For operators of local t y p e we shall now answer a question raised on p a g e 169.
T H E O R E M 3.12. Let P ( D ) be o/ local type and let ~ be an arbitrary domain.
Then the operator P is the closure o/ its restriction to ~9~ N C~176
We note t h a t the restriction of P, mentioned in the theorem, is defined for those infinitely differentiable functions u such t h a t u and P ( D ) u are square integrable.
The value of P u is t h e n of course calculated in the classical way.
PROOF. Using an idea of Deny-Lions [4], p. 312, we shall for given e > 0 and u E OP construct a function v E C ~ such t h a t
(3.9.1) H v - u [ [ < ~ , [ [ P ( D ) v - P u [ [ < e .
Since these inequalities obviously imply t h a t v E L 2 and t h a t P (D) v E L 2, the theorem will then follow. Choose a locally finite covering ~k, k = 1, 2 . . . of ~ such t h a t ~ k c ~2 for every /c, and then t a k e functions ~ k e C ~ (f2k) so t h a t ~ (x)= 1 (cf. Schwartz [28], Thgor~me I I , Chap. I). The function uk = ?k u is in ~Op in virtue of Defini- tion 3.1, and we h a v e
u = ~ u k , P u = ~ P u k
(almost everywhere); t h e series converge since only a finite n u m b e r of t e r m s do not vanish in a compact subset of f2. (However, the second series is not L2-convergent if u ~ O ~ . ) Now L e m m a 2.11 shows t h a t u k E ~ ) ~ , so t h a t we can find a function vk e C~ (f2) such t h a t
(3.9.2) I l u k - v k ] l < 2-k e, I I p u k - - p v k l l < 2 - k e .
I t follows from the proof of L e m m a 2.11 t h a t we m a y assume t h a t v~ has also its support in ~k- Since the covering ~k is locally finite, the series ~ vk(x) converges for every x, and the sum v (x) is in C ~ (~). Using (3.9.2) we obtain
Ilv-ull--< Y IIv~-u~ll <~, IIP(n)v-Pull~llPv~-pu~[l<~,
which proves (3.9.1).
1 6 - 553810. Acta M a t h e m a t l e a . 94. I m p r i m 6 le 28 s e p t e m b r e 1955.
242 LARS HORMA~DER C t ~ A r T E ~ I V
Differential Operators with Variable Coefficients
4.0. Introduction
I n the two preceding chapters we have exclusively studied differential operators with constant coefficients. However, we shall see t h a t the methods of the proof of Theorem 2.2, which is the central theorem in Chapter I I , also apply when the coef- ficients are variable, if suitable restrictions are imposed. I n order to exclude cases where the lower order terms and the variation of the coefficients m a y influence the strength of the operator, we shall only study operators tO of principal type. This means t h a t the characteristics have no real singular points. (When the coefficients are constant, this is equivalent to Definition 2.1 according to Theorem 2.3.) Furthermore, we shall assume t h a t the coefficients of the principal part are real, which means t h a t there is some self-adjoint operator with the same principal part as tO, so t h a t tO is approximately self-adjoint. (It is sufficient to require t h a t tO is approximately normal in the sense t h a t the order of tO tO - tO tO is at least two units lower t h a n that of tO ]).
We do not s t u d y this case here.) The minimal differential operator defined by tO in a sufficiently small domain is then stronger than all operators of lower order, and has a continuous inverse. The same result is true for the algebraic adjoint ~ . Hence, in sufficiently small domains, the equation P u = / has a square integrable solution for a n y square integrable function [. I n the sense of section 1.3 there also exist correctly posed abstract boundary problems for the operator tO. I t seems t h a t this is the first existence proof for differential operators with non-analytic coefficients, which are not of a special type.
4.1. Preliminaries
Let to be a differential operator of order m in a manifold ~.1 I n a local co- ordinate system we m a y write
(4.1.1) t o = ~ a ~(x) D~.
i ztl<rn
Now, if ~v is an infinitely differentiable function in ~ , we have for real t
toe~t~=tm ~ aag~§ 1)
i It is sufficient here to suppose that ~ is a domain in R ~.
G E N E R A L P A R T I A L DIFFEI%ENTIA/~ O P E R A T O R S 243 when t-~ ~ , where ~vz = ~ ~v/~ x ~ a n d ~v~ = ~ : , . . . ~ .
(4.~.2) p (~, $) = Y~ a ~ (x) ~
l~l=m
T h u s t h e p o l y n o m i a l
is a scalar, if ~ is a c o v a r i a n t v e c t o r field, p (x, ~) is called t h e characteristic poly- nomial of p . The coefficients a ~ (I:r = m ) form a s y m m e t r i c c o n t r a v a r i a n t tensor.
T h e differential o p e r a t o r p is called elliptic in s if p (x, ~ ) * 0 for e v e r y x ~ a n d every real ~ = 0 , a n d it is said to be of principal type in ~ , if all t h e partial derivatives ~ p (x, ~)/~ St do n o t v a n i s h s i m u l t a n e o u s l y for a n y x fi ~ a n d real $ . 0.
W e shall n o w deduce some formulas, which replace the more implicit a r g u m e n t s of section 2.4 in t h e case considered here. L e t p (x, ~(1) .. . . . ~(m)) be t h e s y m m e t r i c multilinear f o r m in t h e vectors ~(1) .. . . . ~(m), which is defined b y p (x, ~),
9 9 ~ ~ O : r n .
I f k 1 .. . . . kp are positive integers, k 1 + ... + kp = m, we shall write p (x, ~ (1)k', . . . . ~(P)%) for t h e multilinear f o r m where /c~ a r g u m e n t s are equal to ~(~). S o m e t i m e s we also omit t h e variable x. N o w set for i n d e t e r m i n a t e ~ a n d r]
r n - 1
(4.~.3) ~ R~(r ~ ) ~ = ~ ~ p(~,-~m 1 ~, ~)p(r 1 ~, ~, ~),
t, k = l ] ~ 0
(4.L4)
m - 1
~, k = l j - - 1
~nd T ~ ~ = R ~ k - S t k . E v i d e n t l y T i k = T ~k(x, ~,~) is a s y m m e t r i c tensor which is a homogeneous p o l y n o m i a l of degree m - 1 in b o t h ~ a n d ~. Since
p (~m) = p (~), p ( ~ 1, ~) = 1 ~ ~ ~ p r
mt=l 8~
it is easy to verify t h e following f u n d a m e n t a l p r o p e r t y of t h e t e n s o r T ~k
~p (~) ~p(~) (4.1.5) t=l~ (~t-~t)T~k(~'~)=P(~) ~ ~ P(~)"
The a r g u m e n t s of section 2.6 were based on t h e fact t h a t , in virtue of L e m m a 2.2, there exist polynomials T ik (~,~) satisfying t h e i d e n t i t y (4.1.5), even for a non- h o m o g e n e o u s p o l y n o m i a l p. The simple explicit formulas given a b o v e for T ~k in t h e case of a h o m o g e n e o u s polynomial p, h-uve the n o w essential a d v a n t a g e t h a t T ~ are
244 LARS HORMANDER
G E N E R A L P A R T I A L D I F F E R E N T I A L O P E R A T O R S 245
(4.2.2) f T kk (x, D, D) u ~ d x
= f 2 x k I m (p (x, D)u p(k)(x, D)u) d x + : x ~ F ~ (x, D, D) u'~ dx.
Denote b y ~ an upper bound of [x] in ~. We m a y suppose t h a t ~_-<A, and shall prove t h a t (4.2.1) is valid, if 5 is sufficiently small. If we use the notations
and note t h a t p (x, D ) u only differs from p u b y a sum of derivatives of u of orders
< m, the inequality (4.2.2) and Sehwarz' inequality give
(4.2.4) f T ~ k ( x , D, ~ ) u ~ d ~ < = c a ( l l P u l l Ilullm-i + II~llm-:),
where C is a constant. (We shah denote b y C different constants, different times.) Now we have T ~ k = R k k - S kk, so t h a t (4.2.4) gives, after summation,
(4.2.5) 2R~(O, (0, D,
~ ) - R ~ (~,D, D))u~dx +
1
f.
-t- ~ S k k ( x , D , Z ~ ) ~ d x + C~ (ll P ~ll [[u[lm-l+llullm-12) 9
1
We shall prove (4.2.1) by estimating the terms in this inequality.
The definition (4.1.3) of R ~k shows t h a t
R ~ (0, ~, ~) = ~ (~ p (0, ~)/~ ~)~.
k = l k = l
This is a homogeneous positive definite polynomial, since ] : ) i s of principal type.
Hence we have
R ~ (0, ~, ~)_-> c ( ~ + ... + ~)m-~
k = l
for some positive constant c, and using P a r s e v a r s formul~ (cf. formula (2.5.1)), we thus obtain
(4.2.6) c[u[m_,~<=
f ~R~ (O, D, ~)ur~dx.
I t is easy to find an estimate of the first term on the right-hand side of (4.2.5).
In fact, since the coefficients of R k k (0, D , / J ) - R g k (x, D , / ) ) are continuously dif- ferentiable and vanish for x = 0 , they are 0 (Ix[). Hence
(4.2.7) (R ek (0, D, D ) - R kk (x, D, .D)) uf~ d x < C ~ [Ulm_~ ~.
246 L ~ s tIORMA:NDER
I n order to estimate the integral
f s~(x,D, ~)uadx
we m u s t first integrate it b y parts. F o r m u l a (4.1.7) with ? ' = k shows t h a t(4.2.8) S kk (x, D, / ~ ) u 5
1 =1" ~x ~0 (QJkk(x'D"D)ua)"
= g e (p (x, D) u p(k k) (x, D) u) + G ~ (x, D , / ) ) u ~ + j ~ }Iere
1 " a -jkk
G ~ (x, r 7 ) = - ~ j ~ ~ ~2 (x, r ~),
so t h a t G k (x, D, / ) ) u ~ is a sum of products of derivatives of the orders m - 2 and m - 1 of u. Hence Schwarz' inequality shows t h a t
(4.2.9)
f G ~ (~, D, ~) ~a dx<=Vl~lm_,
1~1~-2.Furthermore, the integral of the last t e r m in (4.2.8) is zero, and using again the fact t h a t
p(x, D)u
differs fromp u
o n l y b y derivatives of order < m of u, we thus obtain(4.2.1o)
fs~(x,n,~)~adxZC(llP~ll+ll~ll~_,)lul~_~.+Cl~lm_~l~lm_~.
I f the two sides of the inequality (4.2.5) are estimated b y means of the in- equalities (4.2.6), (4.2.7) and (4.2.10), it follows t h a t
(4.2.11)
lul~-,"<=c(lipull+liull,o
,)(all~ll~ , + I ~ l ~ ~), ~ c ~ ( a ) . To prove (4.2.1) we have now only to invoke the inequality(4.2.12) I~1~ ,=<cal~l,, ~ e c a ( a ) , ~=1 . . . m,
which is an immediate consequence of L e m m a 2.7 b u t also well known previously (see for example Gs [9], p. 57). I t follows from (4.2.12) t h a t [U[m 2 ~
CSlulm__l
~C6]]ul[m_l, and, since c$~A, t h a t I[ul[~
~C[u],n-1.
Hence (4.2.11)gives with a constant Kso t h a t
Ilullm-?<:K(ll P~II
+II ~llm=~)~ II ullm-~,
(4.2.13) I[ u Jim-1 (1 - K 8) ~ K 6
II P
u [l.Thus the inequality (4.2.1) follows, if K 6 < 1.
I n particular, it follows from Theorem 4.1 t h a t the operator P0 in L 2 (~-~) has a continuous inverse, if ~ is a suitable neighbourhood of the origin. Now let the coefficients of p be sufficiently differentiable, so t h a t p also satisfies the hypotheses
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