; ~
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,]
r..W/{/IOI'YI
D. VAN NOSTRAND COMPANY, INC. 120 Alexander St., Princeton, New Jersey (Principal office) 24 West 40 Street, New York 18, New York
012
PREFACE
D. VAN NOSTRAND COMPANY, LTD. 358, Kensington High Street, London, W.14, England
D.
VAN NOSTRAND COMPANY (Canada), LTD.
25 Hollinger Road, Toronto 16, Canada
©
COPYRIGHT
1965, BY
D. VAN NOSTRAND COMPANY,
D.
INC.
Published simultaneously in Canada by VAN NOSTRAND COMPANY (Canada), LTD.
No reproduction in any form of this book, in whole or in part (except for brief quotation in critical articles or reviews), may be made without written authorization from the publishers.
,. I I r
I
'.!.
.'
J.'
.. \
. ,,,
'1..,.' V -/
'
"
\.,,'
·\I 1 }
1•• , /
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 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 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>;:{\
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 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 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
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\ 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 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/. >
1/2 (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
oo E
==
f
uvdx.
Thus u has the (uniquely determined) weak derivative DO< u == uO 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
" (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 + 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~lTheorem 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;:
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 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 ';,{' 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 ollII:',n -
shows that ~
1-1
n
,
~ 0=
1
1
\c12dx + 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
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, ¢] 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 0 and Ao ~ 0, and for all c,b :€ C~Cn}
constants Co
:RB[c,b, c,b] L collc,bll;',n -
I
,
,i