v

and m. Suppose 0 C U 0; C /- 0

Ix: Ixl < al. Then the L 2

norm of any term on the right in (11.34) is no greater than

K[J ,\D{3-Yt/JI(XWdx J DYrp~/(y). w",/(x - y)dy\ 2]'4 o / E n I

(11.29)

y ~

D"'u/(y),

0,

W"'I(Y) =

0,

y

/

Then (11.28) can be written (11.30)

u/(x)

=

I

H-m

IE rp"'i(Y)W",/(x + y)dy, x ~ 00/. n

=

t/J/(x)

I

H-m

I

rp~/(x - y)w",/(y)dy,

x ~

dy[J \y\2(m-n- 1Y!>\w (x- yWdx]%

KI E

to;,

u/(x)

lylm-n\Y1Iw"'i(x - y)ldyI2]%,

En

where we have again replaced x - y by y in the inner integral, and

have used the homogeneity of DYrp~/. By the Minkowski inequality,

(11.35) is no greater than

Now let t/J/ ( C';(E n ) satisfy t/J/(x) = 1 for x ( 0;, t/J/(x) = 0 for x (.. O/' Since u/(x) = 0 for x (11.30) implies (11.31)

~ K[Jo,dx lf

(11.35)

(0.

0' n

"'/

i

\ylm-n-IYldy[I \w", (x - yWdX]'~

=KI

o.

E

0'

/

I

n

En

(11.36)

Here we have set rp~I(y) = rp"'i(- y). Now let

\y!m-n-\Yldy[I

=KI E

n

x(o'. 1

ID"'u(x_y)12dx]~. /

\x-y(f!

(11.32)

v/(x) =t/JI(x)

Then we set

I

IE rp~/(x - y)w",/(y)dy,

H-m n

x (

En'

x ( O~ and x - y ~ 0, we have Ixj < a and Ix - yl < a, so that

Iy\ < 2a. Thus, since Iy\ ~ m - 1, (11.36) is no greater than

For

174

Elliptic Boundary Value Problems

K(2a)m-IYI-Ij,

Iyl<2"

lyl-n+ldylu.I

'

sec. 12

.... :>:

Klu;lm

m.H

n. •

(11.37)

Ilv; Ilm-l.n :>:

Klulm.n'

Now we need to estimate D JDf3v; for j = 1••••• n and 1f31 = m - 1. In computing D pf3v ; from (11.34). terms arise which can be estimated exactly as above, and we also have a term of the form (11.38)

tf; ;(x)D J In

Df3¢~/x -

y) • wa;(y)dy.

Since Df3¢, al is homogeneous of degree 1 - n, the inequality of Calderon and Zygmund (Theorem 11.4) implies that the L 2 norm of (11.38) is no greater than

Ivllm.n:>:

Kllw "'III O.E

:>: n

Klu;lm.n.

'l'lvll m • E n .<;; v

Therefore. summing over y in (11.34), we have

Thus we have

Klu;.lm.n·

Some Results on Linear Transformations

=

u on

175

Kllullm.n·

n.

Since/this transformation is a bounded transformation from a dense sub$et of H m (0) into H m (E n ), it may be extended in a unique fashion I JO"H m (0). Moreover. it follows that the extended mapping still satis­ fies the requirement v = u almost everywhere on n. Q.E.D. 'l?emarlis. The same argument shows that we can use L p norms (1 < P < 00) everywhere instead of L 2 norms. The only additional tool is the Calderon-Zygmund estimate for L instead of L 2 • We p "" could also obtain estimates for extensions of u preserving Holder con­ tinuity of derivatives of u. Here a slight change appears. since the extension of D"'u t outside n can no longer be W"'i (d. 01.29». but must be an extension perserving the Holder continuity of D"'u ,.•

12. Some Resultson Linear Transformations on a Hi Ibert Space

Combining this with (11.37) yields (11.39)

Ilvlll m' En

This section is devoted to preliminary material needed in the dis­ cussion of eigenvalue problems in section 13. It is divided into three parts.

:>:Klu;lm.n.

Now (11.33) implies that

Part 1. ELEMENTARY SPECTRAL THEORY

Therefore, (11.40) implies Ilvll m• E :>: Kllullm.n'

Definition 12.1. Let A be a linea r transformation defined on a (not necessarily closed) subspace D(A) of a complex Hilbert space X; D(A) is the domain of A. Let R(A) be the range of A; that is, R(A) = {Af: f :€ XI. The resolvent set of A. denoted p(A), is the set of all complex numbers A such that A- A is one-to-one, R(A - A) = X, and the inverse (A - A)-l is a bounded linear transformation on X. Here the notation A stands for the transformation mapping f into At. all f :€ X. The spectrum of A, denoted o{A). is the set of all complex numbers not in p(A).

Therefore, the correspondence u .... v is a linear transformation of Cm(O) n H m (O)into H m (E n ) with the properties

THEOREM 12.1. The resolvent set of A is open, and (A _ A)-l is an analytic function on p(A). If A is bounded, then p(A) ::J

(11.40)

Ilvll

m.

E

n

v :>: K ~ lutl m n. i- 0 •

Next, Leibnitz's rule implies V

2

~ IUII~ • E n :>: Kllullm.n· i- 0 n

IA: IAI > IIAIII.

176

sec. 12

Elliptic Boundary Value Problems

Summing this geometric series, we obtain a very useful estimate

Proof. If Ao ~ peA), then

I~A

A - A = (A O- A) + (A - AO) (12.1)

A).

FrequentlY in dealing with integral equations, we shall consider \~re1ations~f the form

A)-III-I.

f

Moreover, we have the Neumann expansion 00

[1 - (A O- A)(A O- A)-l]-l

=

I (A - A)k(A _ A)-k O k-O O

where (A o - A)-k = [(A o - A)-l]k. Therefore, (12.1) implies that for JA O- AI < II{A o - A)-III-I, A ~ peA), and the continuity of (A _ A)-l

O implies that

=

(A - A)-l = I (A _ A)k(A _ A)-k-l. O k-O 0

in which it is desired to solve for f in terms of g. In case 1 - AA is invertible, i.e., A-I € peA), the solution will be given by f =(1- M)-lg (the case A = 0 is obviously without interest). It will prove to be very fruitful, however, not to analyze the operator (1 - AA)-l itself, but, instead, A(1 - AA)-l. One reason for this is that the range of A(I- AA)-l is the same as the range of A, and we shall frequently have some very precise information about the range of A. Note that

=

Thus, (A - A)-l is expressed in terms of a power series about any point AO ~ peA); that is to say, (A - A)-l is analytic. Finally, if A is bounded and IAI > ItAII, then A - A = A(l _ A-I A).

(1 _ M)-l - AA(I- AA)-l,

so that (12.4)

(l - AA)-l = 1 + AA(1 - AA)-l.

This notion is in the following definition.

and IIA-lAII < 1. Thus, the above argument shows that A ~ peA). Q.E.D.

Definition 12.2. The function (A -

g + Mf,

1 = (1 - AA)(1 - AA)-l

00

(12.2)

<

II(A - A)-Ill-I. o

The operator 1 - (A O- A)(A o - A)-l is invertible if II(A o - A)(A O- A)-III < 1, or, if

< II(A o -

_Ar111

II(A - A)-III::;; 0 ; lAo - AI l '_'. 1-!A o -A!IIA o - A f ll

(12.3)

= [1 - (A O - A)(A - A)-l](A O O

lAo - AI

177

.some results on Linear Transformations

A)-I, defined

resolvent of A. Note that (12.2) implies 00

II(A-A)-lll::;; IIAO-Alkil(AO-A)-lllk+l.

k-O

on

peA), is the

Definition 12.3. The modified resolvent set

Pm(A) of A is the

set of all non-zero complex numbers A such that 1 - AA is one-ta-one, a bounded linear transformation on X. If A ~ Pm(A), then the transformation AA == A(I- AA)-l is the

modified resolvent of A.

R(1 _ AA) = X, and (1 - AA)-l is

Note that (12.4) can be expressed in the form (12.5)

(l - AA)-l == 1 + AAA'

178

Elliptic Boundary Value Problems

sec. 12

Some results on Linear Transformations

Also note that (12.6)

=

m

Thus, if IA - Aol < IIAA WI, we have, as before, o

It should also be remarked that if D(A) = X, and if A ~ peA), then A and (A - A)-l commute. For, in any case, (A - A)(A _ A)-l = 1

and (A - A)-l(A - A) is the identity on D(A). Thus, if D(A) = X, then (A - A)-l(A - A) = 1. Likewise, if D(A) = X and A ~ p (A), m then A and AA commute.

We now state the analog of Theorem 12.1 for modified resolvents.

THEOREM 12.2. The modified resolvent set of A is open, and AA is an analytic function on Pm (A). If Ao ~ Pm (A), then for IA - Ao I < IIA\1\0 WI,AtO,A~pm (A)and

AA= 1 (A-Ao)kAfl, k- 0

0

in the sense of convergence in the operator norm. Also,

IIAA II IIAAlls

0

,

If A is bounded, Pm(A)::J {A: 0 < IAI < IIAWll. Proof. We have =

=

11-1.

IA-Aol
1-\A-A oIIIAAII o

1 - AA

00

((1 - AA)-l = (1- AoA)-l

1 (A - AO)kAk •

k-O

0

-Multiply-i~

by A, we obtain (12.7). The other assertions are then immediate. Q.E.D. In the situation of interest to us, A will be a differential operator, and there will exist some number A in the resolvent set of A. For example, if we are interested in the generalized Dirichlet problem with zero Dirichlet data, X = L 2 (0) and A is defined on H (0). By Theorem o m 8.5, if Garding's inequality holds, then - A ~ peA) for all sufficiently large real A. Also, if is bounded, then (A + A)-I is compact. If we consider the differential operator A + A instead of A, then this differ­ ential operator is itself invertible and has a compact inverse. The' abstract case of this situation is a special case of what we shall con­ sider. A simple result shall now be given.

n

THEOREM 12·3. Let 0 ~ peA) and let T = A-I. Then for A to, A ~ Pm(T) if and only if A ~ peA); if A ~ Pm(T), then

00

(12.8)

[1 - (A - Ao)AA )(1 - AoA). o

A~p (A)~>A-l~p(A).

It should be remarked that if the linear transformation is closed, then the condition in Definition 12.1 that (A - A)-l be bounded is superfluous. For, if A- A is one-to-one and onto, then (A _ A)-l is closed (since A is closed) and defined everywhere on X. Then the closed. graph theorem implies that (A - A)-l is bounded. Like­ wise, if A is closed, then the condition in Definition 12.3 that 0- AA)-l be bounded is superfluous.

(12.7)

179

1 - AoA + (A o - A)A [1 - (A - Ao)A(1 - AoA)-l)(1 - AoA)

0

TA = (A - A)-I. Proof. We have

(A - A)T = AT - AT

= AT - 1. The assertions about Pm(T) and peA) are now easily checked, and then (12.9) is immediate. Q.E.D.

Definition 12.4. A complex number A is an eigenvalue of A if there exists a non-zero vector f such that Af = Ai. A complex number A is a characteristic value of A if there exists a non-zero vector g such that AAg = g.

180

Elliptic Boundary Value Problems

Some results on Linear Transformations

12

Note that every characteristic value of A is non-zero and that Ais a characteristic value of A if and only if A-I is an eigenvalue of A.

T

A

o

181

- _A_ = [T - _A_ (1 - A T)](1 - A T)-1 o o 1-,\,\ o l-AA 0

Definition 12.5. Let A be an eigenvalue of A. A non-zero vector f is a generalized eigenvector of A corresponding to A if for some posi­ = [(1 + '\'\0 )T _ _ A _](1 - A T)-1 o tive integer k, (A - A )kf = O. The set of all generalized eignenvectors 1-,\,\ 0 l-AA 0 of A corresponding to A, together with the origin in X, forms k subspace of X, whose dimension is the multiplicity of the eigenvalue A. Like­ = _1_ (T - A)(1 - A T)-I. wise, if A is a characteristic value of A, then a non-zero vector f is a o I-,\A o generalized characteristic vector of A corresponding to A if for some positive integer k, (1 - AA)kf = O. Therefore, since T - A and (1 - AoT)-1 commute, If we use the notation N(A) for the null space of a transformation A, then f is a generalized eigenvector of T corresponding to A if and only if f ,j, 0 and f is in the set (T __ A_)k = 1 (T _ A)k(1- A T)-k. Ao 1-,\;\ (1-,\,\ )k 0 o 0 00

M(A)

=

U N«A - T)k).

k~

1

Thus, the multiplicity of A is the dimension of this set. Note that this set is a subspace of X, since N«A - T)k) CN«A - T)k+l), k L 1. We shall call M(A) the generalized eigenspace of the operator T corresponding to A. THEOREM 12.4. Let T be defined on X, and let Ao ~ Pm (T). If '\'\0 = 1, then A ~ p(T). If'\'\'o,j, 1, then A is an eigenvalue of T if and only if_A __ is an eigenvalue of T A • Furthermore, the multiplicity

l-AA o 0 of A as an eigevalue of T is the same as the multiplicity of _A_

1-AA o

The relation (12.10) for k = 1 implies that T - A is one-to-one if and only if T A - _A_ is one-to-one. Thus, A is an eigenvalue of o 1-,\,\ o

• T ~ _A_ is an eigenvalue for T A • Moreover, (12.10) implies that l-AA o 0 (T - A)kf = 0 <7 (T A - _A- )kf= O. Q.E.D. o I-M o

Before giving the next definition, some notation will be convenient. . For real and a ~ 0, let

°

2(0, a)

=

Ire iO: r > al.

Definition 12.6. Let A be a linear transformation defined on a as an eigenvalue of T A • Indeed, the set of all generalized eigen­ o subspace D(A) of a Hilbert space X. A vector e iO in the complex plane vectors of T corresponding to A is the same as the set of all general­ is a direction of minimal growth of the resolvent of A if for some pos~ ized eigenvectors of T A corresponding to _A_ . tive a, 2(0, a) Cp(A) and II(A - A)-III = 0(IAI- 1) for A ~ 2(0, a), o l-AA o

Proof. The first assertion follows from (12.6). Suppose AA o ,j, 1. Then by definition of T A (Definition 12.3), o

IAI -+ 00. Likewise, a vector e lO is a direction of minimal growth of the modified resolvent of A if for some positive a, 2(0, a) CPm(A) and

IIAAII = 0(1,\,1- 1) for A ~ 2{0, a), IAI -+ 00.

In general, we cannot expect the resolvent, or modified resolvent, to decay faster al(;>ng any ray than 0(\AI- 1). This is shown by the follow­ ing theorem.

182

Some results on Linear Transformations

Elliptic Boundary Value Problems THEOREM 12.5. Suppose that for some positive constant a and

for some sequence of complex numbers .\ in peA) such that IAn I . . the following estimate holds:

00,

183

i,implies AA = 0 for all A. That is, A(l - M)-I = 0 for all A. Multip1y­ hng this relation by 1 - AA implies A = O. Q.E.D. THEOREM 12.6. The set of numbers e such that e;e is a direction

!of minimfil,growth of the resolvent [modified resolvent] of A is open.

II(An - A)-III .s alAn 1-1.

/

Proof. Suppose that

Then a::: 1. Correspondingly, if for some sequence of complex num­ bers in plit (A) such that IAn I . . 00, the estimate

J

,\,/il(A - AT 111 .s Ji IAI ./,j

IIA A II.sal\I-I

(':for A ~ S(e, a), By (12.3), Il ~ peA) if and we have an estimate

n

holds, then either A

=

0 or a

~

By (12.3) it follows that A ~ peA) if IA - An I < a-ilA n I, and we have an estimate

11(1l- A)-III .s Thus, if A ~

al\l-I II(A-A)-III.s 1-IA-A laIA l- I

n

n

II (Il -

A) -

111 .s 2K IA1-1.

1 a-II\HA-Anl

Now assume a < 1. Then a = (1 + 2f)-I for some positive f. Also all of the circles lA: IA - An I < (1 + f) \\, II are contained in peA), and for A in such a circle (12.11) implies

1 II(A- A)-'ll < (1 + 2f)I A l-(1 + f)IAnl n

K\AI-I

1-IIl-AIKIAI- i '

S(e, a) and if 11l- AI < Y2K- I IAI, then

IIlI.s IAI +

(12.12)

a),

1.

P~oof.

(12.11)

11l- AI < K-IIAI and A ~ S(e,

11l- AI 1

< (1 + - )IAI, 2K

1 ---. flA n I

11(1l- A)-III < 2K + 1

Thus, since such circles cover the complex plane, (12.12) implies (A - A)-I exists and is uniformly bounded in the complex plane. As (A - A)-I is also analytic on peA) = complex plane (by Theorem 12.1), Liouville's theorem implies (A - A)-I is constant. But then A- A is constant, an absurdity. For the proof of the second part, the same procedure as given above shows that if a < 1, then AA is constant. But since IIAA II .... 0, this n

Ill! '

!~ estimate which holds for allll such that

11l- AI < Y2K- I IAI for some

~ S(e, a), An easy computation then shows that l/J is a direction of :tninima1 growth of the resolvent of A if \l/J - el < sin-I 1/2K). A IBimilar argument gives the result for the directions of minimal growth " the modified resolvent. Q.E.D. inf

184

Elliptic Boundary Value Problems

Some results on Linear Transformations

We now give some examples.

185

IIAfl1 S; lese 01 lA-II 11(1- AA)fll·

THEOREM 12.7. Let A be a self-adjoint operator on the Hilbert

~putting in the last inequality f = (1 - AA )-1 g, g an arbitrary element space X. Then any non-real direction elf) is a direction of minimal lin x, (12.14) follows. Q. E.D. growth of both the resolvent and the modified resolvent of A. Further­ We alsolHlve the following result. more. for arg A = a THEOREM 12.8. Let A be a linear transformation with domain I (12.13) [1(A- A)-III ~ lese 011,AI- 1, ';D(A) eX, such that for all f ~ D(A) . /

(12.14)

l IIAAII ~ lese 0IIAI- •

i·.f-m(.4!,f) :2 AoII f11 2,

Proof. Since A is self-adjoint, (Af, f) is real for f <:; D(A). Thus, if A = IAlela, then ~[«A - A)f, f)) = IAI sinO!f,

n.

Therefore, the Cauchy-Schwarz inequality implies IlfW::; IA\-ll esc 01 II(A-A)fllIlfll;

with - ;

<°
A.

. Proof. We have for f ~«A - A)f, f)

dividing by Ilfll, (12.15)

for some real number Ao' Suppose that some number Al is in the 'resolvent set of A, and that ~A1 < Ao' Then the half-plane {A: ~ < Aol is contained in the resolvent set of A, and every direction e lO

II(A- A)-IllS; lese 0IIAI- l .

D(A)

~Allnl2 - ~(Af, f)

=

::; ~AllfW - Ao llf[[2

Ilfll S; lese 01 IAI-1WA - A)llfll.

This inequality shows that N(A - A) = {Ol for all non-real A. Thus, for non-real A, N«A - A)*) = N(A- A) = 101. But the orthogonal com­ plement of the range of A - A is precisely N«A - A)*). Therefore, the range of A - A is dense in X. Since (12.15) implies that the range of A - A is c1 osed (here we use the fact that a self-adjoint transforma­ tion is c1osed), it follows that A - A is one-to-one and onto; and another application of (12.15) implies

~

(~A - Ao)llfI12.

=

(:Therefore, if ~A

< Ao' the Cauchy-Schwarz inequality implies

- 1~ ~«A _ A)f, f)

IlfW

Ao ­

2 < Ao-~A 1 1\(A-A)f[lll f I1

To obtain the r~sult for the modified resolvent of A, note that for non-real A and f ~ D(A) ~[«1- AA)f, Ai))

=

IA\ sin 0IIAfW.

Therefore, applying the Cauchy-Schwartz inequality, dividing by Ilfll,

Ilfll::;

1~

Ao -

A

II(A-A)fll.

186

Some results on Linear Transformations

EIIiptic Boundary Value Problems

187

Since Al ~ peA), (12.16) implies that for f = (AI _ A)-Ig

II(A I - A)-Igil oS

Ao -

19{

Al

Part 2. OPERATORS OF FINITE DOUBLE-NORM ON AN ABSTRACT HILBERT SPACE

IlglI·

THEOREM 12·9. Let T be a bounded linear transformation on a _"Hilbert space X. IflcP l , cP 2 , ... 1and 1t,f;1' t,f;2, ••• 1are orthonormal bases l. 'i in X, then

Thus,

II(A I - A)-III

sA

1

00

_ g{A . o I

OQ

2 IlTcP i l1

2

=

i-I

2

/-1

But (12.3) implies that A ~ peA) if IA I - AI < ll(A I - A)-III-I; there­ fore, A ~p(A) if 1\ - AI < Ao - g{A I • We can then extend this process (see figure), to show that A ~ peA) if g{A < Ao' And then (12.16)

IITt,f;iW

00

i~1 IIT*cPiW =

~ I(TcP i , t,f; oW; i,i- I J

it is to be understood that these sums may be either finite or infinite. Proof. By Parseval's relation, 00

l1TcP/11

i~1 KTcP i • t,f;iW·

2

=

Summing over i gives

~Al

A

o OQ

2

2 II TcP / ~ 1

00

=

i-I

1-1

yields the estimate

II(A-A)-lll

s

19{ ,~A
From this estimate the assertion of the theorem is obvious.

2

=

00

J-I

00

OQ

2

2 l(cP/,

i-I i-I

=

2

2 ~T cP /' t,f;) 1

~ ~

i-I

i-I

T*t,f;JW

1,(T*tj; _, J

OQ

=

2

i_ 1

II T*tj; J_11 2 ,

cP)1 2

188

Elliptic Boundary Value Problems

Some results on Linear Transformations

where we have again used Parseval's identity. Applying this argument to T* instead of T gives 00

I

1-1

Proof. The first relation follows from Theorem 12.9, the second is trivial, and the third follows from Minkowki's inequality. The fourth follows easily:

00

IIT*¢/W = I

J-l

IIT**r,ft

J

11

2

IIIAT 1¢.11 2:$ I-I I IIAI\21I TI ¢I 11 2= i-l 00

00

I

IIAWIIIT\W.

00

=

I

J-I

2 II Tr,ft 11 • J

And the fifth follows from the computation

Therefore all the assertions of the theorem follow.

IIITIAIII = III(TIA)*III

Q.E.D.

Definition 12.7. A bounded linear transformation T on a Hilbert space has finite double-norm if the sums in Theorem 12.9 are finite. In this case, for any orthonormal sequence I¢ l' ¢2""} the double­ norm of T is given by

III Till

[I

=

I- 1

II T ¢I WJ'4 .

=

IIIA*T\III

~

IIA*IIIIITilll

=

IIAII IIIT1111·

Finally, suppose f

~

X. Then

The next theorem gives some elementary properties. THEOREM 12.10. Let T 1 and T 2 each have finite double-norm, and let A be a bounded linear transformation. Then Ti, T 1 + T 2' AT I' and T I A each have finite double-norm, and

f'"

00

f = ~ (I, ¢/)¢/; j"

1

00

. IIIT1111

II laT 1111 IIIT I

+

Tf

= rIIT~III; =

\al lilT 1111

T2111

~

=

I (I, ¢i)T¢/' I-I

for any complex number

a;

The Cauchy-Schwarz inequality shows that

IIIT1111 + III T2111;

IIIAT 1 111

~

IIAIIIIITIIII;

IIIT 1AIII

~

IIAIIII\T11I1;

189

00

1·ITfll:$ II- I IU,

¢/)! ~IT¢/II

~ [I 1(1, ¢)12]'~[ j-

1

I

1- 1

IITIII ~ IIIT1111· =

1\fIIIIITIII.

Q.E.D.

11~¢/WJ'4

190

II

Elliptic Boundary Value Problems

THEOREM 12.11 A transformation having compact.

sec. 12 finit~

double-norm is

Proof. Let {¢1'¢2 ... 1be an orthonormal~Jis for X. Then there exist numbers t/J(i, j = 1, 2, ...) such tlrclf

\

00

T¢. = I }

t/J¢.

/-1

Some results on Linear Transformations

/

IIIT-TNII12=

00

I

l.l-1

It/J1 2-

N

I i,j-!

191

It/J1 2.

Therefore, III T - TN III -> 0, so that Theorem 12.10 implies liT - TN II Since the range of TN is finite-dimensional, it is obvious that TN is compact. But then the fact that II T - TN II -> 0 implies T is compact.

->

O.

Q.E.D. Moreover, Parseval 's relation shows that 00

IIT¢ j 11 2 =

It ii l

I

/-1

2

THEOREM 12.12. Let T and S be transformations each having finite double-norm, and let 1¢1' ¢2, ... 1be an orthonormal basis in X. Then the series

• 00

Therefore, summing on i, we obtain

(12.17)

IIITIW =

!

It;jI2.

/,j-l

If N is a positive integer, let

t ii ,

i, j .:>; N, t/j ­

N­

0, i

> N or j > N.

!

(T¢., ¢YS¢;' ¢.)

/,j-l

}

}

is absolutely convergent and is independent of the particular ortho­ normal basis I¢/I. If the sum of the series is designated tr(TS), then (12.19)

Itr(TS)I::;

IIITIIIIIISIII.

Proof. The absolute convergence of the series and the estimate (12.19) are immediate consequences of the Cauchy-Schwarz inequality, the definition of double-norm, and Theorem 12.9. Note that

Then let TN be the unique linear transformation satisfying

00

TS¢.

=

1

T !

(S¢., ¢')¢j

j-l

J

1

00

T N¢j

= ;

~1 t~¢/.

00

=

! (S¢/' ¢j)T¢.

j- 1

)

Thus, 00

N

I

00

I

=!

t/ j ¢/, l':>;j ~ N,

j- 1 k- 1

(S¢/' ¢j)(T¢j' ¢k)¢k'

i-I

(12.18)

T N¢j =

0, j > N.

Then it is obvious that TN has finite double-norm and

Thus,

(TS¢/, ¢/)

=

00

00

I

! (S¢/, ¢/)(T¢j' ¢k)(¢k' ¢;)

j- 1 k- 1

192

Elliptic Boundary Value Problems

Some results on Linear Transformations <xl

<xl

=

193

j~1 (S¢I' ¢j}(T¢j' ¢I)'

=

I

l,jR 1

(Sl/II' l/Ij}(Tl/Ij' l/II) ,

,by (12.20). Q.E.D.

Summing on i,

,., Notation. The quantity tr(TS) is called the trace of the operator <xl

I I I

(12.20)

"TS. Note that (12.20) can be written

<xl

(TS¢ I' ¢.)

R

1

=

I

l,j

(S¢., ¢ .}(T¢j' ¢ I)'

l

R

11 <xl

Now let

Il/Il l be

another orthonormal basis. Then

<xl

I

l,j Rl

tr(TS} =

<xl

(T¢}' ¢I}(S¢I'

¢) = i,jI Rl (S¢j'

¢)(T¢I' ¢j)

<xl

I

=

IR 1

(ST¢I' ¢I)

<xl

I (T¢I' S*¢I)

=

1-1

=

I

<xl

<xl

I

I (T¢I' l/Ij}(S*¢j' l/Ij)'

R

I

jR 1

by Parseval's relation. Thus, by the absolute convergence of the double series and Parseval's relation again. <xl

<xl

<xl

I

(T¢j' ¢I}(S¢I' ¢.) = I I l,jRl 1 jRl I R

I

(Sl/Ij' ¢1}(T*i/Jj' ¢I)

I (TS¢ I' ¢I)' IR 1

Since any transformation T having finite double-norm is compact, the Riesz-Schauder theory of compact operators applies to T. In particu­ lar, a non-zero complex number A is either in the resolvent set of T ; or is an eigenvalue of T; the non-zero eigenvalues of T have finite ill'. '" multiplicity; and the only limit point of the set of eigenvalues of T is the complex number zero. The next result of immediate interest is Theorem 12.14. But first a lemma is needed.

i'l,

LEMMA 12.13. Let T be a linear transformation on a linear space , X, and let M(A} be the generalized eigenspace of T corresponding to \ the eigenvalue A. Trw subspaces M(A} are independent that is. if ,lA1,... ,Av are distinct, if f l r<: M(A I }, i = 1..... v, and if I f l = 0, then . IR 1 , f j = O. 1 ;::; I ;::; v. r

Proof. The proof is by induction on v. The theorem is trivial for

v = I, but this fact does not help in the proof. We first establish the

lemma for v = 2. Thus, since f 1 + f 2 == O. we may simply assume that

({for some f r<: X. !A

(12.21)

(A 1

-

.T)

(A 2

-

T)

k1

f

=

O.

<xl

j~1 (Sl/Ip T*l/Ij)

k2

f = O.

<xl

=

2 jR 1

(TSl/I j'

l/I j)

If k 1 = O. then it is trivial that f Then the two relations imply (AI - T}g = O.

=

O. If k 1 > O. let g = (AI - T)

k -1 1

f.

194

Some results on Linear Transformations

Elliptic Boundary Value Problems

(A 2

k

T)

-

2

g = O.

Proof. As we have already remarked, the non-zero eigenvalues have

The first follows from the definition of,~ll.nd the second by multiply­ , k -1 ing the second equality in (12.21) by (A~ T) 1 • But now it is seen that Tg

A1g, so that (A 2

==

g = 0, or, (A 1

k

-1

T)

-

1

-

T)

k2

=

g = (A 2

f == O. If k 1 -1

k -2 T) 1 f =

- "k ! 2 - A )''8= 1

= 0, then

f ==

•

O. Since A1 -i- \ ' 0; if k 1 -1 > 0,

then we obtain (A 1 O. Proceeding in this manner, it fol­ lows that f = 0, and the lemma is proved for v == 2.

Now suppose the lemma to be valid for v - I, v 2 3; we then estab­

lish the result for v. We have

k

(A I

(12.22)

T) l f i

-

v

I

(12.23)

f, ==

1_ 1

=

0, 1::;

• I ::;

v;

finite multiplicity and have 0 as the only possible limit point. Ar­ range the non-zero eigenvalues {A j I such that the repetitions of a , given eigenvalue appear in succession. Suppose that A is a non-zero eigenvalue, and that Aj 0 +l, ... ,A.1 + m are the members of {Ajl equal to ,I , 0

A. Then M(A), the generalized eigenspace of T corresponding to A, has dimension m. Obviously, A- T maps M(A) into itself, so we can consider the transformation A which is the restriction of A- T to M (A),. .A is then a linear transformation on M(A). Obviously, the only eigenvalue of A is O. A simple theorem on matrices shows that there is a basis {fjo+1, ••• ,f +m l for M(A) such that the matrix of A with jo 'respect to this basis is triangular. That is,

O.

(12.25)

1

Af.= 1

k

Multiplying (12.23) by (Av - T) v, setting gl == ing (12.23) into account, we obtain kl

(\ - T) g j == 0, 1 ::;

. I -:::;

("-v - T)

k

vf;, and tak­

I'

alfl,io+lS:iS:io+m,

;-jO+1

for certain constants ali. Moreover, ali == 0, since ail must be an eigenvalue of A. Since Ai.1 == (A - T)f 1" (12.25) implies (since also

A=A) j

(12.26) =

j

I

v-I;

V-1

I gi

195

O.

j-1 Tf ==Af I al}fl,jo+l-:::;j~jo+m. j Jj l-j+1

o

,- 1

,

k

By the induction hypothesis, gj = 0; that is, (Av - T) vf = 0, l kv k 1 -:::; is: v - 1. But since (A v - T) f l = (AI - T) l f l == 0, the result for v = 2 implies f J == 0, 1 s: i -:::; v - 1. And then (12.23) implies also f v == O. Q.E.D.

If this is done for each eigenvalue, then the independence of the sub­ spaces M(A) (d. Lemma 12.13) implies that there is a linearly inde­ pendent set

lip

f 2 , ... 1such that

j-1 (12.27) Tf == A/ j + ;~1 a~jfl' j THEOREM 12.14. Let T be a linear transformation on a separa­ ble Hilbert space X, and let T have finite double-norm. Then each for certain constants a~j. non-zero eigenvalue of T has finite multiplicity and the eigenvalues of T can have only 0 as a limit point. If {A j I is the set of non-zero By the Gram-Schmidt orthogonalization process, there is an ortho­ eigenvalues of T arranged in some order, where each eigenvalue is normal sequence {g l ' g 2, ••• 1such that repeated the number of times equal to its multiplicity, then

(12.24)

IjIAJI2:S;

IIITII12.

196

J

I

g = J

I_ 1

,

blJf

I'

bJJ.j, 0

197

Some results on Linear Transformations

Elliptic Boundary Value Problems

LEMMA 12.15. Let P and Q be continuous non-zero projections such that liP - QII < IIPII- 1. Then the dimen­

Iii. 011 a Hilbert space X,

,

ii,sian of R(P) is not greater than the dimension of R(Q). '~'

J

I

fJ =

i-1

:~

JJ b lJ " I' b JJ 1 15 1 -- (b )-1 •

Proof. Suppose that R(P) has larger dimension than R(Q). Now

~R(Q) has dimension at least that of

R(PQ), so it follows that R(PQ) a proper subspace of R(P). Since R(P) and R(PQ) are both closed r'subspaces, there is a unit vector f in R(P) which is orthogonal to ':R(PQ). In particular, f is orthogonal to PQf, so that

f is II

Therefore, (12.27) implies J 1-1

Tg = I bIJ[A f + I aklf ]

i i k- 11k J 1-1

Ilf - PQ f

- bJJA f + I aiJf

J J i.1 2 i

1 CS; IIP(Pf - Qf)11 cs; IIPIIII(P - Q)fll

J-1

J-1

I

= bJJA [bJJg + I blig] + I alJ I 1 J

1-1

1

I

1-1

2

k-1

1 k This contradicts the hypothesis. Therefore, R(P) has dimension not igreater than that of R(Q). Q.E.D. As a corollary we have

t

I a~Jg I'

i-1

~'

for certain constants a~J. The orthonormality of the sequence jg l' g2""} implies

IITgI1 J

2

J-1

lJ 2 la I

=IAIJ 2 + 1-1 I 2:

3

IAJ I2•

I

J

IITg J I1

2

L

I

J

LEMMA 12.16. Let P be a non-zero projection on a Hilbert space, "::and let jp k} be a sequence of projections such that lim lIP k - PII = O• .',~ .

k...."OQ

;Then for all sufficiently large k, the ranges of P k have the same ;dimension as the range of P.

I\P k II < 211PII and also 1 ~IIPk-PII
THEOREM 12.17. Let T be a transformation having finite double­ on a Hilbert space X. and let jAJ } be the sequence of character­ 'atic values of T, each repeated a number of times equal to its mul­ 'iplicity. Then for all A €: Pm (T), the modified resolvent T A has

~rm

Summing over all j, we obtain

fIITIW::?:

cs; IIPIIIIP - QII·

bklg

J-1

AJg J +

=

1 + IIPQW 2 1.

But since f €: R(P), it follows that f = p 2f, and we have from (12.28)

J-1

J

ll 2 =

IAJ J2.

Q.E.D. To finish the main results of this section, an important formula is needed, which will be given in Theorem 12.17. But first a lemma is needed. Recall that a projection on a linear space is a linear trans­ formation P satisfying the equation p 2 = P.

,[finite double norm, and

Tr(TT A) =

A- 1I. ~ J

--l) + tr(T 2) -

A._A AJ J

I ~. J

A2J

198

sec. 12

Elliptic Boundary Value Problems

Proof. First note that since T A = T(l - Al~l~ Theorem 12.10 implies that T A has finite double-nonn. /Let 1<7>1' <7>2, ... 1 be an orthonormal basis in X, and let t ii =JTJp l' <7> i)' Let TN be the trans­ formation defined by (12.18). Asshown in the proof of Theorem 12.11, IIIT- TNIII->O. I. Now TN is essentially a transformatiolfori a Hilbert space of di­ mension N, and so TN has characteristic values \.N' of which there are at most N, counted according to their multiplicity. Moreover, using a triangular representation of the matrix of TN' the formula (12.29) for TN is obvious; thus,

Therefore. Theorem 12.10 implies

IIITA-

Atr(TN(TN)A) =

'2: j

(TN)AIII:s;

[11(1- AT)-III

+

\Al IIITNIII

110-

AT)-III .

·II(I-ATN)-III]IIIT- TNIII· Therefore, by (12.3), lilTA - (TN)A III -> 0 uniformly for A bounded and a fixed positive distance away from characteristic values of T and TN' Hence, (12.31) implies that tr(T N(TN)A) -> tr(TTA), and (12.30) implies

(12.32) (12.30)

199

Some results on Linear Transformations

(

A:" N1 _ A _ A 1• ). 1. JN

Atr(TT ) A

=

lim '2:. ~ N->oo 1 A _A j,N

1_),

Aj,N

the limit being uniform for A bounded and a fixed positive distance Note that in this finite-dimensional case,tr(T~) = '2:l;.~' Since the trace function is linear, we have tr(TT A)

-

from IA.I and lAo1. NI. J Now some results of operational calculus are needed. See, for in­ stance, Taylor, Introduction to Functional Analysis, Wiley, 1958, pp. 287~307. Let A. be fixed. Since the characteristic values of T 1 are isolated, a circle of sufficiently small radius 2f with center A-:1 I contains only the eigenvalue A-: I of T. The result of operational cal­ l culus we shall need is that the operator

tr(TN(TN)A) = tr[(T - TN)T A] + tr[TN(T A - (TN )A)]

Applying (12.19), we obtain (12.31)

Itr(TT A) ­

tr(TN(TN)A)I :s;

lilT - T NIII IIITAIII +

(12.33) ~

IIITNIIIIIITA-(TN)AIII.

I

2m I/1-/\. \ -II -f

(/1- T)- l d/1

1

Now

is a continuous projection onto the generalized eigenspace M(A j ). Likewise, the operator

TA-(TN)A= T(l-AT)-I- TN(l-ATN)-1 = (T - T N)(1 -

AT)-I

+ T N[(l -

AT)-I -

(12.34)

(1 - ATN)-I]

1

\-11

27Ti II /1-/\ . 1

= (T - T N)(1- Uri + T N(l-

AT)-I[l- ATN -

(1- AT)]'

. (1 _ AT )-1 N

= (T - TN)(l -

AT)-I + AT N(l- AT)-I(T -

(/1- TN)-ld/1 -f

T N)(1- ATN)

.,11\

is a continuous projection onto the direct sum of the generalized eigenspaces of TN corresponding to eigenvalues of TN within a dis­ tance of f to A-I. Now since (/1 - T)-I exists for 1/1 - A-: II = f, also, J 1 for all sufficiently large N, (/1- T N)-I exists for !/1- A;I\ = f. More­ over, (/1- T N)-I converges uniformly to (/1- T)-I on the circle

University of

~,~,?,~~ ;n~ton

Liorary

200

Ifl- A~ 1[ = £. All this follows sincell TN - Til .... O. But this implies that the projectj,on(12.34) converges uniformly to the projection (12.33). By the ~o~ollary to Lemma 12.15, it follows that the total multiplicitY9f aii eigenvalues of TN within £ of A~1 is the same as the multiplibty m of A:-1. Moreover, by taking £ smaller I J and smaller, it follows that among the eigenvalues of TN there are precisely m (counted accordingfoinultiplicity) which converge to A/-I. Therefore, the characteristic values {A/,N I can be arranged in such a manner that A/,N .... A/, for all A}' In case T has only a finite number of characteristic values, the remaining characteristic values of TN must tend to infinity. For, if they had a finite limit point, that limit point would necessarily be a characteristic val ue of T. This follows since if (fl - T)-1 exists, then also (A - T)-1 exists for IA - fll < 2£ (some £ > 0), whence also (A - T N)-1 exists for IA - fll < £, all N sufficiently large. Now let

the convergences being uniform in any compact subset of Pm (T). Therefore, F(A) is an entire function of A. We shall estimate F(A) for IAI ::;; r, taking n so large that IA/I·~ 2r, \A/, NI ~ 2r for j 2 n. Thus, by (12.36) IF(A)I ::;; I

\AI

/~n IA / _ A\ IA / 1

Note that since (12.36)

-~ + lim sup I/~n~ J~n Y2IA/12 N .... oo Y2IA/,N 12

::;; 21AI(IIITII12 + lim sup IIITNIW) N ....oo

= 411 I,TIWIAI,

1 A/

­

A = (\ -

A)A/

Letting A -0 0 in this relation, we obtain

1.- -

I /

tr(T2) = a.

2

A/

1 J1A12­

I . __< IIITII1 2 I /

Q.E.D.

(d. Theorem 12.14), it follows that the series in the definition of F(A) converges uniformly for A in a compact subset of Pm (T). By

(12.32) and the previous discussion, we have for any integer n

-

\AI

/~n lA,JoN - A\ IAI /,N

1 - - tr(TT A) - a

I --

and since

F(A) - I

N-->oo

;:;; I.

/ (A/ - A)A/

1 --­ A/ - A

+ lim sup I

by Theorem 12.14. Therefore, F(A) is an entire function satisfying IF(A)I .: :; const IAI for all A, Hence, F is linear: F(A) = aA + b, some constants a, b. Letting A .... 0 in (1235), it follON!s that b = O. Thus, dividing (12.35) by A gives

1 1 F(A) = I/ ( - - - -) - Atr(TTA)' \ - A A/

(12.35)

201

Some results on Linear Transformations

Elliptic Boundary Value Problems

(_1__

J~n A. _ A J

1.-) -lim A/

N....oo

I.~ (__1 J n A/,N - A

1_),

A/. N

Part 3. HILBERT-SCHMIDT KERNELS We shall now give two results identifying the class of operators of finite double-norm in case the Hilbert space is L 2(m, for some measurable subset of E n •

n

__no_._ .•

~

_

_

.

-- __

~

-

202

ElIiptic Boundary Value Problems

Some results on Linear Transformations

THEOREM 12.18. Let K(x, y) be a measurable function on 0 x 0, square integrable on 0 i.e., K ~ L 2 (n If f ~ L 2(0), then /

x/fA"

J

o

K(x, y )f(y

xO).

)J;/

IIITII1 2 = J

(12.37)

Squaring and summing on i,

i

(12.38)

OxO

(T¢ .)(xW

;-1

'.. _-.. .

converges absolutely for almost alI x in 0, and represents an element Tf in L /0). The transformation T is a linear transformation having finite double-norm on L 2(0), and

203

=

i

IJ K(x, y)¢/y) dY12.

;-1

I

0

As remarked above, for fixed x the function K(x, y) is in L 2(0); and J K(x, y)¢ /(y )dy is just the inner product of this function with

o

¢;(Y). Therefore, Parseval's relation implies that (12.38) may be written for almost all x

IK(x, yWdxdy.

00

__

~ IlT¢.)(xW=J

;-1

Proof. Fubini's theorem shows that, for almost all x, the function

K(x, y) is measurable for y ~ 0, and

0

I

IK(x,yWdy.

Integrating with respect to x,

J

o

IK(x, y Wdy < 00.

Thereofre,

00

~

J

o

K(x, y )f(y )dy exists as an absolutely convergent inte­

gral for such x, as a result of the Cauchy-Schwarz inequality; further­ more,

!(Tf)(x)1 2 = IJ K(x, y)f{y)dyI2

o

S;J

o

IIT¢;W =

/-1

J

OxO

IK(x, yWdydx.

Q.E.D. Definition 12.8. An operator of the type given in Theorem 12.17 is said to be an integral operator with Hilbert-Schmidt kernel K(x, y). We now give a converse to Theorem 12.18.

THEOREM 12.19. Let T be a transformation having finite double­ norm on the Hilbert space L 2(0). Then T is an integral operator with a Hilbert-5chmidt kernel.

IK(x,yWdyllfI1 2.

Integrating over 0,

Proof. Let {¢1' ¢2, ..• 1be a fixed orthonormal basis in L 2(0). Let (T¢/, ¢), and let

t ii =

II TfW S; J

OxO

!K(x, YWdydxllfl12 N

KN(x, y)

= .

~ 1 t;j¢ /x)¢ j(Y)'

'.1

Thus, T is a bounded linear transformation on L (n). 2

To derive (12.37), let {¢I' ¢2, ... 1be any orthonormal basis in L (0). 2 Then for almost all x

(T¢/)(x)

=

J K(x, y)¢/(y)dy.

o

Then for N 1 < N 2'

J

OxO

IK N (x, y) - K N (x, YWdxdy

2

1

204

Elliptic Boundary Value Problems =

IS.

f

ilxil

tii¢/X)¢i(YWdxdy

il "

!t 1J

I

/

I

Thus, T 1¢/ = T¢/ for all i. Since 1¢1' ¢2, ... l is a basis, T 1 = T.

Q.E.D.

(x)¢i(Y)¢" (y )dxdy

To complete this section, we give two results showing how certain operations with integral operators can be carried out by performing corresponding operations on the kernels,

1

2 1

i,JSN 2 i>N 1 or i>N 1 00

/-N 00

Since

I

2

It/J1

00

00

I !t 1J 2 +

I

I'

1

1

+l j - l

THEOREM 12.20. Let T be an integral operator on L 2(0) hav­ ing Hilbert-Schmidt kernel K(x, y). Then the adjoint T* is an integral operator on L2
00

I [t ii [2.

j-N1+l/-1

Proof. If f, g € L 2(0), then

IIITII[2 < 00, it follows

=

i ,i- 1

205

= (T¢/, ¢i)'

i>N 1

or

I~~tiitil i' ¢/(x)¢

S

Some results on Linear Transformations

j1,iS N 2

,/>N 1 =

sec. 12

that as N 1 , N 2

->

00, (Tf, g)

=

f (Tf)(x)g(x)dx

il f

ilxil

IK N (x, y) - K N (x, y)[2dxdy 2

O.

->

1

=

f

K(x, y)f(y)g(x)dxdy

ilxil As L2
f

IKN(x, y) - K(x, YWdxdy

->

K(y, x)f(x)g(Y)dydx

ilxil

O.

ilxil

=f

f(x)[f K(y, x)g(y)dy]dx

il

Let T 1 be the integral operator associated with Hilbert -Schmidt kernel K; we then must show T = T l' To see this, note that

=

il

(E, T*g).

Since this holds for all f € L/O), we p.ave almost all x € il (T 1¢" ¢.) I

1

=

f

K(x, y)¢/(y)¢.(x)dydx

ilxil

1

(T*g)(x) = f

= lim f N->oo

K(y, x)g(y)dy.

il

KN(x, y)¢/(y)¢/x)dydx

ilxil

Q.E.D. = lim N->oo

= til

N

II

~

j',i

-

-­

t/ i f ¢/,(x)¢/x)dx • f ¢i'(Y)¢i(y)dy -I

il

il

THEOREM 12.21. Let T i be an integral operator on L/O) hav­ ing Hilbert-Schmidt kernel K /(x, y), i = 1, 2. Then the integral (12.39)

K(x, y)

=

f

il

K I (x, z)K 2 (z, y)dz

206

sec. 12

Elliptic Boundary Value Problems

is absolutely/convergent for almost all (x, y) ~ 0 x 0, and T1T 2 is

Some results on Linear Transformations

(T 1T 2f, g)

=

f(T 1Ti)(x)g(x)dx

=

f [J K 1 (x, z)(T 2f )(z)dz]g(x)dx

=

f

=

fHfK1(x, z)K 2(z, y)dz]f(y)dy\g(x)dx,

207

an integral ((Jperator whose Hilbert-Schmidt kernel is K(x, y). More­

over,I«:.x/~) as defined by (12.39) exists for almost all x € 0, and (\­

(lvi.o)

tr(T 1 T 2)

=

f

"\

K(x, x )dx,

0

the integral being absolutely convergent. Proof. We have seen that for almost all x, K 1(x, z) is in L 2 (O) as a function of z; likewise, for almost all y, K 2(z, y) is in L 2 (0) as a function of z. Thus, (12.39) is convergent for almost all (x, y) €

o x O.

Also, we see that K(x, x) exists for almost all x € O. We next establish the measurability of K(x, y). Let f, g ~ L 2 (0). As we have seen K/z, y)f(y) is an integrable function of y for almost all z. Moreover, f jK/z, y)f(Y)ldy is then in

o

L 2 (O) as a function of z. Therefore, it follows that for almost all x, IK 1(x, z)lf !K 2(z, y)f(y)ldy is integrable, and that

o

[J K /x,

z)1f K /z, y )f(y)dy Idz]g(x )dx

since the integrand is integrable over 0 x 0 x O. Thus, (TJ2f, g)

=

J[K(x, y)f(Y)dy]g(x)dx.

Since this holds for all g, we must have all almost all x (T 1T 2f)(x)

=

fK(x, y)f(y)dy.

Therefore, TIT 2 has kernel K. Finally, we must prove (12.40). First, we have by the Cauchy­ Schwarz inequality

f

o

IK 1(x, z)ldz f !K 2(z, y)f(Y)\dy

0

f IK(x, x)ldx~,f f IK 1(x, y)IIK 2 (y, x)ldydx

o

0 0

is in L 2 (0). But then g(x) times this function is integrable. Since the function K 1(x, z)K 2(z, y)f(y)g(x) is measurable in 0 x 0 x 0, it is integrable over 0 x 0 x O. But then Fubini's theorem shows that its integral with respect to z, which is K(x, y)f(y)g(x), is measurable in 0 x O. Taking f and g to be characteristic functions of measur­ able sets, it then follows that K(x, y) is a measurable function on

~ [J

IKt(x, y)1 2dydx]Y2U

OxO

=

IK/y, x)2dydx]Y2

OxO

III T 1111 III T 2\ \I <

00.

Thus, K(x, x) is integrable over O. Moreover, by (12.20), if 1


Ox O. By the Cauchy-Schwarz inequality,




IK(x, yW ~ f IK/x, zWdz • f IK 2(z, yWdz.

o

n

Integrating this over 0 x 0, it is seen that K ~ L 2(0 x 0). It remains to identify K with the operator TIT 2 and to prove (12.40). Now for f, g € L 2 (0).

tr(T 1 T 2)

= .

~ 1 (T 2
',I

1,


00

(12.41)

I f l,f- l

K 2(x, y)
OxO

208

.f

K 1 (y. x)
OxO

Since the functions


f

K 2(X' y)K 1 (y, x)dxdy

OxO

=

f

o

K(y. y)dy.

209

Spectral Theory of Abstract Operators

EIliptic Boundary Value Problems

Of the various constants appearing in estimates in this section, we wish to isolate the one appearing in the statement of Sobolev's in­ equality. First a lemma is needed, showing that Theorem 3.9 is really a sort of convexity result. LEMMA 13.1. Let m > n/2 and let u :€ Hm(0). Then u can be modified on a set of measure zero so that u :€ CO(n). Moreover, there exists a constant y, depending only on 0 and m, such that for all

x€ n n/ 2m + lui ) lu(x)1 ~ y(lu\1-(n/2m)\ul o m O. Proof. By Theorem 3.9 it is enough to prove the estimate. Accord­ ing to Theorem 3.9,

Q.E.D. 13. Spectra I Theory of Abstract Operators Before giving results for elliptic equations, we can give several abstract theorems in which no mention need be made of differential equations. In these abstract results and in the results of section 14, essential use will be made of Sobolev's inequality (Theorem 3.9) and Rellich's theorem (Theorem 3.8). In order that both these theorems be valid. it is sufficient to assume that 0 is a bounded open set hav­ ing both the segment property and the ordinary cone property. Actually, as the proofs of these theorems show. it is enough to assume 0 is a finite union of disjoint open sets of such type. This is obvious in the case of Sobolev's theorem, since a finite union of open sets which have the ordinary cone property still has the ordinary cone property. In the case of Rellich's theorem, note that the only use of the segment property for 0 was in the application of t he second inequality of Theorem 3.2. But it is obvious from the proof of Theorem 3.2 that the theorem is valid for the case that 0 is a finite disjoint union of bounded open sets having the segment property. We shall be some­ what more restrictive than this, since we shall also need the restricted cone property in Theorem 13.9. We require everywhere in this section and the section following that 0 be a finite disjoint union of bounded open sets. each of which has the restricted cone property. We shall use this assumption throughout these sections without mentioning it explicity.

[u(x)1 ~ Y1r-(m-Yon)( lui m + rml Ul 0)' If lul m ~ \ul o' then, taking r

= 1,

r

2

1.

we obtain from (13.2)

lu(x)1 ~Yl(lul~-(n/2m)\ul::,/2m + \ul o)' If lul m > lul o' then take r m = lulmlul;l, so that (13.2) implies

lu(x)l.::;: Yl(lulmlul~1)-1+(n/2m)(lulm + lul m) =

2Yllult(n/2m)\ul::,/2m

= 2Yl(lul~-(n/2m)lul::,/2m

+ lul o)'

Remark. An easy argument based on Young's inequality shows also that (13.1) implies (13.2). The argument is the same as was given on page 73 to show that the inequality (7.2) implies (7.1). As an immediate corollary we have

LEMMA 13.2. Let m > n/2 and let u :€ Hm (0). Then there exists a constant Ys' depending only on 0 and m. such that, after modifica­ tion of u on a set of measure zero,

210

Elliptic Boundary Value Problems

/~_/(13:3')

lu(x)\ s Ys Ilull::,12m I lull t
X

~

n.

By Lemma 13.4 it follows that these norms all are finite; of course.

IT\ k

Definition 13.1. The smallest constant y s such that (13.3) is

valid for all u ~ Hm (0) is the Sobolev constant of

n.

Likewise, we can prove a similar result concerning interpolation inequalities. This result will not actually be used until the next sec­ tion. LEMMA 13.3. There exists a constant y, depending only on and m, such that for u ~ Hm(O) and 0 s j s m,

lul.j,n s

Y[lul~-(J/m)lul~m +

n

lul o]

.s 2ylult<jlm)llulljlm. m Proof. The proof follows from Theorem 3.3 by choosing the para­ meter l appropriately; the procedure is just the same as given in the proof of Lemma 13.1. Q.E.D.

Now we turn to the consideration of linear transformations which

will later play the role of inverses to di fferential operators.

LEMMA 13.4. Let T be a bounded lineal' transformation on L 2 (0), such that the range of T is contained in H m (f!), some m ~ 1. Then T is a bounded linear transformation of L /0) into H m (0).

Proof. By the closed graph theorem, the result will follow if we show that T is closed. considered as a mapping of L 2(0) into Hm(O). Suppose then that {ukl CL 2(0), Uk .... U in L/O), and Tu k .... v in Hm(O). Then. since T is continuous on L 2 (0), Tu k .... Tu in L 2 (f!). But a fortiori Tu k .... v in L 2(0). Therefore Tu = v. Q.E.D.

Definition 13.2. Let T be a bounded linear transformation on L 2 (f!). such that R(T) cH m (0). For 0 s k s m let

IITll k = ~r2(0).llfl\o.n)ITfllk.n;

ITI k

=sup ITflkro. f~L2(O),llfllo,n-1 ,H

211

Spectral Theory of Abstract Operators

sec. 13

is not a norm, but a semi-norm, if k > O.

THEOREM 13.5. Let T be a bounded linear transformation on L 2 (f!), such that R(T) CHm(f!), where m > n/2. Then T has finite double-norm, and the fol1owing inequalities hold:

!Tlt
(13.4)

IIITI,II.s ylnllh(ITI::,12mITI,t
(13.5)

IIITIII s Ysln\lhIITII::,12mIITII~-
Here \n\ is the Lebesgue measure of n, y s is the Sobolev constant of n, and y is the constant appearing in (13.1). Proof. Let {cP I , cP 2 , ... l be an orthonormal basis in L/O), and let u j = TcP}' j = 1, 2, .... By Lemma 13.1 it may be assumed t~at u j ~

C°(f!). Let al" •• a N be any complex numbers, and let f = j~l ajcPj'

Then, the continuity of the function u. and of u = Tf implies that J

N

u(x) =

2 aju.(x), all

j_ I

J

x ~

By Lemma 13.1 we have for all x !u(x)\

n. ~

n

s Y(ITEI~-(n/2m)ITEI;;,12m + !TEl o) s

y(ITlt
Ilfllo.n'

That is, N

(13.6)

11 j_ I

a.u.(x)!.s y(ITI I -< n/ 2m)\T\nI2m + J

J

0

m

n

N

ITI 0 )[ j_2I la j \2]lh.

This inequality holds for all x ~ and for all complex numbers al, .... a N • If for fixed x we take a j = u/x), then we obtain from (13.6)

212.

Spectral Theory of Abstract Operators

Elliptic Boundary Value Problems N

11

[I lu .(X)\2] 12:::; Y( 1TI I -(n/2m)1 TI n/2m + ITI J-l

1 0 m

0

213

). \l<1-AT)-lll o '::;; 1+ IAIIITAllo'::;; l+C,

A(2(O, a).

Since this holds for all x ( 0, we may square and integrate over 0 to obtain

I

J-l

IIT¢ J 11 O.UnS::y2101(ITll-(n/2m)ITln/2m+ 0 m 2

IITAll m = IIT(1- AT)- l ll m s:: IITll mll(l- AT)-lll o

ITI )2. 0

:::; IITll m(1 + C). Since this holds for all N, we have Therefore, Theorem 13.5 implies T A has finite double-norm, and we 00

I

J-l

IIT¢.11 2 n s:: y2101(ITll-(n/2m)ITln/2m + I TI )2. 1

O.U

0

m

bave an estimate

0

III TAlI! ~ ysIO\ Y2 11 TAII~/2mll TAil t(n/ 2m)

Thus, (13.4) is proved. The proof of (13.5) is ma n/2. Let there exist at least one direction of minimal growth of the modified resolvent of T. Let 1\I be the characteristic values of T, repeated according to multiplicity. Let N(t) be the number of characteristic values of modulus less than or equal to t: N(t) = I 1. Then N(t) = O(t n / m) as t -> 00, IAJI.::;;/

s:: YsIOI Y2 1ITII;/2m(1 + C)n/2mCl-(n/2m)IAI- l +(n/2m) By Theorem 12.4, the characteristic values of T A are precisely

A-I _J_

~;l-A-JIA

N(t):::; 4y;IOIIITII:/m(1 + C)2t n / m, Proof. We have by (12.5)

II

J

1,,?:::; IIITAIW.

1

Indeed, if II TAil 0 ~ C1AI- l for A on the ray

I

A-A

J

I J"

tr(TTA) = I J (A _ A)A

j

J

(1- AT)-1

A -A

.multiplicity of the characteristic .value \ of T. Therefore, applying ~;Theorem 12.14 to T A, we obtain the estimate valid for all A ( Pm(T)

and for all A (Pm(T)

(13.7)

= _1_ ,and the multiplicity of _1_ is the same as the

=

1 + ATM

E<e, t ~ a.

a), then

:,In case IAjl.::;; t and IAI = t, then IAJ - AI s:: 2t. Therefore, if in (13.10) we sum only over those j for which \A J\ .::;; t, we obtain for IAI = t, itA ( 2<e, a),

I

IAJI.::;;/

1
A (2(0, a).

IOIIITII~/m(1 +

c)n/mC2-(n/m)t-2+(n/m)

taking (13.9) into account. Multiplying by 4t 2 , we obtain

214

Spectral Theory of Abstract Operators

Elliptic Boundary Value Problems N(t):;, 4y~lnIIITII~/m(1 + c)n/mC2-(n/m)t n/m •

IB[v, u]1 :;, Kllvllm.nlluI1m,n,

~B[u, u]::;, collull';',n'

Finally, we must prove (13.7). By Theorem 12.17 we have

tr(TT).)

=

Let T be the unique linear transformation of L 2 (n) into

I j (A. _1 A)A + c. j

J

for all A ~ Pm (T) and for a certain constant c. For A ~ obtain from (12.19) and (13.9)

B[v, Tf]

see. a), we

:;, const IA[-1+(n / 2m). -+

00,

A ~ S{e, a). But also I. ( J

\

-

1 A)A j

-+

0

as IAI -+ 00. Therefore, c must vanish. Q.E.D. As a result of this estimate on N(t), we can sometimes obtain lower bounds for individual characteristic values, as we now show. Corollary. In addition to the hypothesis of Theorem 13. 6, suppose also that T A exists for A on the ray 0), and that for such A the estimate II TAllo:;' C1AI- 1 is satisfi~d. If the characteristic values !A j I of T are arranged according to increasing modulus, IA 1 IA21 .s;: •••, then for j = 1. 2, ... ,

See,

IAj \2.

(v, I)o.n

Co

IA j !2 [4y~ln\(1 + C)2]-m/nIITII: 1j m /n. Proof. This is an immediate consequence of the theorem, since

.:S N(IAjp. Q.E.D.

In a sense, this is better than an asymptotic estimate, since the estimate for Aj is valid for all j. However, it gives only a lower bound for IAjl. Under more assumptions we shall obtain asymptotic formulas for N(t) in the next section. Let us now give an example. THEOREM 13.7. Let B[v, u] be a bilinear form over a closed subspace V of Hm(n), such that for certain positive constants K and Co and for all u, v ~ V

jm/n.

(16Y~lnl)m/n

Proof. The existence of T is a consequence of the Lax-Milgram 'theorem (Theorem 8.1). If in (13.11) we substitute v = Tf, we obtain ~(Tf, f) = ~B[Tf, Tf]

2 coIITfII';'.n

1 :;,

j

=

:lor all f ~ L/m, v ~ V. Then the half plane IA: ~A < OJ is contained ;in Pm (T), and the negative real axis is a direction of minimal growth of the modified resolvent of T. If the characteristic values 1A.I of T J are arranged in order of increasing modulus, then

Itr(TTA)I:;, IIITIIIIIITAIII

Thus, tr(TTA ) -+ 0 as IAI

215

2

O.

iSince T is bounded as a mapping of L 2(m into L 2(m, it follows that for sufficiently large \A\, A is in the resolvent set of T. Therefore, "Theorem 12.8 shows that each A with negative real part is in the re­ \:solvent set of T. Therefore, (1 - AT)-l exists for all A such that iRA < 0, since 1 - AT = A(A -1 - T), and ~A- 1 < O. To obtain the es­ '\timate, note that if u = TAf = T(l - AT)- l f, then, since T and (1 - AT)-l commute, (1 - AT)u = Tf, so that u = T(AU + f). Therefore, (13.11) implies that for all v ~ V· B[v, u]

=

(v, Au + f);

216

Elliptic Boundary Value Problems

. sec. 13

i.e.,

\2 B[v, TAf] = (v, ATAf + f),

Now, if we set v B[TAf, TAf]

=

=

~A < O.

TAL, then

All TAfl1 ~ + (TAf,

,',

f).

Therefore,

coIITAfll~::;: ~B[TAf, TAf] =

~AII TAfl1 ~ + ~(TAf, f)

s; ~AIITAfll~ + IITAf 11011 f ll o' Dividing by II TAfl1 0' we obtain

IITAlloS:

1 Co -

~A

,~A
Therefore, if A is a negative real number, IITAll o s:

1_ ::;:1­ AlAI

~_ o

Spectral Theory of Abstract Operators Co

(16y~If!I)2m/n

217

j = 1, 2, ....

j2m/n

'

Proof. From (13.12) we have (Tf, f).2 0 for all f so that T is a positive compact self-adjoint operator. We now use a well-known

result that a positive self-adjoint operator admits a positive self­ adjoint square root. More precisely, since T is compact there exists a (unique) self-adjoint positive and compact operator S such that S2 = T. Also the seqqence of characteristic values of S is given by the sequence IA~21 (positive square root). We observe that since I Tf t. 0 for f t. 0 (an immediate consequence of (13.11)) also Sf t. 0 for f t. O. From this it follows that the range of S is dense in L 2(0). Using now (13.11) with v = Tf we have: B[Tf, Tf] = (Tf, f) = (S 2 f, f) = IISf\\~.

Putting Sf

=

g, we then get

collSgll,;,

=

coIITfll,;,::;: B[Tf, Tf]:;: Ilgll~.

Since by the previous remark elements g = Sf are dense in L 2(0) it foll ows from the last inequality that the range of S is contained in H m (0) and that

.

Thus, it is seen that the hypothesis of the corollary to Theorem 13.6 is satisfied with C = 1. Since it follows from (13.12) that II Tll m S; C~l, the estimate for IAil is a consequence of the estimate in the corollary. Q.E.D. In case the operator T in Theorem 13.5 is self-adjoint, the result can be improved to obtain the ·correct" order of growth of A}' We have Theorem 13.7. bis. Suppose that the conditions of Theorem 13.7 hold and that in addition the bilinear form B[u, v] is Hermitian symmetric so that the operator T defined by (13.11) is a self-adjoint compact operator. If the characteristic values IA.I of T (all of which are posiI tive) are arranged in an increasing order, then

IISll m S; c~~. Consider now the modified resolvent SA along the positive imaginary axis. By (12.14) we have: IISAllo s: IA\-I. Thus it is seen that the hypothesis of the corollary to Theorem 13.6 (for SA) is satisfied with C = 1. Using this and the estimate for IISll m, recalling that 1All is the sequence of characteristic values of S, we obtain our theorem as a consequence of the estimate in the corollary. In Theorem 13.5 it is reasonable to expect that if m is much bigger than n/2, then more can be expected of the transformation T. This is in fact the case, as will now be shown. First, a lemma is needed. LEMMA 13..8. The open set f! x f! is a finite union of disjoint open sets, each of which has the restricted cone property.

218

EIIiptic Boundary Value Problems

sec. 13

Proof. Since n is a finite union of disjoint open sets having the restricted cone property, it is obviously sufficient to verify that the Cartesian product of two cones contains a cone. By shrinking the cones, we may assume that the two cones in question have the same height and same opening. Thus, let ~o and ""0 be unit vectors in En' let 0 < < 17/2, let h > 0, and let

e

C!

e,

I~: ~. ~0'21~1 cos

=

C2 =

I.,.,: ""'''''0'2

l~ ~hl,

~ = O. Moreover, I(~,

I \

We shall show that C C C!

I(~,

'2 X

.,.,)1

cos 01' \(~, .,.,)\ ~

hi.

So let (~, .,.,) ~ C; then

If

~

l

+

InI 2 )Y2 cos e!.

i I I

(13.13)

I~\

0;:;'

(1 +

\

ElL'\ % y'1 cos e - flJ .

I~12 )

!

\

~I

If a is any real number greater than one, then (1 +

t2)'~ a -

t

L~,

Applying this with a

=

1.- . ~o '2 )2 cos

2

I~I

so that ~ . ~ 0

'2 I~I

cos

y2 cos e1

e.

-

0

[fnIK(x, xWdx]'~:::; y[(ITI:;,/m +

~ t::;

eland t

00.

=

1 = lcos 2e 1

1.,.,111 ~\, =

cos

y is a

IT*I:;,/m)ITlt
we obtain

e,

This relation obviously holds also if

constant depending only on

n and m.

Remark: We state here without proof that under the conditions of the theorem the conclusion (13.13) could be replaced by a much stron­ ger result. Namely, it could be shown that K(x, y) is actually con­ tinuous and bounded in n x n and that (13.13)'

0, then, using the Cauchy-Schwarz inequality,

1.- . ~

=>.,., ~ C 2 ' so that

I,

C 2 and the proof will then be complete.

(~. ~l +.,.,. ""o)TY2 ~ (1~12

Therefore,

THEOREM 13.9. Let T be a bounded linear transformation on

L 2 (n), such that R(T) CHm(n), where m > n if n is odd, m > n + 1

if n is even (equivalently, m > 2[n/2] + 1). Suppose also that

R(T*) C H m (n). Let e= m - [n/2] - 1. Then T is an integral opera-

where I(~, .,.,): (~,.,.,). v 0

:::; h.

n n

T'~(~O' ""0)' and let C be the cone =

obviously implies \~l

tor with Hilbert-Schmidt kernel K(x, y). The function K belongs to He(n x n). Moreover, the trace of K on the diagonal of x exists in the sense of Theorem 3.10, and

Vectors in En X En will be denoted (~, .,.,). Choose the angle e! such that cos 2e 1 = cos 2e, 0 < e! < 17/2, let v 0 be the unit vector v 0 =

C

.,.,)1:::; h

(~,.,.,) ~ C '9 e~ Ct. Likewise, (~,.,.,) ~ C CCC 1 XC 2 • Q.E.D.

e, \.,.,\ ~ hi.

\.,.,1 cos

Spectral Theory of Abstract Operators

!K(x, y)\::;

y[<\T\;;,/m

+

IT*I;/m)

ITI~-
ITl o]'

where y is a constant depending only on nand m. Also for the con­ tinuity of K and for the last estimate to hold it is enough to assume that n possesses the ordinary cone property and that m > n (also for n even). On the other hand from the proof of the theorem that follows one can deduce easily (using Sobolev's inequality and a dimensional argument) that K is continuous and satisfies (13.13)' if e> n.

Proof. It follows from Theorem 13.5 that T has finite double-norm. From Theorem 12.19 it is then seen that T is an integral operator with a Hilbert-Schmidt kernel K(x, y). It remains to verify that K ~ He(n x n) and that the estimate (13.13) holds. Let

r = max

IITI~/mITI~l/m, IT*I~/mITI~l/m,

. Then we have

1l.

220

• ID""u(x)ls; yr-(m-~n-H)
ITl m~ rmlTlo'

(13.14)

\T*l m ~ rmlTl o,

(note:

ITlo = IT*lo)

N

I

Let 1 = 1 a.¢., u I

I

=

I

I

I

[1 i-

1

J-I

n,

and using the relation D""u.I =

+ rmlTI

0

)2.

Therefore, as this holds for all positive integers N, we find

I

IIIT""III:S;: ylnlV,r-(m-V,n-H>(ITl m+ rmITl o)'

cecn). n

By (13.14) this implies

I (13.16)

IIID""TIII

=

I

ID""u(x)1 S; yr-(m-~n-H>Clulm.n

n,

+ rmlu!o.n).

But by Definition 13.2 this inequality implies since

IIIT""II ~ 2Ylnl~r~n+HITlo'

Since T"" = D""T, it should be expected that the kernel K""(x, y) of T"" is given by the formula K""(x, y) = D:K(x, y). This is indeed true, if this derivative is interpreted as a weak derivative in L 2 (O x as we now show. To see this, take first for a test function in n x n any function ¢(x, y) = a(x)b(y), where a ~ C:;"(O), b ~ C:;"(O). Then

D""u(x) = 1 ap""u .(x).

By Sobolev's inequality we also have the estimate for all x ~

TI

n.

'

lIIT""¢.W ~ y2Inlr- 2(m-V,n-H>(ITI m I 0

N

=

~

1

J- 1

u

J

ID""u/x)!2J'4 ~ yr-(m- 4n-H >(ITl m+ rmlTI 0)'

i-I

1 aJu/x),

~ 1.

i-I

IN

N

since r

0

I

After squaring and integrating over D""T¢i = T""¢i' this becomes

TI = 1 aJu., where al' ...,aN are complex

cecn)

J- 1

m

This inequality is valid for all constants a 1 , ... ,aN , and for all x Choosing a, = D""u ,(x), we obtain

numbers. By Sobolev's inequality it follows that. after modification on a set of measure zero. u ~ and u ~ Therefore, if a J is a fixed index with lal S; f. we have for all x ~

(13.15)

N

+ rmlT\ )( 1 \a \2)~.

=

e,

=

I

N

KCy, x). Now consider the operator T"" which maps 1 ~ L 2 (0) into D""TI. If lal ~ then since TI ~ Hm(O), it follows that D""TI is in Hm-H(O)' and m - lalL m - f = [n/2] + 1> nl2. Therefore. T"" is a bounded linear transformation of L/O) into Hm-f(O)' since T is bounded as a transformation of L 2 (0) into Hm(O); d. Lemma 13· 4. Thus. we may apply Theorem 13.5 to T"". if lal :s;: e. It follows then that T"" is an integral operator with a Hilbert-Schmidt kernel K""(x, y). We could certainly estimate III T""III by using (13.4) and (13.5). but we shall find it better to repeat the proof of Theorem 13.5 in the present situation. Suppose then that 1¢1' ¢2 .... ! is an orthonormal basis in L 2CO). and let u. = T¢,.

u(x)

rmITl o)\Ifllo'

11 a.D""u.(x)1 ~ yr-(m-~n-I"" >
J-I

+

By definition of 1 and by (13.15). this may be written

1 S; r.

By Theorem 12.20, T* has the Hilbert-Schmidt kernel K*(x, y)

221

Spectral Theory 01 Abstract Operators

sec. 13

Elliptic Boundary Value Problems

,

:";~lj

I

'ff:)"

J1.

f K""(x, y)¢(x, nxn

y)dxdy = f a(x)dx

n

f K""(x, n

n),

y)b(y)dy.

_

222

sec. 13

Elliptic Boundary Values Problems

Since K'" is the kernel corresponding to T'"

=

we can obviously carry out all the preceding lines of the argument with T replaced everywhere by T*. Since T* has the kernel K*(x, y) K(y, x), the results already obtained show that K(y, x) has all the weak derivatives D:K(y, x), lal ~ £, and we have an estimate analo~ gous to (13.16),

D"'T, it follows that

for almost all x I<: 0

f

K"'(x, y)b(y)dy

o

=

(D"'Tb)(x).

Therefore,

f

(13· 19)

K"'¢dxdy

=f

OxO

(_l)Hf D"'a • Tbdx.

o

K"'¢dxdy = (-l)H

OxO

f

D"'a(x)dx

IID:Kllo,Oxo

~ 2yl01'/:'r'/:'n+HITI0'

IID;Kllo,Oxo

~ 2yI01'l:.r'l:.n+ H ITl o'

K(x, y)b(y)dy

e

K(x, y)D"'(a(x)b(y))dxdy

OxO

f

f

0

(-1)\'" I f

= (_l)H

2y 10\ 'l:.r'l:. n+HI TI O'

e

0 =

~

Here, the kernel of D"'T* is precisely K(y, x). But this means that K(x, y) has weak derivatives D;K(x, y), lal ~ t Moreover, (13.16) and (13.19), when combined with the formula (12.37) for the double-norm of an integral operator, show that for la[ ~

But T has the kernel K(x, y), so this relation can be expressed

f

1\ ID"'T*III

D:

aD"'(Tb)dx

0 =

223

Spectral Theory of Abstract Operators

Therefore, a fortiori all pure derivatives of K(x, y) of order ~ exist weakly and satisfy such estimates. But then since by Lemma 13· 8, x 0 is a disjoint union of open sets having the restricted cone property, and since the corollary to Theorem 11.10 obviously holds in such open sets, we obtain the result that K €: H£(O x n), and that

"

o

KD:¢dxdy.

OxO Therefore, it fall ows immediately that

(13.17)

f

K"'¢dxdy

=

(-l)H

f

OxO

(13.20) KD:¢dxdy

OxO

=

I aj(x)b/(y),

i- 1

t

where aj' b I<: C:;'(n). One can then show that (13.17) holds for all 1.1' j test functions ¢ on 0 x O. In order not to impair the continuity of ••.. . .•. .: the proof, this will be shown as the final stage of the proof. " Thus, it follows that K(x, y) has all the weak d e r i v a t i v e s :

y~, la.1 ~ e. ~e also wish to sho~ that ~(x, y) has.all t~e weak denvatIves DyK(x, y), \aj ~ t It IS precIsely for thIS pomt that

D:K(x,

we need the n"nmption, of the 'heo'em on the ,djoln' T' of T. Fo,_

£.

Now we are in a position to apply the trace theorem (Theorem 3.10). In our case the basic open set is 0 x 0, of dimension 2n, and we wish to consider the trace on the intersection of this set with the manifold II = I (x, x): x I<: En I, of dimension n. Thus, since K I<: He(O x n), in order to apply the theorem we need that 12(2n - n) =

N

¢(x, y)

pS

I

for any test function ¢ of the form

(13.18)

!Klp,Oxo ~ Ylr'l:.n+P!Tlo'

:t

e>

tw. ,.or, tha.t m - [n/2] - 1 > ~/2, or, that m > n/2 + [n/2] + 1. But p hIS IS pr~clsely the assum t.lOn ~ade on m. Therefore, Theorem 3.10 applIes. Thus, K(x, x) IS umquely determined for x I<: 0, and we have an estimate corresponding to (3.19). Since da, the Lebesgue

2

1

Ir

me.asure_on the diago~al of En X En' in our case is just 2'/:'ndx, (3.19) (wIth a - 0) becomes In our case

=

~I I

224

I

[f IK(x, x)12dx]~ ~ Y2r-(e-~(2n-n»(IKle,oxo + relKI o.OxO)

II

sec. 13

EIIiptic Boundary Value Problems

I~I+I~I~N

o

~ Y3r-(e-'4n)(r~n+eITI 0 + r~~nITI 0)'

[J IK(x, x)\2dx]~ ~ 2Y3rnl T\ O'

o

(13.22)

f

= (_l)H

K"'¢Ndxdy

OxO

To finish the proof we simply need to use the definition of r. If r = 1, then (13.13) is immediate. If r > 1, then r is equal to one of the numbers ITI;/ml TI;l/m, IT*I;/ml T[~ 11m; thus, rn ~

a~, ?7t;;(x)e27T1".~t;;{y )e27Tiy·Tf.

By choice of t;;, t;;(x)e27T1"'~ and t;;{Y)e 27TiY '1] are each in C,;;,(m. Thus, ¢N(x, y) has the form (13.18), and it follows that (13.17) holds with ¢ replaced by ¢ N :

by (13.20). Therefore, (13.21)

ITI::,/m\TI;n/m

+

¢(x, y) = Ie ae e27Ti("'~+Y'Tf) '0, Tf '0, Tf

'

I~,Tf(27Ti()'" a~'Tfe27Ti(X'~+Y'Tf),

Now consider the function ¢N(X, y) = t;;(x)(;{y)

I

I~I+\ ?71~N

at:

,?7

KD:¢Ndxdy.

OxO

00

we have the fol­

¢N .... t;;(x)t;;(y)¢(x, y),

D:¢N .... D:(t;;(x)t;;{y)¢(x, y)).

Therefore, (13.22) becomes

f

K"'t;;(x)t;;(y)¢dxdy

= (-l)H

OxO

1&;'

KDO«t;;(x)t;;{y)¢)dxdy;

OxO

"

or,

I·.~... (13.23)

II

f

f

K"'t;;(x)t;;{y)¢dxdy = (-l)H

supp(¢)

f

KD:(t;;(x)t;;(y)¢)

supp(¢) dxdy.

;y)

where I~,Tf is taken over all ~, Tf with integral components. Also, D:¢(x, y) =

f

Now it follows from Leibnitz's rule that as N .... lowing limits in L2
IT*I::,/m\TI;nlm.

Applying this to (13.21), it follows that (13.13) is valid also in this case. Finally, we have to fulfil the promise made earlier to establish (13.17) for every ¢ ~ C,;;,
225

Spectral Theory of Abstract Operators

e27Ti("'~+Y'Tf)

If (x, y) ~ supp (¢) cO' x 0 ' , then on a neighborhood of (x, y), ~x) '= 1 and t;;(y) '= 1. Therefore, (13.23) is the same formula as (13.17). Q.E.D. We shall later need an expression relating the dependence of the constant Y in (13. 13) on the dimensions of O. More precisely, we have the CoroIIary. Under the assumptions of Theorem 13.9, let y be the constant in (13· 13). IE a

>0

and if the assumptions of the theorem

hold for the open set aO and an operator T(s) on L/an}, and if K(B)

·.-.---------::::~--=--"'..,."--.-'"-:=--~~--=-:-------;,:_=.-~._=--.-----

.

.":::~~ ~--=.:_~--=--

226

..---,,'-=::------.-----~~_:.=_--,.-.--~=_-___=?--:-_~~-=-:',:::::_'""--~~~~:~~~~~~~~~~~~~~,~;;:~~~=:~~~~~~~~',~~~'~~.~~~~.::~:-;;~;;.;;~~~:~~~~~~~~~~~~~~~~~~~~:~~~~;,7.;,~~~~-~~,:~~~~:~

Elliptic Boundary Value Problems

sec. 13

is the kernel corresponding to T(a)' then an /

2

[Jan IK (a) (x, x)1 2dx]'4 -< y[an(IT (a) \n/m + m

Spectral Theory of Abstract Operators

227

Therefore, ITl k = akIT(a)lk. Likewise, IT*lk = akIT(a)lk' Substi­ tuting these relations in (13.13), it follows that if K(x, y) is the kernel of T,

\T* In/m) IT ll-(n/m) + (a) m (a) 0

(13.25) [J IK(x x)12dx]'~ < y[an(i T

n'

IT(a)lo]' Proof. For a function u(y) on u'(x)

(Tf)'(x)

=

thus, =

u'(ax)

u(x),

=

x

(n.

Clearly, u ~ H k(n) ""=> u' (H k(an); and

lu' Ik.an

(13.24)

=

[ aI/'n-kl 2 u k.n.

Now we shall define an operator on L 2 (n) which corresponds to T(a)' Precisely, for f ( L 2(n) let Tf be given by the expression (Tf)'

T(a/'.

=

Since R(T (a») C H m (an), it follON:s that Tf ( H m (n). Furthermore, it can be easily checked that (T*f)'

Now if k

=

=

III o.n

ak-l~nl(Tf)llk,an

a-I~nlf'

=

I O.an

a k IT(a/'lk.an

Ifllo.an

In/m + T* In/m)IT II-n/tn + IT I] m (a) m (a) a (a) 0 •

(Tf)(a - I x)

J K(a- Ix, n

~f(~d~

. ~f'(a~d~

=

JnK(a-1x,

=

J

=

(T(a/')(x)

=

J

an

an

K(a-Ix, a-Iy)f'(y)a-ndy

K(a/x, y)fl(y)dy.

Therefore, K(a/ x , y)

an

0 or m, and if f ( L 2 (n), we have from (13.24)

ITflk.n

I

J

T(a/"

=

(a)

Finally, suppose K(a)(x, y) is the kernel of T(a)' Then for f (L 2 (n) and y = a~

n let

x ~ ani

u(a-1x),

=

-

=

IK(a/x, xWdx

a-nK(a-Ix, a-Iy). Thus, a- 2n IK(a- I x, a- I x)[2dx

=

J

=

a- nJ IK(y, yWdy.

an

n

Compare this with (13.25), and the result follows. Q.E.D. THEOREM 13.10. In addition to the hypothesis of Theorem 13.9, assume that there exists a direction e j () of minimal growth of the modified resolvent of T. Let IA I be the sequence of characteristic J values of T, repeated according to multiplicity. Then for any A which is not a characteristic value of T, the modified resolvent TA

228

·sec. 13

ElIiptic Boundary Value Problems

for IAI ....

(13.27)

[J IKA(x, xWdx] '4 = O([AI- 1+(n/m»

o

00,

arg A = O. Also, for A not a characteristic value of T,

f KA(x, x)dx

o

=

A tr(ATT A) = I J (A _ A)\

I _1_ . JA - A J

J

Proof. If A ~ Pm(T), then T A = T(1- AT)-l satisfies the same con­ ditions as T. For, obviously R(T A) eR(T) eHm(n). Since T and (1- AT)-l commute,R«TA)*) = R(T*(l- AT)-l.) eR(T*) eHm(n). Therefore, by Theorem 13.9, T A has a Hilbert-Schmidt kernel K A~

He(O x 0), and we have the e s h m a t e . (13.28)

229

Thus, it remains to prove the formula (13.27). For this, note first that the corollary to Theorem 13.6 implies that IA} I ~ const jm /n for sufficiently large j, since at most a finite number of characteristic values have modulus no greater than a. Since m >n, this implies that the series IJI\\-l converges. By Theorem 13.6, we then have for A ~ Pm(T)

is an integral operator having Hilbert-Schmidt kernel KA(x, y), and

K A ~ HE
(13.26)

Spectral Theory of Abstract Operators

[fOIKA(x, xWdx] '4

S Y[(ITAI';,,"m + I(T A )1::./ m)\T A lt(n/m)+ ITAl o].

=

I

1 -.!) (A - A- A

J

J

1 1 = I1 . - - I 1.A. A _A

(13.29)

iI

J

J

J

Now (l - AT)TA = 0 - AT)T(1 - AT)-l

Now on a ray S(O,a) we have an estimate ITAl o ~ C1AI- I . By (13.8). IITAll m S IITll m (1 + C) for such A. Also. (T A)*= T*(l- AT)-l., so

that for A on S(O, a)

= (l - AT)(l- AT)-IT = T,

so that

II(TA)*I 1m ~ I\T*ll mll(1-AT)-I·ll o

(13.30)

= IIT*llmIIO- AT)-lilo

ATTA = T A - T.

By Theorem 13.9, there exists f K(x, x)dx. Since Theorem 12.21

o

S IIT*\lm(l + C), since (1- AT)-l = 1 + ATA. Thus, (13.28) implies for A ~ S(O. a)

[J !KA(x, xWdx]'~

o

s y[(1 + c)n/m(IIT\\n/m + IIT*lln/m) • Cl-(n/m) m

tr(ATTA)

m

\AI-l+(n/m) + C\AI- l ]

S const IAI-l+(n/m).

implies that the trace of ATTA is the integral of its kernel over the diagonal of 0 x 0, (13.30) shows that =

f [K A(x, x) - K(x, x )]dx.

o

Therefore, (13.29) implies that for all A ~ Pm(T) f KA(x, x)dx

o

=

I. -\-1 1I\J-A

+ c,

=-~-::='~;~~~=~~~~='==..==---==:~-~~.c:E=-~g"=-"'o-=-t'7~~.f.:-'="""~-::=:==~-=O::=~~~~~~~

230

~~~~=,:~~~i~~~~~~~~_";=c_===-~~~~~~1-ili.~;~;"i~:~~~~~~,~~~~~~~~~~~~~';:;:~~~~'=~(:;'~~~~:;-€~;~;:i~;;'::'~;;;::~E3;:;t~~~~~~-~::2~#.g:~~-"f::'~T~~.=;~~~,:;;'i;;'~~~~~2~~=~~~~~~~';;~;i~*0~Z;~-;-':;~~_:&~t::'fu-~~l;;~-;~~:'==

sec. 14

Elliptic Boundary Value Problems

where c is independent of A. Finally, we show that c = O. First, the estimate (13.26) shows that I KA(x, x)dx -+ 0 as IAI

n

1~)2el~(~Wde:

IE (1 +

(14.1)

n

-+ 00,

arg A = O. Next, we show the same result for 'I}A - A)-I. By J Theorem 12.6 there are no characteristic values of T in an entire

converges. Moreover,

angle about SW, a); thus, if IAJI >a, then larg \ - 01 > 8 for some positive number 8. Now, if z = re iO and w = se i ¢ are the polar forms of two complex numbers, and if 8.::;; 10 - ¢I ~ %77, then Iz - wl 2 = r 2 ­ 2rs cos(O - ¢) + s2 = r 2 sin 2 (0 - ¢)+ [r cos(O - ¢) - sF 2 r 2 sin 28, or, Iz - wl2 \zJ sin 8. If %77'::;; 10- ¢I.::;; 77, then [z - wl 2 ~ r 2 + s2 ~

(14.2)

r

2

17.

,

or Iz - wi:?:: Iz\. Thus in any case \z - wl2 Izi sin 8 if 8.::;; Applying this inequality to A. and A, we have

IluWe

I,'"

1\ - AI::: max (IAJI,

Therefore, if \

IID'l(ul\~E

n

=IE

eO
From this relation, (14.2) follows upon addition over all a. lal .: ; £. Conversely, suppose (14.1) is satisfied; notice then that ("~(~ is square integrable over En if la\ .: ; Therefore. (i~"'~(~ is the Four­ ier transform of a certain function u'" in L /E). Then Parseval's relation implies that if ¢ r;:: Cc;(E n),

\'i \ I:,

I From this it follows easily that I.(A. - A)-1 -+ 0 as [AI -+ 00, A r;:: S(O, a), 1 1 since we know that IjlAjl-l converges. Therefore c = O. Q.E.D.

-

E

uD"'¢dx =

I

n

~./"'".

E

uWD"'¢(~de: n

~(~(i~O<¢(~de:

=I E

Part 1. PRELIMINARY RESULTS ON FUNDAMENTAL

SOLUTIONS

' = (i~O
e.

a)

14. Eigenvalue Problems for Elliptic Equations;

The Self-Adjoint Case

H.::;;e

Proof. If u € He(E), then

,

II _1_ < const + csc 8 I _ _ 1 1 j A.1 - A - IAI JLJomax(!A.I, IAI) 1

En

lation implies

10-4>\'::;;

SW,

n

I

/AI) sin 8.

is outside this angle for j 2 j 0' then for A r;::

I eo<\~(e:Wde:.

I

=

E •

1

(13.31)

231

Eigenvalue Problems; The Self-Adjoint Case

I

=

n

(-1)

E

=

(i~"'~W¢(~de:

HI

"

(-l)H

n

I ~~(~¢(~de: E

n

The results of the preceding section allow considerable sharpening in the case of operators T associated with elliptic differential opera­ tors. Before considering such cases, we need a preliminary discus­ sion of the concept of fundamental solution. Before beginning this discussion, we need the following characterization of He(E n)'

Thus, u has the weak derivative D"'u = u'" in L 2 (E n ).

LEMMA 14.1. Let u r;:: L/E n ) and let ube the Fourier transform of u. Then u r;:: He(E n ) if and only if the integral

fore, u € We(E n ) = He(E n ). Q.E.D. Let A(D) be a homogeneous elliptic operator of order stant coefficients; thus,

= (-l)H

I

u"'(x)¢(x)dx. E

n

lal

~

e.

There­

ehaving con­

232

A(D)=

~

Eigenvalue Problems; The Self-Adjoint Case

sec. 14

Elliptic Boundary Value Problems

233

e

Therefore, since > n it follows that (A(ig) - A)-l is in L 2 (E tI ) and even in L 1 (En)' Therefore, (A(ig) - A)-l is the Fourier transform of a function in L 2 (E n ), say (211)n/2F A(x). By the Fourier inversion formula,

a.,D"',

I"'I-e and for some positive Eo,

IA(i(1 = IA(g)\ L Eo\(\e

(14.6)

FA (x)

=

11<'(

I

(211)-n

E

e

for real (. Assume that > n and that A is a complex number such that the equation A(ig) = A has no solution for real (. Our aim is to solve the equation

n

e d(.

A(it) - A

Since (A(ig) - A)-l is integrable, it follows not only that FA(x) is in L 2 (E n ), but also that FA (x) is a continuous function on En' More­ over, FA(x) decreases rapidly as ~ 00. For, since

Ixl

(14.3)

(A(D) - A)u = f

~)"'FA(/:\ = D~FA (g),

(-IX

in case f (LZ(E n ). We shall indeed find a function u (He(E n ) such that (14.3) is satisfied in all of En' The technique is similar to that employed in Theorem 5.1, in which f and u were periodic. There we could use techniques of Fourier series. Since now f is not.periodic, we shall have to apply Fourier transform techniques. Suppose first that u (He(E n) is a solution. Since the Fourier trans­ form of D"'u is

A

D"'u(g) =

it follows from the Fourier inversion formula that for all a (14.7)

The justification of this formula comes from the fact that D~A(ig) ­ A)-l is square integrabel for every a. Since D~A(ig) - Afl is inte­ grable. the relation (14.7) implies that for all positive N and for some positive constant C N ,

~

(ig)"'u(g),

A);;(g)

=

IF A(x)1

fcg).

Therefore,

(14.4)

;;w

(-ix)"'F A(x) = (211)-n IE e!1C'(Dg(A(ig) - A)-ld(. n

it follows upon taking the Fourier transfonn of both sides of (14.3) that (A(ig) -

<,1

CN 1+

IxlN

, x (E •

N

By the definition of FA, (14.4) can be written =

.,.

~1 . fcg)

(14.8)

Since AUg) - A f, 0 for real (, it follows from the ellipticity of A that for all real (

v

;; = (211)n12FAl.

Now the rapid decrease of FA implies that the convolution FA *f has the Fourier transform

/"0.

(14.5)

~

_---=1_ < const IA(ig) - AI - (l + Itn

FA *f

e.

~ ~

= (211)n/ZFAf.

Therefore, (14.8) implies that u and FA *f have the same Fourier trans­

~~----=;:.;.-

J!.'"J:~~

234

sec. 14

EIIiptic Boundary Value Problems t

form, and must therefore coincide. That is

u(x) =

(14.9)

f

FA(x - y)f(y)dy. E

n

What we have shown is that if u ~ He<E) and if u satisfies (14.3), then u is given by the formula (14.9). Conversely, suppose that f ~ L 2 (E n) and that u is defined by (14.9). By retracing the above argument in the opposite direction, it follows that u ~ L 2 (E n ) and, that its Fourier transform satisfies (14.4). Therefore, the estimate (14.5) implies that (l + 1-gI)rl;(~1 ::; const IR~I, so that

f

(l + E

1~)2rl;(~12d~ ~

E

Thus, Lemma 14.1 implies that u :€

~

(AW) - A)u

= (A(i~

IR~12d~.

f

const

n

Eigenvalue Problems; The Self-Adjoint Case

bounded. Since FA(x) is continuous, the kernel FA(x ­ y) certainly has a trace on the diagonal of n x n, and this function is obviously the constant FA (0), since x = y on the diagonal. Recall that we have assumed that ACi~ - A f- 0 for real~. If A has the polar form

A=

/e iO, r = IAllIC > 0,

this condition is obviously equivalent to the condition that ACi~ f­ e;() for real ~; here we use again the homogeneity of A. If we set ~ = ry in the formula (14.6) with x = 0, then

(14.12)

FA(O)

=

n

Hr(E n ).

(2rr)-n

f

1 rndy En zl'ACiy) _ zl'e iO

= (2rr)-nr n-C

Then it follows that

=

f,

=

is in

u(x)

He<m

=

f

n

FA(x - y )f(y )dy,

(A(iy) _ eiO)-ldy. E

n

n

x~

n,

and satisfies

(A(D) - A)U = fin

f

This formula will be needed in some later computations. Note that, taking the integral over of the trace of FA(x - y), we obtain

IlullrE ::;constllfl!o.E' • n n

The function FA (x) will be called the fundamental solution for the operator A(D) - A. Now we shall investigate the formula (14.9) for the solution of (A(D) - A)U = f in the case in which f :€ L 2 (m for some bounded open set n. Regarding f as being extended to En by setting f = 0 outside n, then we see from the previous results that the function (14.11)

(A(iy) _ eiO)-ldy n

(2rr)-nIAI (n/Cl-l

so that u is a solution of (14.3). The formula (14.2) also implies that

(14.10)

f E

~~

- A)U

235

n.

The relation (14.11) represents an integral transformation of L 2(n) into Hr(m, having a Hilbert-Schmidt kernel FA(x - y), since n is

f

n

FA(O)dx = InIIAI(n/Cl-l(2rr)-n

f

1 dy

EnA(iy)-e;()

We shall obtain analogous expressions in Part 2 of this section. Two more preliminary results are needed before giving the main re­ sults of this section. Note first that the condition, A(i~ i- A for ~ real, actually implies that the values A(i~ for real ~ omit an entire angle in the complex plane. Thus, for some 0> 0 it follows that for real ~ A(i~ f- /l if larg /l- arg Al ::; O.

LEMMA 14.2. Let A(D) be a homogeneous elliptic operator of order r having constant coefficients, and let Eo be a positive constant such that lAW I L

Eol~IC, ~ real.

236

sec. 14

Elliptic Boundary Value Problems

Eigenvalue Problems; the Self-Adjoint Case

Let 5 be a positive number such that

= csc5

A(it) ~ elf)

s

\A\(k/f)-l[wll(A -A)ullo.o + C1\u\lf-I.O]'

1(1

s Ion 0

IACit) - AI L sin5 max (IA(it)\, IAI).

~IDccullo E ,

= n

11~llo E '

""

ll(it)cc~11

and ~ == Ion 0'. Let v = ~u. Then v€: Hf(E n ), and

rule

(A - A)v = A(~u) ~ A~u = ~Au

IIDOC(~u)\\o E .s: csc5 Eok/fIAI(k/f)-I[\I~(A- A)ull o E +IIBull o E ] , n ' n • n S csc5

The lemma then follows from the relation n

A(it) -A

•

E

n

l{l k

111>11 0 nsf,u

S csc5

E~k;fIA\(k;f)-III~lo E •

IAI

II(A -

A)1>ll o n, .u

for all 1> ~ C:;"(U), A ~ =:(0, a). Then A '(X, it) - e iO ~ 0 for all x ~ 0, ~ real.

Thus, (14.14) implies

n

IIDOC(~u)11 0,0'

°

1~lk

sin5 \A(it)\k/f\A\ I-(k/f)

IID"'ull o•E

s

LEMMA 14.3. Let A(x, D) be a differential operator of order f in 0, having continuous leading coefficients and bounded, meaSUr­ able lower order coefficients. Assume that for a certain and posi­ tive numbers C, a,

s csc5 E~k/fIA\ (k/f) -I.

(14.15)

IIDCC(~u)11 0,0'

Q.E.D. We can give a simple condition that A'(x, it) omit a certain complex number when x varies over 0 and ~ takes on all real values.

IA(it) - AI k/f\A(it) - A\I-(k/f)

.s:

=

•

la\ = k

.s:

- A)u 11 0 ,0 +C 111111 le-I ,0]'

O.E

IIDOCul1 0.0'

(it)cc

E~k/eIAI (k/f )-1[11 (A

n

= II(it)CC(A(it) - A)-I~lo Now (14.13) implies for

+ Bu - A~U,

where Bu involves derivatives of ~, and derivatives of u of order less than f. Thus, by (14.15) with lal = k and with u replaced by ~u,

Suppose first that u ~ Hf(E n ), and let f = (A(D) - A)u. Then f ~ L 2 (E n ),

and it fo llows from Parseval's relation and (14.4) that

(14.14)

n

(14.15) may be applied to the function v. Note that by Leibnitz's

Proof. By (13.31) we have for real ~ and for larg A - 001 < 5 (14.13)

E •

Now suppose only that u ~ He(U). Then let ~ ~ C~(U) satisfy

for 10 - 001 < 25 and ~ real. Let 0 ' ccO cE n • Then there exist positive constants w = w(E o' 5, f, n) and C = C(E o' 5, f, n, 0, 0 ' ) such that for all u :€ Hf(U), for 0 S k s f, and for larg A- 00 I < 5 lulk.o'

E~k/f\AI (k/e)-III(A - A)ull 0

237

n

Proof. Let rf; ~ C~(U) and let

~"r::.~

~':.~.'-.-.~:,-"""_~

238

sec. 14

Elliptic Boundary Value Problems ¢(x)

=

eit~.xt/J(x),

239

Eigenvalue Problems; The Self-Adjoint Case

t> O. Part 2. EIGENVALUE PROBLEMS FOR ELLIPTIC EQUATIONS

For ,\

t£(}O, Leibnitz's rule implies that as t

=

(A - ,\)¢

-->

00,

=

(A'(x, D)eit~.x - '\eit~.X) • t/J(x) + O(t£-I)

=

(A'(x, itt! - '\)elt~.xt/J(x) + O(t£-I)

=

t£(A'(x, it! - eiO)eitl;.xt/J(x) + OU£-I).

Thus,

\[¢I \ ~,n

=

In \t/J(x)\2dx II
<£

- 1,\1 2

Now we are ready for the fundamental result of this section. Be­ cause of the generality of the situations to which the theorem applies, the theorem requires quite a long statement. As in section 13, it will be assumed throughout the remainder of section 14 that is a finite union of disjoint open sets, each of which has the restricted cone property.

n

THEOREM 14.4. Hypothesis. T is a bounded linear transfor­ mation on L/m such that both the range of T and the range of T* are contained in H ,(U), where m' > n if n is odd, m' > n + 1 if n is even. The directi; e iO is a direction of minimal growth of the modi· fied resolvent of To There exists an open set no contained in n and an elliptic operator P(x, D) of order m' in no of the form P(x, D)

I

=

a",(x)D'"

\ocI-m' =

r;2 [t2£

I

t 2£ =

Letting t

I

-->

00,

c2 I

n

\A'(x, it! _ e iO \21t/J(x)\ 2dx + 0(t 2£-I)]

n

\A'(x, it! _

eioI2 \t/J(x)1 2 dx + OU- 1 ).

where a", ~ C°(Uo).

1° F or x ~ no' for all real ~, and for all complex ,\ ,J 0 such that arg ,\ = 0, P(x, it! ,J '\.

2°

we obtain

[c 2 I A '(x, it! - e iOl 2 -1]

n

1t/J(xWdx L O.

For any XO ~ no and any positive f, there exists a neighbor­ hood U of xc, U c no' and a constant C f ' such that if

P o(D)

=

c,;(m,

Since this holds for all t/J r;;: it foIl ows that it must also hold hold for all t/J ~ L/m. Therefore, since A'(x, it! is a continuous function of x,

C 2\A'(x, it! - e lO , 2

I IIIIII 'Ill,

II

'.1

so that for all x ~

1 ~ 0,

n and all real ~,

IA'(x, i~ - eiOI .2 C Q.E.D.

-

1.

then for all f

(14.16)

I

I,\I-m' ~

L

a",(xO)D"',

/m

l\PoTf - fll o• u ~

fllfllo.n + C l/ Tf ll m '_I,n

11P~T*f - fllo,u ~ where

PS is

f

fllfll o•n + Cf IIT*fll m '-l.n

the formal adjoint of Po'

240

Elliptic Boundary Value Problems

sec. 14

Conc1bsion. For A ~ Pm(T), TA is an integral operator with a Hilbert­ Schmidt kernel K A~ Hecn x n), where = m' - [n/2] - 1. The kernel K>.. has a trace K>..(x, x) ~ L 2 (n) on the diagonal of n x n. For >.. ~ E"[ -> 00,

e

[f

n

In K>..(x, x)dx

=

the following. For any positive f, there exists a neighborhood U' of xo, U c no and a constant C~, such that for all f ~ L 2 (n) and for all I>"1 sufficiently large, arg A =

e,

II (P 0

-

>..)T>..f - fll o.u ~ fllfll o.n + C;II TAfllm-l.n

II(P~ - >..)(T>..)*f - fll o• u ':;; fllfllo.n + C;II(T>..)*fllm_l.n·

cl>"l(n/m')-1 + 0(1)..1 (n/m')-l),

o

241

Proof. We first show that Hypothesis 2° implies that we also have

(14.16)'

!KA(x, x)l"dx]'I, = 0(1)''1 (n/m')-l),

Eigenvalue Problems,' The Self-Adjoint Case

Indeed, observe that

where c is a constant depending only on P, e, and no' Indeed, if

Prix) = (217)-n

IE nP(x,

(Po - >..)T>..f - f = PoT>! - (l - >..T)-lf = P oT(l - AT)-lf - (1- >"T)-lf.

dg

i() _ e i "

,

and if no is the. subset of no consisting of points whose distance

Given f> 0 let U be the neighborhood of Xo for which (14.16) holds. Then, applying this inequality to (1 - >"T)-lf, using the above rela­ tion, we obtain

I

from all o is greater than

c = lim

0.... 0

I

I

0, and if no c D8 cc no' then II(Po - >..)TAf - fll o• u .:;; fll(1 - AT)-lfll o.n + Cfll T>..fllm'-l.n

Pe(x)dx.

::;; flll- AT)-I\lollfllo.n + CfIITAfllm-l.n·

nl)

Remark: Some care was taken in the definition of c, since the fol­

lowing proof does not show that pix) is absolutely convergent over no' In reality, however, the function Pe(x) is bounded in no' This follows from a much stronger version of Theorem 14.4 (compare re­ mark at the end of Theorem 13.9) which yields pointwise asymptotic results. Thus, under the conditions of the theorem, one can show that K>..(x, y) is continuous and bounded in n x n for A ~ Pm(T). Moreover, if x ~ no' y ~ no' x I y, then as 1>"1 .... 00, >.. ~ E(e, 0), K>..(x, x) = pe(x)I>..\(n/m')-1 + oC\>"j(n/m')-I),

Using (12.5) we have for [>"1 sufficiently large, arg >..

II11 i.

.

1

, I

Iml

l

III ·!i II

while KA(x, y) = o(I>..\(n/m')-I).

e,

11(1- >"T)-lilo ~ 1 + \>"IIIT>..llo':;; 1 + K where K is some constant. This combined with the preceding in­ I equality yield the first part of (14.16)' (one replaces f by f' = f(l + choosing U' as the neighborhood which corresponds to f' in Hypoth­ esis 2"). The proof of the second inequality in (14.16)' is similar. I I , Let now no be fixed and let n' cc Let f >0. Then there exists a collection of congruent non-overlapping cubes Q~, ..., Q~, such that

Kr

no'

N

II

=

if &= U Q;) then n ' c&c I- I

,

no, and the estimates (14.16)' are valid

in QI' the cube concentric to Q: having twice the side length of Q;. Let xl be the center of Q;, and let b be the length of the side of Q;. Let arg >.. = By the results of Par t 1, there exists a fundamental solution FA(x) for the elliptic operator with constant coefficients, P(x i , D) - A. For arg >.. = IAI sufficiently large, the modified re­

e.

e,

242

Elliptic Boundary Value Problems

solvent T A exists. Suppose f ~ L2(O~); extend ito be zero outside Define an operator S~ on L2(O~) by the formula

0;.

SAi = T Af - F ~ *f. Now R(TA) cR(T) CHm,(n), and, as we have seen in Part 1, FJ...*f~ Hm,(E n ). Thus, if we restrict our attention to 0:, R(SJ...) CHm,(O;). By Lemma 13.4 and by the estimate (14.10). it follows that SJ... is a bounded linear transformation from L 2(O:) into Hm'(O~). The assumed relation between m' and n shows that the hypothesis of Theorem 13.10 is satisfied. so it follON:s that the Hilbert-Schmidt kernel KA(x. y) of T A exists. and that KA(x, x) exists as an L 2 func­ tion in n. The same remarks apply to the kernel GJ...(x, y) of the opera­ tor SJ..., by Theorem 13.9. It should be noted in this case that the ad­ joint of SA is given by the formula SA *f

=

T~f -

J

FA(Y - x)f(y)dy,

Q1

GA(x, y)

KA(x, y) - FJ...(x- y),

=

GJ...*(x, y)

=

K~(x, y) -

FA*(x - y) ,

where GJ...* and K~ are the Hilbert-Schmidt kernels for the operators (S;{)* and (TA)*, respectively. Let y be the constant of (13.13) which applies to a fixed cube of side 00' Then for b < 8 0 , we have by the corollary to Theorem 13.9 (14.18)

(b/8 0 )n/2

Eigenvalue Problems; The Self-Adjoint Case

[J IGJ...(x, xWdx]~ Q'

(14.19)

\SJ...flk S ITAflk + IFA*fl k •

For large

11..1.

IITAll oS

are

1..= e,

we have by hypothesis

KIA[-l.

Therefore. by (13.8) it follows that for such A

IITAllm's IITllm,(l + K). Combining this with (14.10) implies

IIS~JI!m'.Q. s

[[ITllm,(l + K) + const] IlfIIO,Q.' I

Thus, IISAll m' is bounded independently of implies

I

I,

,

I

, 1

-III

~

y[(b/oo)n(ISAI;:,l,m' + ISA*I;:,l,m')ISJ...lt Cn / m') + ISJ...l o]·

A.

Therefore, Lemma 13.3

ISAl k S const ISAI~-Ck/m')IISAlr~l,mu (14.20)

S const

ISA It

Ck / m ').

By Lemma 14.2, we obtain for f

~

L 2 (Q:).

IFA*flo,Q: s IAI-1[wllfllo,En + const IIFA*fllm"E) ~ const

IA[-lllfIIO,Q'.' I

where we have used (14.10). Thus. (14.19) implies

I

i.

243

We now need to estimate the semi-norms of SA appearing in this ex­ pression. First we shall obtain some rough estimates. By the triangle inequality and the definition of SJ.... for f ~ L2(O~) we have

I

and FA*(x) = FJ...(- x) is the fundamental solution for P*(x l , D) - X. Thus, R(SA*) CHm,(O;), and Theorem 13.9 can be applied. Therefore, corresponding to the operator definition of Sf... is the equa= tion for the kernels: (14.17)

sec. 14

[SAl os

ITAl o + const

11..\-1

244

sec. 14.

Elliptic Boundary Value Problems ::; const \AI- l • ~

Thus, by (14.20) we have for 0

Eigenvalue Problems; The Sell-Adjoint Case

By the rough estimate (14.21), this implies

ISAllk,Q~ ~ IAI(k/m')-1[2wlllfllo'Q~+ ClIAI-l/m'Illllo,Q;l.

k.:s; m' - 1

ISAlk.:s; const /AI(k/m' )-1. Therefore, we have for Therefore, for sufficiently large IAI it follows that (14.21)

245

IA\

sufficiently large, arg

A= e,

!Silk.:s; 3Wl!A!(k/m')-I.

I\SAll m'-1 ~ const IAI-l/m'.

The same argument shows also that for such A,

Now we shall obtain some sharp estimates, using the inequalities (14.16)'. We have for I ~L2(Q~)

I(SA)*lk.:s; 3WlIAI(k/m')-1.

II(P(x l , D) - A)TAI - Illo,QI

,-,

= II(P(x i , D)-A)(TAI-KA*I)llo.QI

The proof is now almost completed, except for some easy, but lengthy, computations. Substituting the estimates just obtained into (14.18), we find, using (14.17),

[J IKA(x, x) - FA
(Q/ Oo)n/2

Q'

i

= II(P(x

l

,

D) - A)SVllo'

QI ::; y[(b/o o)n3wl2\A!(n/m')-1 + 3Wl!A!-ll;

.:s; tWllo. Q. + c;IITAfllm'-I.11

or,

I

~ [H c;ITAI~/m'IITAllm',11]lllllo.Q~

x) -FA(O)!2dx]~

[J IKA(x, Q' I

t:

::; [l + ClIA\-l/m'l\llllo.Q'!

::; 6YWl(b/o o)n/2I A (n/m'l-

I

I

where we have used again Lemma 13.3; we are now using Cl as a generic constant. It follows that for sufficiently large IAI, arg A= (14.22)

\ I( ~Xl, D) - A)SJJII

°

•

Q

i

.:s; 2l\ lill 0

e,

•

1

Now we shall again apply the estimate in Lemma 14.2, taking 11 = QI' 11' = Q;, u = SAl. We obtain, using (14.22),

Q1

I

i

IAI,

after squaring,

x) - F,{
Next, summing over all the cubes (14.23)

ISVlk,Q~.:s; IAI(k/m')-1[w2llllllo,Q~ + ClIISAlllm'-I,Ql

+ 3YWl(b/oo)-n/2IAI-I.

Therefore, we obtain for sufficiently large

J ,IKA(x, Q"

1

JQ' IKA(x, x) i

Q;,

we obtain since b n N

F,{(OWdx::; (l2yw)2l2o-onl1_ U /

1

IQ:I,

=

Q'\ !A!(2n/m , )-2 I

246

sec. 14

Elliptic Boundary Value Problems

s c(21.\1 (2n /rn')-2.

Therefore, the Cauchy-Schwarz inequality implies

E

=

Therefore, the triangle inequality applied to (14.25) shows that

(Pk, i?f) - eiO'j-ldg

1.\!(n/rn')-1(21T)-n J n

lim sup II.\II-(nlm'l J K.\(x, x)dx - J 1.\\400 00

!.\\(n/m'l-lpe<x i ).

arg

Thus, (14.23) becomes, after multiplying by

CiOo-&1 '1,.

°0-&

1

=

IKA(x, x)ldx ~

I.\II-(n/m'l J

We shall use C as a generic constant for the remainder of the proof. Now we utilize the formula (14.12) for F (0): Fl(O)

247

Eigenvalue Problems; The Self-Adjoint Case

Os

A.=e

Pe(x)dxl

1.\1 2-(2n / m'),

s

CIOoI11,( +

1

ClO o _ &1 1,

+ IJ&Pe(x)dx - J&Pe(x)dxl

2

II JQ,.II.\!1-(n/m')K.\(x, x) - POk)1 dx S C(2.

(14.24)

I

+ i

0;

Now let Prix) be the step function on 00' equal to Pe(x ) on and equal to zero on 00 - uo; = 00 - &. Then (14.24) can be written

Now the integral J Pe(x)dx exists, and

0' o

(& = UQ;)

[J&I 1.\\ l-(n/m'lK.\(x,

x) - Pe(xWdx]'I,

O{--& !Pe(x)!dx,

J Pe(x)dx

&

is an approximating

.

Riemann sum for the integral J&Pe(x)dx, By choosing the cubes

s C(.

0;

smaller and smaller, the above inequality becomes

By the Cauchy-Schwarz inequality,

If

IAI1-Cn/rn')KA(x, x)dx -

&

J.

lim sup 1.\1 400 arg

Pe(x)dx!

&

I 1.\II-(n/m

I

lJ

00

A.=e

Pe(x)dxl ~

K A(x, x)dx: - J

Os

+

Since 0' was any set satisfying 0' ceo tive number, the last inequality implies

S 1&1'1,[J~ \A\I-(n/m'lKA(x, x) - Pe(xWdx]'I,

lim sup IAI400

(14.25) S CjOO\'!,f.

arg

Applying Theorem 13.10, it follows that K.\ ~ He(O x 0), and for

111 'I

=

11.\\ I-(n/m'l! 00

A.=e

[J !K.\(x, xWdx]'I,

°

s C.

!Pe(x)ldx,

K.\(x, x)dx - J

Os

a, and since ( was any posi­

Pe(x)dx\

s CIO o- 0al'l,

\.\1

s CIO o - 0 0 1'1,.

e,

!.\jl-(n/m')

J

O'-D' o

S J&II.\II-(n/m')KA(x, x) - Pe(x)!dx

large, arg .\

CIOo I'1,( + C10 0

Since IAII-(n/m'l J

KA(x, x)dx is independent of 0, and J

°0

Os

Pe(x)dx

-

0'1'1,

248

Elliptic Boundary Value Problems

sec. 14 00

is independent of A, it must be the case that the two limits

f b

lim

f

IAI~oo B

rg

(14.28)

lim

f

Os

(A + t) 2

dA

ba

+ af

= -

b+ t

b

foo ~_ dA b (A+t)2

00

(14.29) exist and are equal. Q.E.D. Before giving the proof of our main application of Theorem 14.4, we need a Tauberian theorem of Hardy and Littlewood, which we state without proof. For a proof see Karamata, Neuer Beweis und Verallgemeinerung der Tauberschen Siitze, welch die Laplacesche und Stieltjessche Transformation betreffen, Journal fur die reine und angewandte Mathematik 164 (1931): pp. 27-39. THEOREM 14.5. Let o{A) be a non decreasing function for A > 0, let 0 < a < 1, let P be a non-negative number, and suppose that as t t ~ 00

=

~ + at a- I foo

b+t

Therefore, (14.28) becomes (14.30)

r-'

00 Aa ba f dA=- +~ t a- I _ at ab+t b (A+t)2 sin 17a

(14.27)

foo do(A) = fb do(A~_ o(b) + foo ~ dA. o A+ t 0 A+t b + t b (A + t)2

!

bit

a-I

L - d/l. /l + 1

foo do(A) _ Pta-I = foo _1_ [o{A) _ P sin 17a AB]dA A+ t b (A + t)2 17a __1_ [o(b)- p sin 17a ba] + fb do(A) b+t 17a 0 ~

_ P SIn

ITa

17a

t a- I f 0

bit

a-I

/i-_ _ d/l.

/l + 1

.

Given (> 0, there exists a positive number b such that for A ;;;'·b 100A) _ P sin 17a Aa\ < (A a•

17a

Thus, we obtain 00 do(A) _ pta-II < ( foo ~ dA + (~ + fb do(A) o A+ t b (A + t)2 b+ t 0 A+ t

If

+

.

IP SIn 17a I t a - 1 f 17a

"1: 1

f o

o

Another integration by parts shows that III

1

Multiplying this relation by P sin 17a and combining with (14.27) . ld s 17a yle

.

The converse ("Abelian theorem") of this result is valid, even if we omit the assumption of monotonicity of o(A). Because its proof is elementary, we include it here. Thus, we assume the asymptotic be­ havior (14.26) of the function o{A). For every positive number b, we obtain by integration by parts the formula

d/l.

a-I

00

o(A) = P sin 17a Aa + O(A a). 17a

L..:.

blt/l+1

f ~ d/l=_17_. o /l + 1 sin 17a

foo do{A) = Pta- 1 + o(t a- 1). o A+t

(14.26)

Aa-I

dA.

A + t

Now a familiar formula concerning beta functions is

Pe(x)dx,

Then as A ~

00

Setting A = tIl in the last integral, we obtain

and

8~0

Aa

KA(x, x)dx,

00

!l.=e

249

Eigenvalue Problems; TheSel f-Adjoint Case

Therefore, using (14.30) again,

0

bit

a-I

_/l_ d/l. /l + 1

250

sec. 14

Elliptic Boundary Value Problems

lim sup t l -

a

/-+00

da(A) - IIo"" A+t

P t a-l\ -< f l'1m sup t l (->00

a

I b

00

\

a

_1\-

(A + t) 2

dA

17a

=f-~-

sin 7Ta Since f is arbitrary, the converse of Theorem 14.5 is proved. One further comment should be included. The Tauberian theorem is usually stated in the case that P > O. But, having the result for P > 0, it is trivial to derive the result for the case P = O. For sup­ pose o{A) is non-decreasing, and

(0

do{A) o A+ t

=

Eigenvalue Problems; The Self-Adjoint Case

251

n is odd, n' = n + 1 if n is even. In case m' .:s; n', suppose that there

exists an odd positive integer k such that k n '1m', the coefficients

of A are in C
Then the spectrum of (1' is discrete, and the eigenvalues of (f have finite multiplicity. Let IAJ I be the sequence of eigenvalues of (1' counted according to multiplicity. For A > Glet N+CA) be the number of non-negative eigenvalues AJ.:s; A, and let N_(A) be the number of negative eigenvalues A. ::<: - A. Then

>

I

NiCA) = ciAn I m' + oCAnl", I ) as A ->

"",

oCt a-I). where c± = (27T)-n

Then o{A) + Aa is non-decreas'ing, and the converse proposition just proved shows that

( ' d[o{A) + Aa] o A+ t

7Ta sin 7T8

t a-

l

+ o(t a -

l

).

Therefore, the Tauberian theorem shows directly that a(A) + Aa = Aa + O(A a ),

I

(Uj;(x)dx,

o

and (UiCX)

=

lIg:

0 <.± A'CX, if) < 11/.

Remark. Since (1 is self adjoint, its spectrum is real, by Theorem 12.7. Also, all powers (j'k of (f are again self-adjoint transformations on L 2 CO), a well-known fact about self-adjoint transformations.

or, a(A) = O(A a).

Proof. By Theorem 12·7, CA - (1')-1 exists for all non-real A, and

all non-real directions e iO are directions of minimal growth of the resol vent of (1'. Moreover, since DC(f) cH, Co), it follows that (A _ (ff 1 Therefore, the result of Theorem 14.5 is also demonstrated in the l maps L 2 CO) into Hm,Co). Therefore, CA is compact, by Rellich's case P = O. l' theorem. Therefore, the Riesz-Schauder theory of compact operators The spectrum of a linear transformation on a Hilbert space is dis­ implies that the spectrum of CA ~ (f)-1 consists only of eigenvalues crete if every complex number in the spectrum is an eigenvalue. of finite multiplicity, whose only possible limit point is the number THEOREM 14.6. Let A(x, D) be an elliptic operator of order 0, which is also in the spectrum of (A - (f)-I. By Theorem 12.4, m' in 0, having continuous leading coefficients and bounded, mea­ this implies that the spectrum of (f is itself discrete, and the eigen­ surable lower order coefficients. Let A be symmetric over C~(O) in values of (f have finite multiplicity. the sense that for all ¢, l/J (C~(O), (A¢, l/J)o,O = (¢, Al/J)o,n. Therefore, there exists a real number Ao such that 1. 0 (p(Cf). By con­ (This holds, in particular, if the coefficients of A are sufficiently sidering (1' - Ao instead of (1', we may now assume that T = (1'-1 exists smooth, and A coincides with its formal adjoint.) Suppose that there as a bounded linear transformation on L 2 (n). By Theorem 12.3, exists an unbounded self-adjoint transformation (f on L 2 (fl), such (A _(f)-1 = - T A, so it follows that all non-real directions are direc­ that C~(O) C D(a) C Hm' (0) and (fu = Au for u ~ D(Cf). Let n' = n if

ch-

252

EIIiptic Boundary Value Problems

sec. 14

tions of minimal growth of the modified resolvent of T. This also fol­

lows immediately, as noted above, just from the fact that T is self­

adjoint. Note that R(T) = D(Cf) cHm,(O).

Let us now examine carefully the case in which m' ~ n'. Then, if

k is the integer guaranteed by the hypothesis, Tk = (Cf k)-t, so that

• R(T k) = D(CfIt) C Hkm ,(0). By the smoothness assumptions on the co­ efficients of A, the operator A k exists as a differential operator, and the coefficients of A k are continuous and bounded in n. Moreover, the principal part of A k is just (Ak)~X, ~

= (AI(X,

(14.31)

A kTkf

=

fll o,n s IIBTkfl\ o,n

(14.32)

S CI/Tkfllkm'_t,n.

Now let Xo r;:: n, let U be a small neighborhood of xc, such that the co­

efficients of P vary by less than f in U. If Po is the operator P with

coefficients evaluated and frozen at xc, then (14.32) implies

II PoTkf - fl IO,u s

~I( Po - P)Tkfll o,u + C\ ITkfllkm I_I,U

.:s; YfllTkllkm,llfllo,n + ClITkfllkm'_I,n,

where Y depends only on m and n. But this is precisely the first estimate of (14.16), with m' replaced by km', the order of p. Likewise, the estimate for the adjoints of the operators can be derived. Next, if e 1e is not real, then e 1e is a direction of minimal growth of the resolvent of the self adjoint operator Cf, as we have remarked. Therefore, for all u r;:: D(Cf) and for arg ,\ =

e,

Ilull o,un .:s; const 1,\/

i-

WCf -

'\)ull 0 ,un.

In particular, since D(Cf) :::> C;;"(O), we have for all ¢ r;:: C~(O)

11¢llon~const II(Cf-,\)¢llon.

,

,

Now let no = n, and let the operator P of Theorem 14.4 be the princi­ pal part of Ak. Thus, Ak = P+ B, where B is an operator of order

less than km' having bounded coefficients. By (14.31),

!I PTkf -

(14.33)

I

f.

253

.:s; YfIlTkfllkm',n + CIITkfllkm'_I,n

~)k.

Therefore, A k is an elliptic operator of order km'. Our aim is to apply Theorem 14.4 to the operator Tk. We shall set k = 1 in case m' > n'. Since in any case R(Tk) C Hkm ,m) and km' > n', and since Tk is self­ adjoint, it remains to verify the hypotheses 1°, 2° of that theorem, as we now do. Since T k = (Cfk)-t, it follows that for any f r;:: L 2 (0),

Eigenvalue Problems; The Self-Adjoint Case

I

I I

1,\1

'

Therefore, Lemma 14.3 implies A'(x, i~ - e ie Ie- O. This also implies that (A '(x, i~)k - e ie Ie- 0 for real ~, so that condition 1° of Theorem 14.4 is verified. Note that in our case it even follows that AI(X, i~ is real for all real ~. Thus, we can apply Theorem 14.4 in the case of the operator Tk. In particular, we wish to apply the asymptotic formula for the kernel K,\ of the operator (T k ),\; we have from Theorem 14.4 (14.34)

I,

f

K,\(x, x)dx = cl,\1 (n/km')-l + 0(1,\1 (n/km')-l).

n

wbere c is given in accordance with the formulas given in the state­ ment of the theorem. Now since the characteristic values of Tare just the eigenvalues I,\J I of Cf, the characteristic values of Tit, counted according to multiplicity, are 1,\:1. By Theorem 13.10, we then have

f

n

K,\(x, x) dx = I

_I_

J ,\k_,\

J

254

sec. 14

Elliptic Boundary Value Problems

2. __1__

Comparing this with the asymptotic formula above, we have for arg A == e fixed, e Ie not real,

2 _1_ J

== CIA!(n/km')-l + o(IAI (n/km')-l)

Ak _ A

as 1..\1

Eigenvalue Problems; The Self-Adjoint Case =

JA~k+t

255

f"" dN(A 1/ 2k )

A+t

0

'

J

so that (14.39) becomes

-> "".

J

f"" dN(A1/ 2k )= c t(q/2k)-1 + 0(t(q/2k)-I).

In particular, let us take for simplicity t> 0, we have

2. __1_

(14.35)

J

Ak

== ct(n/km')-l

e ==!!.2

Thus, letting A == it,

2

A+ t

o

,-, Applying the Tauberian theorem of Hardy and Littlewood (Theorem 14.5), it follows that

+ o(t(n/km')-l).

it

_

J

(14.40) N(>.,1/2k)

=

c

2

sin(17q/2k) Aq / 2k + 0(Aq/2k),

17q/2k

Using the formulas derived in Theorem 14. 4, or, replacing A by A2 k, (14.36)

c ==

f

p(x)dx,

n

\

I

I,

where (14.37)

p(x) ==(277)-n fEn [A '(x,

dg

i~]k _

i

.

The problem is now to use this information to find expressions for N+(A) and N_(A). Let q == n/m'; 0 < q < k. Then, if c 1 and c 2 are the real and imaginary parts of c, it follows from (14.35) that

I"

(14.41)

N(A)

=

We now must separate the parts N+(A) and N_(A) out of their sum N(A). This is just the reason we had to assume k is odd: the sign of A!< is the same as that of A.J in this case. Now J

2

At J

=

A2k + t2 j

JA2k+t2

2 Af + 2 A.>o A2k + t2 A.
j

J

AJk A 2k +t 2 J

\

2. _~ = c t(q/k)-l + O(t(q/k)-l),

(14.38)

2kc sin(77q/2k) Aq + O(Aq). 2 17q

=

1

f"" _yX

o A+t 2

dN.(A 1/ 2k )-f"" 0

_..JA_

dN_(A1/2k)

A+t2

J

!

= c2t(q/k)-1 + Q,t(q/k)-l).

t

= f""

1 A2k + t2

o

J

\.,

Divide the latter relation by t, and replace t 2 by t, to obtain

2 __ 1_ J A2k + t

(14.39)

=

c2t(q/2k)-1 + 0(t(q/2k)-1).

I

J

" II

=

N+(A) + N_(A), the number of AJ such that

1\1 ~ A.

Then

A+ t

2

= C 1 t(q/ k)-l

(14.42)

dN(A1 / 2k )-2 f"" 0

~

dN_(A 1/ 2k )

A+ t2

+ oCt (q/k )-1),

by (14.38). Let

_ N(A) =

Let N(A)

~

f

A

o

y'p.dN(ll l /2 k).

, I

',i'

j

II

Then, by an integration by parts and the asymptotic formula (14.40),

~~

~~~'.----.:...-,"--

256

--~~~~~~~~;o-~~~~~~J~!~~~~~~:_~~~~~~~~~

257

Eigenvalue Problems; The Self-Adjoint Case

Elliptic Boundary Value Problems

Therefore, Theorem 14.5 implies

~

N(A) = VAN(A1/ 2k )_!h

I

A

/l-'I,.N(/l1/2k)d/l

. (17q

~

o

N (A)=!h[c tan(17q/2k)-c] -

. ( q/2) ---'!- + 1=2kc S1n17k [A 2k 2 2 17q =2kc sin(17q/2k)

2

17q

-!hI

A

1

q

q

2

17)

S1n\2k +2"

J!l

1

1)

-S.+1-

,,41­

\2k 2 +0(A2k 2) 1\

"\2k + 2

,

/l-z+fi"d/l]+o(>/'1C+2)

0

1

= Y2[c2 sin (17q/2k) - c 1 cos (17q/2k)]

I\t d(/lq/2k)+0(A i1 +t )

Iq

.!!+t

1)A2k

17 ' 2k +"2

0

~+1

+ o(A'11i: q

But from (14.43) it then follows that

1

q , An + z Z = c sin(17q/2k) ~+ o(A2k+ ) 2 -17 2k + 2 .

I

N _(A ~k) = const +

I A /l- y,­ 2dN_(/l)

o

= !h[c2 sin (17q/2k) - c 1 cos (17q/2k)]

Thus, the converse of the Tauberian theorem implies

2).

q1

1)'

17(2k +2'

(" ~ dN(A1/2k)",,(" dN(A) o A+ t2 0 A + t2

.I

_- c sln. (/2k) 17q t sin(li +-i)

q

1

2(fIcZ)

A

I

q

/l-~d/lfi

+1­ 2

+ 0(Aq/2k)

o

+ 0(t 2 (-rJ(q .1­2»)

2

= C

2

= [c2 sin (17q!2k) - c 1 cos (17q/2k)]..!- Aq/2k + 0(Aq/2k). 17q

tan(17q/2k)t(q/k)-l + o(t(q/k)-l).

Therefore, (14.42) implies

(14.44)

2/"" ~ dN_(A1/ 2k )=[C tan (17q/2k) _ c,]t(q/k)-l + O(t(q/k)-l). 2 o A+ t 2 In order to apply the Tauberian theorem, let (14.43)

A N_(A) = I V/ldN_(/l1/2k). o

The previous asymptotic formula then shows that, after replacing t 2 by t,

41

~ C 1]t 21-t)

+ oCt 2l-

t).

N_(A) = [c2 sin (17q/2k) -c 1 cos (17q/2k)]..!- Aq +o(Aq). 17q

Since N = N + + N_, it follows from (14.44) and (14.41) that (14.45)

~

/"" _1_ dN_(A) = !h[c2 tan(17q/2k) o A+ t

Replacing A by A2k, this relation, becomes

N+(A) = [c2 sin (17q/2k) + c 1 cos (17q/2k)]..!- Aq + o(Aq). 17q

Since q = n/m', the theorem will be proved if we now compute c 1 + ic 2 = c. This we proceed to do. As we have remarked, A'(x, i~ is real for real~. Let x be a fixed point of n and let p(~ = (A '(x, i~)k. Then (14.37) can be written in terms of its real and imaginary parts as

258

sec. 14

ElIiptic Boundary Value Problems (277)n p(X)

=

dt

Eigenvalue Problems; The Self-Adjoint Case

p(x) = igp(x)

f E p(g)- i n

= =

f En

peg) p(g)2 + 1

dt + i f

1 En

km 'v(l)

77q 1 . i(277)-n2w±(x) 2k sin (77ql2k)

dt.

p(g)2 + 1

Therefore, it follows from (14.36) that

For 0 < t < "" let VCt) be the region It: Ip(g)1 < tl, and let v{t) be the measure of Vet): v(t) = lV(t)\. Then the homogeneity of A' shows that ~r(t} = t 1 /km 'V(l), so that vet) = t n /

c

1

+

iC

2

=

i(277)-n 77q _ _l __ k sm (77q/2k)

_ 77q

k

= t q/ k v(l).

1 sin (77q/2k)

(277)ng p(X)

=

f"" _1_ dv(t)

N ±(A)

o t 2 + 1 q

= - v(l) f k

"" t(q/k)-l

-t2 +

0

wi(x)dx

n

IC±.

2"

k

0

s + 1

Case 2. m' is even. In this case peg) = (A '(x, ig))k is even, and the ellipticity of A implies peg) does not change sign. Let z be the common sign of p(e> and A'(x, ig). Then

ds

77 , !L v( 1) ----:;rq

(14.46) =

= c±Aq + o(Aq),

so that the theorem is proved in case m' is odd.

dt

1

= q v(l) 1 f"" s(q/2k)-1

(277)-n:i{p(x)

=

sin 2k

2k

Case 1. m' is odd. In this case peg) is odd, so that :i{p(x)

f En

=

by (14.29) (we have substituted s = t 2 ). It will be convenient to treat two cases separately.

z

peg) dg

p(g)2 + 1

f "" - to

=

t2 +

lIt:

=

z 9. v(1)

IpW/ < 1/1

~

=

211t:

=

211t:

=

2w+(x)

- z 2k

0 < peg) < 1/1 0 < A'(x, ig) < 1/1 =

=

=

t2 +

dt

1

77 (77 q 77) sm 2k + 2

v(l) -.

z ~ v(l) _ _ 77 2k cos (77q/2k)

as in (14.46). Also, v(l)

t / k f "" 0

2w_(x).

Thus, (14.46) implies

dv(t)

1 q

O. Also

k =

f

Therefore, c 1 = 0 and sin (77q/2k)c2 = (77q/k)ci' Hence, (14.44) and (14.45) imply

Therefore,

v(l)

259

Ilg: Ip(g)/ < 111

260

Elliptic Boundary Value Problems

=

Ilf.

0 < zp(g) <

111

=

II';:

0 < ZA'(X, ig) < 111

A '(x, i';2) < 0, then there is a curve connecting ';1 and ';2 which does not pass through'; = O. Since A'(X, ig) is continuous, it must happen that A'(X, ig) = 0 for some'; on the curve, a contradiction of the ellip­ ticity of A. Likewise, in case n = 1 and m' is even, A '(", ig) has constant sign for each fixed x. (Note that our theory certainly is valid in the case n = 1, that is for ordinary differential equations.) Therefore, in either of these cases it follows that if is connected, either w+(x) or w_(x) vanishes identically. Therefore, either the positive or negative eigenvalues predominate in number. This is not the case for the exceptional situation of an ordinary differential equation of odd order. For example, the eigenvalues of the problem

= w z (x);

also,

n

W_z(x)

= II';: - 1 < zA'(x, ig) < 011 = O.

Thus. (14.36) implies c1 +

261

Non-Self.Adjoint Eigenvalue Problems

sec. 15

iC 2

r

= (217)-n[Z !L

2k cos 17

q/

+ i -.!L 2k

2k

)

Inw z (x)dx

17 s-1·n-(-17q/~2-k)

Hence, cos (17q/2k)c1 = z(17q/2k)c z and sin (17q/2k)c2 = (17q/2k)cz' so that (14.44) and (14.45) imply =

('n - 'nZ)czA q + O(A q),

N +(A)

=

('n + iflZ)czA q + O(A q).

As c_ z = 0, we find for

Z

N _(A)

=

C_A q + O(A q),

N +(A)

=

C+A q + O(A q),

du i dx

In w)x)dxJ

=z-.!L 17 c +i-.!L 17 c. 2k cos (17q/2k) z 2k sin (17q/2k) z

N_(A)

!

=+

u(O)

=

=

AU,

U(17),

are all the even integers. This brings up another question. Even in the cases n 2: 2 or m ' even, it can actually happen that there are infinitely many eigenvalues of each sign. For second-order problems this indeed cannot happen, but there are examples involving the biharmonic operator in which there are infinitely many eigenvalues of each sign.

15. Non-Self··Adjoint Eigenvalue Problems

n

We still shall assume to be a finite union of disjoint bounded open sets. each of which has the restricted cone property. THEOREM 15.1. Let A(x, D) =

operator or order 2m in n, having real leading coefficients; let A be normalized so that A'(X, ig) > 0 for x€: n, .; real, .; ~ O. Let a", €: C 1"'1 and let A * be the formal adjoint of A. Let l1 be a closed, densely defined operator with domain in L 2(m, such that

"em,

and likewise for Z = -. Q.E.D. Some comments are in order concerning the theorem just proved. In case n 2: 2, A '(x, ig) has constant sign for each fixed x. The proof of this is like that of Theorem 4.1, but simpler. If A'(x, i ';1) > 0 and

2 a",(x)D'" be an elliptic i"'\S2m

1°

D(l1) U D(l1*) CH 2m (m;

262

sec. 15

Elliptic Boundary Value Problems 2°

for u ~ D(a), au = Au;

for u ~ D(a*). a*u = A*u;

3°

all directions except the positive real axis are directions of minimal growth of the resolvent of a,.

4°

in case 2m < n + I, there exists an integer k such that k;> (n + 1)/2m, the coefficients of A are C(k-1)2m*(n), and D( (fk) LJ D«a*)kk:H 2mk(n).

Then the spectrum of a is discrete and the eigenvalues of a have finite multiplicity. There are only a finite number of eigenvalues outside the angle larg Al < (, for any positive (. If {A I is the sequence J of eigenvalues of a, counted according to multiplicity, and if for A> 0, N(A) is the number of A. such that :RA. < A, then J J ­ N(A) = c oAn / 2m + 0(An/2m), where Co = (217)-n

f I{~: n

a-

a

a-I.

11 ["

111 1

,I!III

il!

< 11[dx.

Proof. The proof is quite similar to that of Theorem 14.6 except that the condition of self adjointness is replaced by other restrictions. Let A o ~ pea). Then R«A o - a)-9= D(a) cH 2m(n). so that Rellich's theorem implies that (A o - a)-l is a compact transformation of L 2 (n) into L 2 (n). Therefore. it follows that (A o - a)-l has a discrete spectrum. with eigenvalues of finite multiplicity. Hence. has the same properties. Note that this property of the spectrum of requires only that the resolvent set of be non empty. By considering the operator Ao instead of itself. it is seen that we may assume that ~ pea). and we shall thus consider the compact operator T = The assertion on the location of the eigenvalues follows immediately from Theorem 12.2. since. for any ( ) 0, there exists a number A( such that the set {A: \AI > \ . larg AI > d is contained in the resolvent set of Note that the conditions on m, n, and k are no different than those made in the statement of Theorem 14.6, concerning m', n', and k,

°

III

A'(x, ig)

a.

a

a a

263

Non-Self-Adjoint Eigenvalue Problems

since now m' = 2m is even. As before. if 2m :2 n + 1. set k = 1. We need to verify the hypothesis of Theorem 14.4. applied to the operator Tk. Note that our assumptions 1° and 4° imply R(Tk) = D(a k ) CH 2mk(n). and R«T k )*) = R«T*)k) = R«a*)-k) = D«a*)k) c H2mk(n). and 2mk > n if n is odd, 2mk > n + 1 if n is even. Also, our assumption 3°. together with the relation (Tk)A = - (A -

a )-l, k

implies that except for the positive real axis, all directions are directions of minimal growth of the modified resolvent of Tk. We take = and for the elliptic operator P(x, D) we take that opera~ tor whose characteristic polynomial is (A'(x, if»)k. Our assumption that A '(x, ig) is positive shows that 1° of Theorem 14.4 is verified trivially. Finally, 2° of Theorem 14.4 is checked in exactly the same manner as in the proof of Theorem 14.6. Therefore. we may now apply the conclusion of Theorem 14.4 to our case. Combining the estimate of Theorem 14.4 for the trace of the opera­ tor (Tk)A with Theorem 13.10, we obtain just as in the proof of Theorem 14.6 the asymptotic relation for fixed arg A.

no n,

~. __1_ = clA\ (n/k2m)-1 + o(IA\ (n/k2~)-1).

J

Ak _ A

J

for IAI -> 00, arg A ~ 0. Now let q_=rr/2m and let A = tain for t -> 00 . /

(15.1)

-

t, t> 0. to ob­

~ _1_ = ct(q/k)-l + O(t(q/k)-l). J

A~ + t J

Now let AJ = IlJ + ill J be the Cartesian representation of the complex The result that the eigenvalues are eventually located in number A.• J IlJ -> 0. There­ any angle about the real axis implies that Il J -> 00 and O fore. it follows that A~ = 1l~(1 + iV/IlJ)k = Il; + Ilr (v JIIIlJ)' and so if IV/IlJ\ < (for j> jOt then

v/

I ~

_1_ _ ~

J>J o Ak + t J

_1_ I ~ ~

J>J o p,k + t j

IIlJk _AklJ

I

J7J O (Ak J

1

t)(lI k + t) rJ

264

Non-Self-Adjoint Eigenvalue Problems

Elliptic Boundary Value Problems v(t) $;

yE I

yEI j

lIe:

p(~

< til,

/l:

i>J o

$;

=

265

A~

---­ (Af + t)(/l: +

t)

then we obtain, as in the proof of Theorem 14.6, p(x) = (217)-n

1 +t .

J ~ En pW + 1

J

From this it is a consequence of (15.1) that as t

I _1_ = ct(q/k)-l + J

-->

=

00

(2)-n

Joo

_1_ dv(t). o t+1

O(t(q/k)-l).

fl.: + t v(t)

= t n / 2mkv(l) = tq/kv(l),

By definition of N(A), this may be written this becomes by (14.29)

foo

_1_ dN(fl.l/k)

o

fI. + t

=

+ o(t(q/k)-l).

ct(q/k)-l

p(x) = (217)-n

Now we may apply the Tauberian theorem, Theorem 14.5. The re­ sult is that =

N(/ll/k)

=

c sin (17q/k)

/lq/k + O(/lq/k).

(217)-n

~v(I)

Joo

k

0

t(q/k)-l

t+1

dt

17q v(l).

k sin (17q/i)·

17q/k

Therefore, Replacing fl.l/ k by A, this becomes (15.2)

'N(A)

= ck sin

(17q/k)Aq +

O(Aq), A ...

17q

F

00.

ck sin (17q/k) = (217)-n v(l)dx

17q /11

=

Thus, all that is left is the computation of c. According to Theorem 14.4, with = 17,

< Illdx.

Q.E.D.

11

where

If we let p(~

A'(x, i~

When this expression is compared with (15.2), the theorem is proved.

J p(x)dx,

p(x) = (217)-n

J lIe: 11

e

c =

(217)-n

de + 1 JEn (A'(X,i~)k

= (A'(x, i~)k,

and let

.

Remark. To obtain results similar to those of Theorem 14.6, we need only aSsume in Theorem 15.1 that all directions except the positive real axis and negative real axis are directions of minimal growth of the resolvent of 6. Then we get asymptotic expressions for the num­ ber of eigenvalues AJ such that 0 < ~AJ < A (or such that - A < ~AJ < 0).

266

sec. 15

ElIiptic Boundary Value Problems

Non-Sell-Adjoint Eigenvalue Problems

First, suppose the transformation (10 of Theorem 15.2 is self~ad­ joint. Theorem 12.7 implies that for non-real A,

The question now arises concerning the hypothesis 3° of Theorem 15.1. Note that Lemma 14.3 implies that a direction e/() is of minimal growth of the resolvent of (1 only if A'(x, it> omits the value e/(). In case (1 is self adjoint, then the remark made above allows the result of the theorem. Of course, this is not important, since we already had obtained Theorem 14.6 for the self-adjoint case. What is important, however. is that Theorem 15.1 applies to perturbations of self-adjoint operators. This concept shall now be discussed. First, a preliminary result is needed.

Ilullo.n~ IgAI- I II(A-(1o)ull o,n, u~D«(1o); or, equivalently, (15.4) I''':;

II(A - (10)-111 0 ~ IgAI- I •

Therefore, for u ~ D«(1o) it follows from (15.3) that

THEOREM 15.2. Let (10 be a closed, densely defined linear operator in the space L 2 (n), such that D«(1o) CHm,(n). Then there exists a positive number C such that for all u ~ D«(1o)

Ilullm',n ~ C[II(1oull o,n + Ilullo,n]

~C[II(A-(10)ullo,n+[Alllullo,n+ Ilullo,n]

Ilullm',n ~ C(II(1oull o,n + Ilullo,n)'

(15.3)

~ C[II(A - (1o)ull o,n + qAI + l)jgAI- I II(>' - (1o)ull o,n]

Proof. Let G CL 2 (n) x L 2 (n) be the graph of (19; that is, G

=

l(u, (1ou): u ~ D«(1o)l.

=

Since (10 is closed, G is a closed subspace of L/n) x L/n). Con­ sider the mapping of G into Hm.(n) given by (u, (1ou) ~ u. This is a closed linear transformation of the Hilbert space G into the Hilbert space Hm,(n). For if

I

m

,(n),

then certainly uJ ~ v in L 2 (n), so that v = u. By the closed graph theorem it follows that the mapping (u, (1ou) ~ u is a continuous map­ ping of G into Hm' (n). But this is precisely the statement of the theorem. Q.E.D. This theorem is of considerable interest in other ar eas than the one at hand, and we shall have more to say about it later. For the time being, however, we wish to apply this result to perturbation of self­ adjoint operators.

III i

C[l + LAI + 1 ] II(A - (1o)ull o·n. IgAl ~

Therefore, II(A - (10)-11 k';I-~Cl + 21 AI

/j IgAI

(u j ' (1ou j ) ~ (u, (1ou) in L 2 (n) X L 2 (n), u.~vinH

267

By the convexity result of Lemma 13.3, this estimate together with (15.4) implies r,'

(15.5)

II(A-(1

)-111, m -

o

m

I

<'C (1+IAI)I-(1/ ') ~ I -"'-'---'--IgAl

with a different constant C I ' Now suppose that (1 = (10 + $, where (10 is a self-adjoint transfor­ mation in L 2 (n), with D«(1o) cHm,(n), and $ is an operator of lower order. Precisely, assume D(~) ~ D«(1o) and that ~ satisfies an estimate

268

Elliptic Boundary Value Problems

(15.6)

sec. 15

II$ullo,o ~ C z llull m'-I,o,

Non-Self-Adjoint Eigenvalue Problems

(,\ - (1)-1

This inequality allows us to extend $, if necessary, so that D($) :J

Hm '-1 (0). For'\ not real,

= (,\ -

(,\ - (10)-1(,\ - (1)

=

c:;

Z

----'------'':7----­

Ig,\1

2C z ll('\

I ~

-

(1o)-lll m'_1 ~ 1,

II (,\ -

(1)- III 0 ~ 211,\ - (10)-111 0

,:"

.

< 2

-~

That is.-if­ (

I

(15}O) = (,\ -

(10)-111 0[1 + 2C z ll('\ - (1o)-lll m'_I]'

so that we obtain

If this quantity is less than, say, 1/2, then 1 - (,\ - (10)-1$ is an in­

vertible mapping on H m '-I(O), and its inverse has norm less than 2.

Therefore, since ,\ - (1

11('\ -

Since the expression (15.8) is less than 1/2, (15.5) implies

C C (1 + 1,\!)I-(l/m'> 1

(10 )-111 0[1 + II $(,\ - (1)-111 0]

c:; 11('\ - (10) -111 0[1 + Czll('\ - (1)-lll m'_I]

1 - (,\ - (10)-1$;

this equation must be interpreted as being valid on D«(1o) = D«(1). Now $ maps H m'-I(O) into Lz(O), and (,\ - (1or 1 maps Lz(O) into D«(1o) CH m'_I(O)' Therefore, (,\ - (1or 1$ is a mapping of Hm '-1 (0) into Hm'-I (0); the estimates (15.6) and (15.5) imply that the norm of (,\ ~ (10)-1$, considered as a mapping on H m'-I(O)' is bounded by

(15.8)

II(,\ -

(10 - $,

= ,\ -

and, multiplying by (,\ - (10)-1 gives

(15.7)

(10)-1 + (,\ - (10)-1$(,\ _ (1)-1.

Hence, by (15.6) and (15.9)

II (,\ - (1)-11 \0 ~ ,\ - (1

269

(10)[1 - (,\ - (10)-1$]

Ig,\1 > 2C 1C Z (l + \'\I)I-C 1 / m '>,

then ,\ ( p«(1) and we have the estimate

on D«(1), then there exists

(15.11)

(,\ _ (1)-1 = [1 - (,\ - (10)-1$]-1(,\ - (10)-1.

1-' Moreover, the estimate of 2 for the norm of [1 - (,\ - (10)-1$]-1, as an

operator on Hm '-1 (0), shows

(15.9) 1'-1

1\(,\ - (1)-lll m'_1 ~ 211('\ - (1o)-lll m '_1

Multiplying (15.7) by (,\ - (1)-1 yields the relation

I

I-

11(,\-(1)-111 0 ~2/lg'\l.

One can easily see that there exists a constant K such that if

(15.12)

Ig,\l:::: K max

(1:R,\1 1-(l/m'>,

1),

then (15.10) is satisfied. A pictorial description of the region de­ scribed by (15.12) is given in the figure; we have shown on the first quadrant.

270

EIIiptic Boundary Value Problems

sec. 15

Ilull,;,',o ~ C[IIAull~,o

~>..

271

Non--SeU-Adjoint Eigenvalue Problems + \Iull~,o),

u ~ C~(O).

Thus, the quadratic form IIAull~ 0 = f

,

0

K

I

lui':;;,m'

1{3~m'

I"

,

a.>..

1

2

Aollul\~,o,

u ~ D(C(o),

then the argument given above allows one to show that al so the nega­ tive real axis is a direction of minimal growth of the resolvent of Cl. Cf. also Theorem 12.8. We now wish to examine the abstract Theorem 15.2 and consider its consequences.

ilill i,

THEOREM 15.3. Let A(x, D) be a differential operator of order m' on 0 having continuous leading coefficients and measurable, l~ caIly bounded lower order coefficients. Let Cl be a closed, densely defined linear operator in the space L 2 (0), such that C~(O) cD(Cl) c Hm ,(0), and (1u = Au for u ~ D(Cl). Then A is eIIiptic. Proof. By Theorem 15.2, there exists a positive C such that

a (x)8{3(x)D U u(x) • DfJu(x)dx

0

a

t,

2

la/-m' IfJ -m'

aa(X)8fJ(X)(ft{3

> 0,

or,

I

ii::

Remark. I~ also (10 satisfies an estimate

J

satisfies G~rding's inequality. Therefore, the necessity of G~rding's inequality, Theorem 7.12, implies that for real non-zero

Obviously, our results imply that any non-real direction is a direction of minimal growth of the resolvent of Cl. We have actually done much better, becuase our estimate (15.11) holds in a region of the A plane which includes all rays E(e, a) with e ie not real, and a sufficiently large, depending on e. And since the regioo (15.12) is in p(C(), it fol­ lows that all the sufficiently large eigenvalues of Cllie in the • para­ bolic" type region. This is of course a much more restricted behavior of the eigenvalues than the conclusion of Theorem 15.1 can give.

(Cl ou, u)o,O

I ! a (x)Dau(xWdx Ia 1<_m ' a

a

2 aa(x)t / 2 > O.

"112 I-

m'

That is,::4.'(x, e1 f. O. Q.E.D. A consequence of Theorem 15.3 is that if A(x, D) is not elliptic, and w~!wish to close the operator defined by applying A to functions in C(~(O), then the extended operator (1 will have a domain which includes .f<1nctions not in Hm ,(0). For u ~ D«(1), the function Au still exists weakly, in the sense of Definition 2.2, but now Au can no longer be computed by taking the individual strong derivatives of u and forming the appropriate linear combination of them. In the discussion above of operators of the form (1 = (10 + ~, where (10 is self-adjoint and ~ is of lower order, we obtained estimates for A - (1 just having an estimate (15.3) involving (10' Of course, we also had to use the estimate II (A - (10)-111 0 ~ 1!L\1-1, valid for arbitrary self adjoint (10' We shall now show a general procedure for obtaining estimates containing a parameter A, using only estimates not involv­ ing parameters. These latter estimates will be for an operator in a space of dimension one higher.

2n

sec. 15

Elliptic Boundary Value Problems

.::; C11'elttAu - r2meIO'elttu\lo,r

(15.14)

THEOREM 15.4. Let A(x, D) be an elliptic operator of order 2m in n, havinlJ leading coefficients in CoeD) and lower order coefficients bounded and measurable in Let (:I' be a closed, densely defined operator in.the space L 2(!1) su:..h that D«(:I') is the closure in H 2m (n) of a linear subspace M of c 2 m(n), and (:I'u = Au for 1.1 :€ D«(:I'). Suppose that for some real 0, and for all real ~ and all x :€ n, A'(x, i~ i= e iO• Let D: = D"" and D t = a/at, and set

273

Non-SeU-Adjoint Eigenvalue Problems

n.

+ylluI1 2m - 1 ,n +yr2m-lllullo,n, for a certain constant y. Now also

\'eittu(x)\ 2

T"'

1 J. I"" I+a-2m

=

2 m ,I

'\D""DaWt)e ittu(x))1 2dxdt

x

T"'

1

L(x, D , D ) = A 1 (x, D ) _ (_1)m e iOD2m, x x t t

J~ J

1

L

H+a-2m -~ n

where A 1 is an operator havinlJ the same principal part as A, and having bounded measurable coefficients in Then L is elliptic in r=nxE 1 • Suppose that for all v ~ C2m(f') such that vex, to) ~ M for each fixed

n.

I

H+a-2m

to and vex, t) == 0 for It\2 1,

t

In""Da(e 1ttu(x))\2dxdt x

t

r 2a J \D""u(x)\2dx n x

2m I r 2aI1.1 122m-a,n.

a-a

(15.13)

Ilvl12m,r S C(IILvllo,r + Ilvllo,r), Thus, if r L 1,

for some positive C. Then for u ~ D«(:I'), arg large,

A=

0, and

IAI

sufficiently

II'eittu(x)l~m,r L r4mllull~,n ',~

Ilull on

,H

= (~.

Moreover, if the resolvent set of (:I' is not empty, then e iO is a direc tion of minimal growth of the resolvent of (:1'.

III

i,1

if.

I'(t)eittu(x)l 2

m

,r s c[11'(t)e 1ttA 1 u -

j

[r 4m lluW O,n

+r211uW2m _l,n ]'I,~C1IAu-r2mrIOull 0,1

T"'

+ yy2[r4m-21Iull~,n + Ilull~m_l,n]'1, .::; Cy211Au -

+ II'eittullo,r]

r2meiOuil O,n

+yy2r- 1[r 4m Ilull ~,n +r2111ull ~m-l,n]~'

(_1)meiOuD2m('eitt)l\ T"' t 0,1

I I

i

r 4m \l ull ~,n +r21\u\1 ;m-l,n'

Therefore, (15. 14) implies

I

G

Proof. Let' be a function in C""(E 1) satisfying I'(t)! s 1 for all t. '(t) == 1, It I ~~, and '(t) == 0, Itl2 1. First, let u ~ M, and let vex, t) = '(t)eittu(x), where r is a positive parameter. According to the hypothesis, (15.13) is satisfied by the function v. Thus, for r r .... "" we have by Leibnitz's rule

II

lul~m-a,n

"

!

2C II(A- u)ull ~ o'n. IAI

~-

2m

+r 2}1

Thus, if r is sufficiently large,

[r4mllul\~,n + r21Iull~m_l,n]~'::; 2C1IAu - r2meIOullo,n.

=",,,=!~~~--=;;:.:~~~~;~~~~~~t"!~~~~~~~~:;~~~~~,,,:,~~~,-c,'~~~,,~~l;~j~"
274

Elliptic Boundary Value Problems

sec. 15

~ 2Cr- 2m IIAu -

Thus, if A = r 2m e

r

2m

e

e/ ull 0.0'

ie, this becomes

Ilullo.o ~ 2C1AI- 1 11(A

-

A)ullo.o,

IAllarge,

which is the required estimate. In case u is an arbitrary element of D(a), let lukl eM be chosen such that Uk .... U in H2m (O). Then aUk = AUk .... Au = au in L 2 (0), so that the inequality for u follows from the proved inequality for Uk' Finally, suppose that there exists a complex number Ao in pea). Then (A o - a)-l is a mapping of L 2(O) into D(a) eH 2m (U), and so is compact by Rellich's theorem. Thus, the spectrum of (A o _ a)-l is discrete. Hence, the same holds for the spectrum of For arg A = 17, IA!large, it follows from the estimate obtained above that A - (f is one-to-one, so that A is not an eigenvalue of Since the spectrum of a is discrete, A c pea). This estimate then implies

mil

(m

a.

a.

°

II(A - a)-III ~ 2CIAI- 1 •

ie

1

I I'

!

Au

E in

=

\, D"'u

=

0

on

I

(Dirichlet form for A, it follows that A can be written in the form I

.'

A(x, D)

Daa

1

=

al~m

l f31':::;m

af3

(x)Df3,.

d. (8.9). Now let

a~f3(x)

=

'h[aaf3(x) + af3a(x)],

and

I.

'1Ii,",1

Here A(x, D) shall be a strongly elliptic operator of order 2m in

n.

T*, so that a is self­

.~!.Suppose next only that A'(X, i(j is real. By passing to some

0,

an, lal : ; m -

=

ad)oi~t. Theorem 14.6 then applies to our case immediately.

a.

n,

(IS. IS)

(m.

1. In case A is formally self-adjoint, then T

Thus, e is a direction of minimal growth of the resolvent of Q.E.D. We now shall give some indications of how the abstract theorems on eigenvalues can be applied to specific problems. First, we discuss the Dirichlet problem. For convenience, we shall assume that the opera­ tor has. infinitely differentiable coefficients in and that 0 is of class Coo. Of course, these restrictions are far more than is needed, and we could easily keep track of the regularity actually needed. But for sim­ plicity, let us assume the infinite regularity. Assuming zero Dirichlet data, the Dirichlet problem can be stated:

275

By G~rding's inequality (Theorem 7.6) and Theorem 8.2, it follows

that the GOP for A + A has a unique solution, for certain complex

numbers A. If A is suitably normalized, A can be any sufficiently

large real number. We shall replace A by A + A, and thus assume

o (15.15) has a unique solution u ~ Hm (0) for every E in L /0). Let u = TE. By Theorem 9.8 it follows that u is actually in H,.m (0). Thus, R(T) e H 2m(O)' Since the same results hold for the formal adjoint A * of A, and since the solution operator associated with A * is just the adjoint of T*, it follows that R(T*) e H 2 CO). For the operator a we can take T- 1 • Then D(a) = H 2 Co) n (0), and au = Au for m ma U ~ D(a). For, clearly D(a) = R(T) eH (0) nH (0). Conversely, 2m o m suppose u ~ H 2 m (m n Hm (0). Let aE= Au. Then E~ L 2 (m, and, trivially, u = TE. Thus, H2 m (monm H (0) e R(T), and (fu = aTE = E= Au. Likewise, D(a·) = H 2 m n Hm We also need to check the behavior of powers of T. Again, we shall appeal to the regularity theory of section 9, and prove that R(Tk) e H 2mk(O)' For, suppose this has been groved for k, and let u = T k + IE. Then u = TTkE, so that u ~ H 2 m (m n Hm (m and Au = TkE. Again, . Theorem 9.8 implies that u ~ H 2m+2mk(O)' since TkE ~ H 2mk(m. Now there are three cases we shall discuss.

Hence, Ilull 0.0

Non-8elt-Adjoint Eigenvalue Problems

Ao(x, D)

=

1

a'~m

lf31':::;m

Daaof3Df3.

a

276

Elliptic Boundary Value Problems

Then A 0 is formally self-adjoint (d. (8.11)), since a~f3 = a~a' and A and A o have the same principal part. For, Leibnitz's rule implies A~ =

2

ja/-m

1/3

=

A'(x, (j _ (_I)m e i8r 2m.

Then

a~f3Da+f3

(-I)mL(x, ~, r) = A'(x, i(j _ e i8r 2m •

a f3 Y2[aaf3 + a/3a]D +

lf31-m 2 a f3Da+f3

lal-m a 1f3I-m A'.

Thus, A = A o + B, where A o is formally self-adjoint, A~(x, i(j is real, and B is an operator of order less than 2m. Moreover, the strong o ellipticity of A implies by means of Garding's inequality that, with suitable normalization, (Aou,

L(x, ~, r)

277

-m

al:m

=

Non-Self-Adjoint Eigenvalue Problems

sec. 15

u)o,n::2 YoEollull;',n -

J\llull~,n

:2 - Aollu\I~,n.

Therefore, we can apply all the results dealing with operators of the form ct o + ~, and obtain, in particular, that all directions except tb,e./ positive real axis are directions of minimal growth of the resolvent of ct. Compare the remark preceding Theorem 15.3. Also, we have "parabolic" condensation of eigenvalues along the positive real axis, as discussed above. 3. Finally, consider the case in which we assume only that A is strongly elliptic. By suitable normalization it may be assumed that IA I (x, i(j: x ~ real I fills oli: precisely the region IA: Iarg AI ~ 8 0 1, where 8 0 < Y217. In this case, all directions not in this sector are directions of minimal growth of the resolvent of ct. To prove this, we shall apply Theorem 15.4. Let 8 be any number satisfying 8 0 < 8 < 217 - 8 0 , and let

n, (

It is clear that for such 8, the values (_I)mL(x, ~, r) lie in a cone properly contained in an open half space of the complex plane, so that L(x, D x' D /) is actually strongly elliptic. Therefore, G~rding's in­ equality holds for L in a cylindrical domain n x E l ' so that (15.13) is verified. Moreover, since ct = r- 1 and r is compact, it follows that the spectrum of ct is discrete; in particular, the resolvent set of ct is not empty. Thus, Theorem 15.4 applies to our case. We conclude this section with a brief discussion of real second­ order problems. We shall assume that n

n

A= .~laIPPj+ l~laIDI+a r,l

and that-A'(x, (j 2 col~12 for real~. In the case of the Neumann prob­ lem, the boundary condition has the form aufav = 0, in which aufav " I is the conormal derivative corresponding to A. In case A is formally \ self-adjoint, we obtain immediately asymptotic expressions for the '~umber of eigenvalues less than t. But in any case, this problem is jalm0st self-adjoint, as in 2. above. Therefore, all but a finite number of eigenvalues lie in a parabolic region IA: ~A > q \A\ ::;; CI~AI'~I. This is a result first obtained by Carleman. In the case of an oblique derivative problem, we can show again that all directions except the positive real axis are directions of minimal growth. For this we consider the operator L = A + e 18D; and apply Theorem 15.4. In this case the estimate (15.13) follows from remarks made at the end of section 10. It should be noted that it is not known in this case whether the eigenvalues condense in a parabolic fashion along the positive real axis.

278

Elliptic Boundary Value Problems

Completeness of the Eigenfunctions

Sec. 16

279

2H

III TAIII < /k .

16. Completeness of the Eigenfunctions

In this section we shall state conditions under which the general­ ized eigenvectors of an eigenvalue problem span the Hilbert space, or, at least, a certain subspace of the Hilbert space. First we shall state some theorems on the growth of analytic functions.

Proof. By a modification of the classical Fredholm theory of inte­ gral equations, we have a formula,

tlA

T \ = - - , A (p (T),

THEOREM 16.1. Let F(A) be an entire complex-valued function, and let !F(>') I ~ C 1!A[-1

where 0(>')

1>'I~rk

W(A)I:s; C 2 e

=

m

1"" 0m>.m is a complex entire function, and tlA =

m~O

on two rays making an angle and let max

Q(A)

1\

TT/ a

at the origin. Let r 1 < r 2 <. •• ,r k

.... "",

~,

1"" tl Am is an operator-valued entire function, the convergence of m-O

80

where (3 < a. Then W(>') I :s; C11AI-1 everywhere between the two lines. THEOREM 16.2. Let F(>.) be an entire complex-valued function of finite order p. Then for any positive { there exists a sequence r 1 < r 2 <... ,r le .... "" such that min IF(\ _/!H IA\-r I\)\>e k •

m

the latter series being in the double-norm sense. Moreover, the fol­ lowing estimates are valid: =

1,

10ml:s; (e/m)m/ 2!11 Till m;

~"~

=

T,

,/-­

Illtlmlll:S;

ev,(elm)1I1/211I T lllm+ 1 •

k

The second of these theorems is a direct statement of a theorem on p. 273 in Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press (1939); the first is a Ph~agmen-Leinde1of theorem, and can be found in a slightly different form on p. 177 of the same book. Now we give an estimate due to Carleman. LEMMA 16.3. Let T be an operator of finite double-norm on a Hilbert space, and let { be a positive number. Thenfihere exists a sequence r 1 < r 2<... ,r Ie .... "", such that the modified resolvent T>. exists on the circles IA: IAI = r kl, and for IA[ = r k'

Also, T A exists precisely when 8(>') ,j, O. For a derivation of these formulas, see Zaanen, Linear Analysis, 1st ed. (1953), Amsterdam­ Groningen, pp. 261-274. From these estimates we obtain

18(A)I:s; 1 +

1"" (e/m)m/211I T lll ml>.lm.

m-1

Now let Il = Yzelll TII1 21>.1 2, and split this sum into the odd and even terms, to obtain "" 10(A)1 ~ 1 + 1

k

""

k+ Y,

~ + 1 ----'---Il_

k~lkk

le-o(k+Yz)k+Y,

280

Completeness of the Eigenfunctions

Elliptic Boundary Value Problems ,

00

~ 1 + (2/l) ~ + ~ k-I ~ 1 + (2/l)

=

'/,

2

"k

,00

k

c-+/l~ ~!!:­ kk k-I kk ,00

k

+ (1 + /l ~) ~ !!:­ k-l k!

1 + (2/l) 'I, - 1 - /l'1, + (1 + /l'I,)e /l

~ (1 + (2/l) 'I, )e/l.

Therefore, 18 (A)1

~

2

2

(1 + e'I,I\ITI11IA\)e 'l,eIIITII1 IAI .

Likewise, one sees that

III~AIII.s;

IIITIII +

I

m-I

~ e'I,IIITIII[l +

e'l,(e/m)m/ 2 jIIT\\lm+I\Alm

00

~ (e/m)m/211I T lll mIAlm]

m-I

for r k sufficiently large. Q.E.D. If T is a linear transformation on a Hilbert space, let sp(T) be the

closed subspace spanned by the generalized eigenvectors of T, and

let sp'(T) be the closed subspace spanned by the generalized eigen­

vectors of T which correspond to non-zero eigenvalues. In case

Tf = ,\f and A ,j 0, then it is clear that f ~ R(T). Thus, we have the

trivial relation sp'(T) c R(T). Therefore, if R(T) is not dense in the

Hilbert space, it cannot be expected that sp'(T) is dense. The most

" that could ever be expected is that sp' = R(T). We shall now give an abstract theorem containing a sufficient condition for this to occur. This theorem is found in Dunford and Schwarz, Linear Operators, II, Interscience, 1963. THEOREM 16.4. Let T be an operator of finite double-norm on a

Hilbert space X, such that there exist N directions of minimal growth

of the modified resolvent of T, where the angle between any two

adjacent rays is less than Y2*. (Thus, N L 5.) Then sp'(T) = R(T).

Proof. The Riesz-Schauder theory of compact operators has as a

consequence that the analytic function (A - T)-I has poles at the non-'

zero eigenvalues of T. Thus, if Ao is a non-zero eigenvalue of T,

the Laurent expansion of (A - T)-I about Ao has the form

k

(l6:})

~ (e'I,IIITIII + eIIITII12IAI)e'l,eIIITII12IAI2. Hence, both 8(A) and ~A are entire functions of order not exceeding 2. Applying Theorem 16.2 to 8(A), we obtain r 1 < r 2 <... ,rk ... 00, such that

=

2+(

<e

rk

=

~ Bv(A - Ao )-V

v-I

00

+ ~ Av(A _ Ao )V,

v-o

I

AV' commute with T. Therefore,

1 == (A - T)-I[(A - Ao ) + (A o - T)] k

~

V-I

00

B (A _ A )-v+ 1 + V

0

k

r k we have

IllTAIlls (e'I,IIITIII + elllTll1 2r k)e

(A - T)-l

/~here all the operators B v '

2+ '1,(

min 18 (A)[ > e -r k

IA1- r k Therefore, for IAI

281

~

v-o

A (A _ A )V+ 1

v 0

00

+ ~ Bv(A o - T)(A - Ao )-V + ~ Av(A o - T)(A - Ao)V V-l v-o

'l,elilTII1 2 r 2 r 2 +'I,(

2 ke k

. =

k-I B k(A o - T)(A - Ao)-k + ~ [Bv(A o - T) + Bv + 1 ](A - Ao)-V + B 1 + Ao(A o - J V-l

282

Elliptic Boundary Value Problems 00

+ ! v-I

[Av(A o - T) + A V_ 1](A - AO)V.

From this it follows that

(16.3)

283

Completeness 01 the Eigenlunctions

sec. 16

= T[

k

00

B A- 1(A- 1 _ A;I)-V + ! A A- 1(A-l _ A~I)V]. v v-I v V-I

!

Now

;.. - A A-1(A- 1 _ A-I )-V = A-Ie 0 )-V o ,\A

Bk(A O - T) = 0,

o

Bv(A O- T) = - B V + l'

1.

V = 1, ... ,k -

= A~AV-l(_I)V(A -

Therefore,

= (_I)VA v [A

o

(16.2)

(A o - T)k+ I-vB v = 0,

Now suppose I

II;:

k

0

V-I 1 =(_I)I/AV! (V- )A/!:{A-A )V-I-I1(A_A )-V o p._ 0 {f 0 0 0

!

(A-AO)VAVI.

=

v- 0

The relation (16.2) implies that = =

0

00

(A-Ao)-VBvl+

v-I

(A O - T)B kl

+ (A _ A )]V-l(;" _ A )-V

v = 1, .•. ,k.

X. By (16.1),

(A_T)- I/= !

0

;"o)-V

viI (_I)V(V - 1) A/1+1/(A _ A )-P.-l p.-o P. 0 0

I

(_I)V(V

p.- 1

0,

P.

=i);..f v- (A - Ao) 1

-p..

Therefore, (l6.3) maybe written (A O - T)2B k-l 1 = 0, v I ' !k 1 (_I)V(V - )AP.+V-ITB (A _ A )-p. + T (A) A- v-I p.- 1 11-10 VOl' r

/­

T -

'I ( A0 -

T)

k

lJ-tf=.1)\ • r:

//

where T 1 (A) is analytic at Ao' Rearranging this series, we obtain T

k k 1 ! (A - A )-11( 1 (_I)V(V - )Ap.+v-lTB ) + T (A) A p._ 1 0 v-p. P. - 1 0 V 1

=

k

(16.4) =- 1 CueA -- Ao)-P. + T 1(A) . P.-l

r

T;.. = T(1 _ AT)-1

=

A- 1T(A- 1 _ T)-1

Here, as above, it holds that for any I II;: X, Cp.1 is either zero or a generalized eigenvector of T corresponding to the eigenvalue A;I. This follows from the corresponding fact for Bvi and the formula for

CIl"

284

Elliptic Boundary Value Problems

Now we shall prove that sp'(T) == R(T). Since we already know sp'(T) c R(T), it is enough to show that R(T) C sp'(T). This is equivalent to showing that the orthogonal complement of R(T) contains that of sp'(T). Thus, assume that g is orthogonal to sp'(T) and that f ~ X. The theorem will be proved once we show that (Tf, g) == O. Consider the function F(A) == (T Af, g).

This function i~ii~alytic on Pm(T), and, as we have seen, has only poles at the c1}aracteristic values of T. Let Ao be any characteristic value. By (16.4), ,:': follows that for A near Ao' "\ k

F(A)

==

(

"'~1 (C",f; _~)(A - Ao)-'" + (T 1 (A)f, g).

As we have shown, C",f is either zero or is an element of sp'(T). Since g is orthogonal to sp'(T), in any case we have (C",f, g) == O. Therefore, F(A) has a removable singularity at Ao ' Since this holds fOr any Ao not in Pm (T): F(A) is entire. On any of the N rays of minimal growth of the modified resolvent, of T, (F(A)I ~ IITA[\

Ilfllllgll

sec. 16

Completeness of the Eig,enfunctions

For applications of this result to elliptic problems, the following

theorem is useful.

THEOREM 16.5. Let er be a closed, densely defined operator in the space L 2(m, such that D(er) cHm,(IJ). Suppose that all but a finite number of directions are directions of minimal growth of the resolvent of er. Tn case m' ~ Yw, suppose there exists an integer k > n/2m' such that D(er k) c H km ,(0). Then speer) == L 2 (m. Proof. Let Ao I;: peer). Then (er - AO)-1 maps L 2 (0) into D(er) c H m I (m, and is therefore compact, by Rellich's theorem. By con­ sidering er - Ao instead of er, it follows that it is sufficient to sup­ pose 0 l;: peer). Let T == er- 1 • Clearly, Sp'(T) == speer). In case m ' > Yw, then Theorem 13.5 implies T has finite double-norm. Since TA == - (A - er)-I, the directions of minimal growth of (A - er)-1 are also directions of minimal growth of T A. Therefore, Theorem 16.4 implies sp~T) = R(T). But R(T) = D(er), so R(T) = L 2 CO). Thus, speer) == Sp'(T) = L 2 (0), and the result follows in this case. Now suppose m' ~ 'hn, but k > n/2m'. ThenR(T k) = D(er k) CHkm,(O), so that again Theorem 13.5 implies Tk has finite double-norm. Also, R(Tk) is dense in L 2 (0), since R(T) is dense and T is continuous. That is, R(Tk) = L 2 (0).

Let wlJ""w k be the k roots of unity, and let z be any complex num­ ber such that Izl < 1. Then

== O(IA\-I). k

2+f .

k

Z2w/1-w l z)-I=Z i-I

Now let f be a positive number such that the angle between any two such adjacent rays is less than 17/(2 + f). By Lemma 16.3 there exist numbers r k, r 1 < r 2 <... ,rk -> "", such that for IAI == r k IITAII ~ II/TAIII < e

=

2

""

WI

""

2

w;zi

i-O

i-I

k

~ (~ W~+I)Zi+l. i-O i - I I

rk

But now the Phragmen-Lindelof theorem, Theorem 16.1, implies that !F(A) I ~ C1AI- 1 between two such rays. Therefore, since such angu­ lar regions fill the complex plane, W(A)I ~ C 1 \AI- 1 for all A. There­ fore, F(A) is bounded in the plane, and must be constant, by Liouville's theorem. Since F(A) -> 0 as IA\ -> "", F(A) == 0; i.e., (TAf, g) == O. Let­ ting A -> 0, we obtain TA -> T, so (Tf, g) == O. Q.E.D.

285

Now k

~ 1_ 1

W i +1 = I

lk, if j + 1 divides k,

0, otherwise.

Therefore, k

Z ~ w i- 1

/1 - Wi Z)-1 ==

""

~ kz Vk V-I

= k Z k(1

_ Zk)-I.

286

sec. 16

Elliptic Boundary Value Problems

k

Multiplying this relation by 1 - zk k

n (l -

i_ 1

w.z), and dividing by z. 1

k

n (1- w.z) = k z k-l.

~ WI

(16.5)

=

i-I

jral

J

iii

Although (16.5) has been derived under the assumption 0 < Izi < 1, it mus t actually hold for all complex z, since both sides are just polynomials in z. Therefore, if we replace z by the operator zT, a corresponding relation is obtained: k

~ WI

(16.~)

( Tk)

'0

tel

k

k- I t(l / k)-l e IO(1-k)/k ~ w T 0

= J-l J w.tl/kel Ik.

1

=

te 10,

n (1- w.zT) = k z k-lTk- l •

i-I

lal

\

Let e iO be a complex number all of whose kth roots are on rays of minimal growth of the modified resolvent of T. The assumptions of the theorem imply that all but a finite set of the numbers e lO possess this property. Indeed, we didn't need to assume so much, but only enough to insure that such numbers e iO determine at least five lines such that any two adjacent lines make an angle less than ¥27T. Letting z = tl/kelOlk in (16.7), we obtain

Therefore, if i\

k

287

Completeness of the Eigenfunctions

1

i'l'i

il(Tkh \I:;; r

l

k

I

1\

A[(1l k )-1 ~

.

liTw.t 11k e iOlkl1

J-l 1

Since also \

:; k- l li\! (1/k)-1

k

n (1 -

i-

i- 1

wJzT) = 1 - zkTk,

J

1

:; Cji\I(1lk)-IIi\I- l /k

it follows that zk €Pm(Tk) ~> wJz € Pm(T) for all j. Assuming that zk ~ Pm (T k ), multiply (16.6) by

(1- zkTk)-1

=

nk (1- w JzT)-1

J- 1

to obtain k

~ w/1 - w.zTfl = k z k-lTk-I(1_zkTk)-I. I_ 1

::; C\i\I-I.

Therefore, e 10 is a direction of minimal growth of the modified resolvent of Tk. Applying Theorem 16.4 to Tk, it is seen that sp'(Tk) = R(Tk) = L 2 (n)· . Now we shall show that sp'(Tk) C sp'(T). From this it will follow that sp(ct) = sp'(T) = L 2 (n), and the proof will be completed. Fils t, k if a is any positive integer, the rational function (z - w.)-OC can be

n

i_ I

I

Finally, multiply both sides of this by T to obtain ~ w.Tw z = kzk-I(Tk)

I-I

I

I

• zk

This relation holds for all complex z such that zk € Pm (Tk).

1

decomposed into its partial fraction expansion. Thus, for certain com­ plex numbers C/V we have k

k

(16.7)

k

~ Clw tllkelOlkl-1

n (z J-I

k

wJ)-OC =

oc

~ ~ c/V(z - wl)-v. I-lv-I

-'-- ",'--.

-"'" .. ~

__

-"'";.."--

",.

-,... -

__._

288

-=--:_~ ,'~_.

--,--- - .,.,,-

.

---'''-'-''-~''-=-''"~.-'",,".

--,:::·-=--''''--=;:-=-;;;~~-:'::_::-_··~:-··_=-':'~-=~"f=~~~~1~~:;~=:~~~==:=£:~~~-i.~EE~~S~,~~£:~;:';:~'~~}~-'!~::E~:~~~~~~T±:akT~~~~!;~~~~"E~~,;::~:¥~;~~~~~~'a~~~*:a~;~~~~~Jf~~~~

sec. 16

Elliptic Boundary Value Problems

'"

!

j-

Q.E.D.

k

1 cw(z - w)"'-V Il (z - Wj)'" == l. I V-I

j- I

j'1'i

Since the left side is 'a polynomial in z, we obtain a corresponding identity if we replace z by the operator z-IT. And then, multiplying the resulting expression by z"'k, we obtain '"

k

!

(16.8)

j-

I

!

k

cwzV(T - w/z)"'-V II (T - w.z)''' = z"'k, j_ I j'tj

V- I

1

a re lation which holds for all complex z. Now suppose f is a generalized eigenvector of Tk corresponding to a non-zero eigenvalue A. Thus, for some positive integer a, (Tk - A)"'f = O. Let z be any complex number such that zk = A. Then k Tk_A= Tk_ zk= Il (T-w.z). j_ I 1 Thus, k

Il (T - wjz)"'f

= O.

j-I

This implies that for any i, i = 1, ...,k, and any k

(T - wiz)V[(T - wiz)"'-V Il j_ I

(T - w.z)"'f]

=

v, V = 1, ... ,a, 0,

1

j'l'i

so that the vector k

(T - w/z)"'-V Il (T - wjz)"'f j- I

j'1'i

is either zero or is a generalized eigenvector of T corresponding to

289

w/z. Hence, this vector belongs to sp'(T). But then (16.8) implies z"'kf ~ sp'(T), and, since z -I 0, f ~ Sp'(T). Thus, spl(Tk) C sp'(T).

k

Multiplying this relation by Il (z - w.)"', we have an identity for all. j-I 1 complex z, k

Completeness of the Eigenfunctions

I~ _,I

This result may be applied to the Dirichlet problem for elliptic operators of order 2m. For example, if the principal part A' satisfies either the condition that A' (x, if) is real, or the condition that [arg A'(x, if)1 S a, where a < 'f.!17 and 2m> )1m, then the generalized eigenvectors span L 2 (0).

291

Index resolvent, 177

resolvent set, 175

Riesz-Schauder theory, 103, 281

INDEX a priori estimates, 67, 148

Aronszajn's theorem, 171

global regularity, 103, 124

Green's fonnula, 92, 134, 144

biharmonic operator, 96, ISO, 261

boundary value problems, 134

Hermitian, 96

Hilbert Nullstellensatz, 161

Hilbert space, 2, 175

Hilbert-Schmidt kernel, 201, 204, 219

hypo elliptic, 66

Caldero'n Zygmund kernel, 151

Caldero'n's extension theorem, 171

Cauchy Riemann operator, 47, 71

characteristic, 134

characteristic value, 179, 214

classical solution, 92

coercive quadratic form, 142, 159

coerciveness results, 158

compact operator, 103, 190

compact support, 2

completeness, 278

cone property (ordinary and re­ striCted), 11

conormal derivative, 136

convolution, 153

strong derivative, 3, 4

strongly coercive form, 142

strongly elliptic operator, 46

support, 1

Tauberian, 248

segment property, II

test function, 2

self-adjoint, 230, 250

trace of a function, 37, 38

semi-norm, 2

trace of an operator, 193

Sobolev constant, 210, 211

Sobolev's inequality, 32, 40, 208

Sobolev's representation fo.nnula, 151 uniformly elliptic, 45

162

weak compactness, 41

spectral theory, 175

weak derivative, 4

spectrum, 175

weak limit, 41

lI-smooth, 51, 52

inner product, 2

integral operator, 203

interior regularity, 111

interpolation theorem (inequality),

23, 35, 133

Laplace equation, 91, 93

Lax.Milgram theorem, 98, 215

Leibnitz rule, 9, 10

Lewy's example, 67

linear functional, 99

local existence, 47

local regularity, 51

difference operator, 42

direction of minimal growth, 181, 212

Dirichlet (bilinear) form, 94, 120

Dirichlet boundary conditions, 90

Dirichlet data, 92, 93

Dirichlet integral, 91

Dirichlet problem, 90

double norm, 188

eigenfunctions, 278

eigenvalue, 179

elliptic operator, 45, 46

ellipticity constant, 78

formal adjoint, 51, 95

formally self-adjoint, 96

Fourier series, 25

Fourier transform, 72

Fredholm al,ternative, 102

fundamental solution, 230, 234

o Garding inequality, 71, 78, 84, 86

generalized characteristic vector, 180

generalized Dirichlet problem (GDP),

98

generalized eigenvector, 180, 282

generalized solution, 93, 98

global existence, 90

290

mixed boundary value problems, 148

modified resolvent, 177

mollifier, 5

multiplicity, 180

natural boundary conditions, 143

norm, 2

normal derivatives, 91

non-self-adjoint problems, 261

null space, 102

oblique derivatives, 136, 277

ordinalY diff erential equations, 261

overdetermined elliptic system, 63,

65, 66

Parseval's identity, 72, 79

partition of unity, 7, 8

perturbation, 266

Phragmen-LindeHif theorem, 278

Poincare~s inequality, 73, 75

principal part, 45, 47

quadratic form, 78

Rellich theorem, 30, 99, 208

regularity theorems, 58

regularity up to the boundary, 103

Universitv Of \'~'?~h inf1ton Librar I

.

''­

I