PETER SJOGREN On the adjoint of an elliptic linear differential operator and its potential theory

On the adjoint of an elliptic linear differential operator and its potential theory

PETER SJOGREN University of G6teborg, Sweden

1. I n t r o d u c t i o n

I n her thesis [4], R.-M. H e r r 6 develops Brelot's axiomatic potential theory.

Within this theory she constructs an adjoint potential theory satisfying the same axioms. She applies this to the potential theory associated with an elliptic linear second-order differential operator L. When the adjoint operator L* exists in the classical sense and has H S l d e r - e o n t i n u o u s coefficients, the adjoint potential theory coincides with t h a t of L*. I n Section 3 of this paper we generalize this fact to the case when the coefficients of L are assumed to be locally ~-I-ISlder continuous and L* is defined in the sense of distributions. This result easily implies some properties of supersolutions of the equation L * u = 0 proved b y L i t t m a n [5]. H e shows that t h e y satisfy a minimum principle and have some approximation properties.

Under the same assumptions, we prove in Section 4 that the distribution solutions of L * u ~- 0 are locally ~-I-f61der continuous. In Section 5 we obtain a formula for I-Ierv6's L*-harmonie measure of a domain co. This measure is shown to have an area density given simply b y a conormal derivative of the Green's function of L in ~o. :Finally, we prove a :Fredholm t y p e theorem for the I)Mchlet problems for 15 and L* in a given domain.

The author wishes to t h a n k Professor Kjell-Ove Widman for his valuable help in the preparation of this paper, in particular for giving the ideas of several of the proofs presented.

2. P r e l i m i n a r i e s

Suppose we are given a domain ~2 0 C R n, n > 2, a n d a differential operator L u = aiJuij -~- b~u~ + cu,

154 PETER SJOGREN

d e f i n e d in D 0. W e a s s u m e t h a t a (i ~ a jl, t h a t L is elliptic in Y20, a n d t h a t t h e coefficients are locally ~-HSlder c o n t i n u o u s , for some ~ w i t h 0 < ~ < 1. As I-Ierv6 shows, we can let t h e C (2) f u n c t i o n s u satisfying

Lu

= 0 be t h e h a r m o n i c f u n c t i o n s in B r e l o t ' s a x i o m a t i c p o t e n t i a l t h e o r y p r e s e n t e d in B r e l o t [2, 3]. I n this w a y we o b t a i n a p o t e n t i a l t h e o r y satisfying B r e l o t ' s A x i o m s 1, 2, a n d 3' (see I-Ierv6 [4]). W e write ~>L-harmonic~>, ~>L-potentiab>, etc. w h e n we r e f e r to c o n c e p t s o f this t h e o r y .

T o m a k e possible t h e c o n s t r u c t i o n o f a n a d j o i n t t h e o r y , we m u s t l i m i t ourselves to a d o m a i n D C z9 0 w h e r e a p o s i t i v e L - p o t e n t i a l exists. D e p e n d i n g on t h e coef- ficient c of t h e o p e r a t o r , Y2 m a y be chosen in t h e following w a y , as s h o w n b y l-[erv6 [4, p. 562].

1. I f c < 0 a n d c ~ 0 , we m a y t a k e z g c t 9 o a r b i t r a r y . 2. I f c ~ 0 , we m a y t a k e a n y b o u n d e d z9 such t h a t ~ c z 9 o.

3. I f c is a r b i t r a r y , a n y x 0 C D O has a n e i g h b o u r h o o d w h i c h is a n admissible D.

F r o m n o w on we f i x such a n .(2. I n t h e sequel o), 0~1, 9 9 9 will always be s u b d o m a i n s o f z9 or z9 0.

W e follow t h e n o t a t i o n of B r e l o t a n d K e r v d a n d write / ~ for t h e b a l a y a g e d f u n c t i o n o f a n o n n e g a t i v e L - s u p e r h a r m o n i c f u n c t i o n v a n d a set E c ~9. I f 05 is c o m p a c t a n d c o n t a i n e d in $2, a n d f is d e f i n e d a n d c o n t i n u o u s on ~9, t h e n t h e solution of t h e D i r i c h l e t p r o b l e m for L in ~o w i t h b o u n d a r y values f is d e n o t e d b y H y . A p o i n t x 0 E ~ is called L - r e g u l a r for ~o if

Hy(x)-+f(Xo)

^as

x ~ Xo,

x C o9, for a n y c o n t i n u o u s f. As s h o w n b y H e r v d , x 0 is L - r e g u l a r if a n d o n l y if it is r e g u l a r in classical p o t e n t i a l t h e o r y .

K e r v d [4, P r o p . 35.1] c o n s t r u c t s an L - p o t e n t i a l Py in D w i t h s u p p o r t {y}

a n d such t h a t t h e m a p p i n g

(x, y)--~ Pz(x)

is c o n t i n u o u s for x, y C D, x # y.

T h e s u p p o r t o f a p o t e n t i a l P is d e f i n e d as t h e c o m p l e m e n t of t h e largest o p e n set in w h i c h P is h a r m o n i c . T h e f u n c t i o n _Py(x) is a f u n d a m e n t a l solution of L in ~9.

K e r v d [4, T h e o r e m 18.2] shows t h a t a n y L - p o t e n t i a l P in D can be r e p r e s e n t e d as

P(x) =~ f P:~(x)d#(y)

(2.1)

for a u n i q u e p o s i t i v e m e a s u r e # in s T h e s u p p o r t of # coincides w i t h t h e s u p p o r t of t h e L - p o t e n t i a l P .

F o r a n y b o u n d e d ~o of class C (1'~) a n d such t h a t 05 c Y2 t h e G r e e n ' s f u n c t i o n is g i v e n b y

G+(x, y) = _P,(x) - - H77(x).

T h e f u n c t i o n G ~ can be used t o solve a b o u n d a r y v a l u e p r o b l e m , as follows. I f f is c o n t i n u o u s in 05 a n d locally K S l d e r c o n t i n u o u s in ~, t h e u n i q u e solution o f t h e p r o b l e m

L u = f

in ~, u = 0 o~1 0oJ

O N T H E A D J O I N T O F A N E L L I P T I C L I N E A R D I F ] ~ I E R E N T I A L O P E R A T O R 155 is given b y

For this see Miranda [6].

u(x) - _ f Go (x, y)f(y)dy.

( o

(2.3)

To construct the adjoint potentials, Kerv~ uses the concept of completely- determining open set in /2, which is defined in Herr6 [4, p. 451]. I f o) is L-completely determining, ]-Ierv6 defines the L*-harmonic measure a~' for o~ at y E o~ by the equation

The left side of (2.4) is an L-potential in /2, so because of (2.1), the measure @' is uniquely determined by (2.4). tterv6 now calls a function L*-harmonic in % if it is continuous there and satisfies

f

u ( y ) = u(x)d x), y C m , for a n y L-completely determining m such t h a t o5 C ml.

I-Ierv6 shows t h a t the L*-harmonic functions satisfy the axioms of Brelot's potential theory. I n the adjoint t h e o r y the function P*(x) =- Px(y) is a potential with support {y} and plays the role of Py(x). Following Kerv6's notations, we shall write ~>L*-superharmonic~>, ~>L*-potential~>, etc., for concepts pertaining to this adjoint theory. F r o m the definition it can be proved t h a t the property of L*- harmonicity in ~o is independent of the domain /2 considered, ~2 ~ (o. I f the adjoint operator

02 - 0

L * u -- axiOxi (aqu) -- ~ (b~u) @ cu (2.5) exists in the classical sense and has ]-[51der-continuous coefficients, then the L*- harmonic functions are simply the solutions of L * u = O. I n the general case, we interpret (2.5) in the sense of distributions, for a n y locally integrable u.

For each s > 0 we fix a nonnegative C (~) function w~ in R n, with support contained in {Ix I < s}, and such t h a t

We let H(x, y) defined b y

f %(x)dx = 1.

be the fundamental solution of the operator aq (y ) a2 / Oxiax i

] 5 6 P E T E ~ S J O G ~ E N

H(x, y) -- / ~ : w . . log ¹ (~a~i(y)(xi -- yi)(x i -- yj))-'/' if n = 2, 2~ Y A ( y )

1

2-~on V A ( y ) (~aq(y)(x, -- y,)(xj -- yi)) (2-n)/2 if n > 2.

(n

H e r e ~o= is t h e a r e a of t h e u n i t s p h e r e in R", a n d A ( y ) a n d d e t e r m i n a n t a n d t h e inverse, resp., o f t h e m a t r i x (aq(y)).

(aii(y)) are t h e

3. The fundamental equivalence

W e s t a r t w i t h a p r e l i m i n a r y r e g u l a r i t y p r o p e r t y o f t h e d i s t r i b u t i o n solutions of L * u = O in a d o m a i n c o C s

L]~MM~, 1. I f u E L~oo@) satisfies L * u : 0 in the sense of %'@), then u coincides a.e. in co with a continuous function.

Proof. A s s u m e n > 2, a n d t a k e a f u n d a m e n t a l solution F ( x , y) of L in ~o.

L e t U be a r e l a t i v e l y c o m p a c t o p e n subset of co, a n d p i c k y E U so t h a t

f

lu(x) -- u(y)idx = o(~ n) (3.1)

B~o

a s e --> 0, w h e r e ~ , is t h e ball {x: Ix - - y[ < ~}. Choose ~0 E ~(eo~/~el2) e q u a l to 1 in U ~ B ~ . I n B(~B(,/2 we let t h e d e r i v a t i v e s o f ~0 s a t i s f y ~01 ---- 0(~ -1) a n d

~ 0 q = O ( ~ -2) as ~ - - > 0 , a n d outside U we t a k e ~ i n d e p e n d e n t o f ~ a n d y.

Since x - + 7(x)F(x, y) is a C (2) f u n c t i o n , we conclude t h a t f u(x)L,~(q~(x)F(x, y))dx = O.

:But L~F(x,y)----O for x 4 : y , so

~, B~ (3.2)

f .

y)dx + ] y))dx = o.

B e e o ~ U

W e k n o w t h a t F(x, y) = O(Ix -- yl 2-'~) a n d t h a t

F ( x , y) -- H(x, y) = O(ix -- y]=-2-,).

(Cf. M i r a n d a [6, pp. 18--20]). H e n c e , (3.1) implies t h a t t h e t h i r d t e r m in (3.2) is o(1) as e - + O, a n d t h e f i r s t t e r m equals

O N T H E A D J O I N T O F A N E L L I P T I C L I N E A R D I F F E R E N T I A L O P E R A T O R 157

fu(y)aq(y)qaii(x)H(

^{, y)dx + o(1). (3.3)}

B 9

B y m e a n s of a n i n t e g r a t i o n b y p a r t s , we f i n d t h a t t h e s e c o n d t e r m in (3.2) e q u a l s t h e s a m e e x p r e s s i o n , e x c e p t f o r a f a c t o r - - 2. ]~ut H is a f u n d a m e n t a l s o l u t i o n o f a~J(y)O~/Ox~Oxi, a n d ~ 0 - 1 c a n b e c o n s i d e r e d as a f u n c t i o n w i t h c o m p a c t s u p p o r t in B , so t h e i n t e g r a l in (3.3) e q u a l s u(y). L e t t i n g e - - > 0, we g e t

u(y) =

f y))d

^(3.4)

o , i

co\ U

for a.a. y i n U. Since

E(x, y)

is continuous i n

(x, y)

for x ~ y, the integral i n (3.4) is a continuous f u n c t i o n of y i n U, and the lemma is proved for n > 2.

I f n = 2, we i n t r o d u c e a n e w v a r i a b l e x 3 a n d p u t M = L + a2lOx~ a n d v(x, x s ) = u(x) in w• T h e n M is elliptic, a n d v satisfies M * v = 0 in t h e sense o f ~ ' ( e o • since for ~ C ~(o~• we h a v e

f f f + f f xjx =o

o) (o

T h u s we c a n m a k e v a n d h e n c e also u c o n t i n u o u s b y c h a n g i n g t h e m on null sets, a n d t h e p r o o f is c o m p l e t e .

Remark. As w e shall see l a t e r , u is in f a c t I~51der c o n t i n u o u s , a n d t h e r e f o r e t h e p r o o f o f (3.4) holds also in t h e t w o - d i m e n s i o n a l case.

THEOREM 1. Let co ~ f2. A locally integrable function u in co satisfies L * u = 0 in the sense of ~'(a~) i f and only i f u coincides a.e. in a~ with a function which is Z*-harmonic in co. Similarly, u is locally integrable in co and satisfies L * u ~ 0 in the sense of ~5'(co) i f and only i f u coincides a.e. in a~ with a function which is L*-superharmonic in o~.

_Proof. S u p p o s e u is L * - h a r m o n i c in co, a n d l e t ~0 E ~(eo). T a k e ~o 1 a n d o~ 2 s u c h t h a t

s u p p ~ o C ~ I C ~ I C ~ 2 C ~ 2 C o ~ ,

a n d let (0 2 b e b o u n d e d a n d of class C 0'~). If u ~ O, define v = (R*~'~ _{x - - u}

/(o s

w h i c h is the b a l a y a g e d f u n c t i o n of u in t h e Z * - p o t e n t i a l t h e o r y in 0) 2. T h e n v is a n L * - p o t e n t i a l in co 2 w i t h s u p p o r t c o n t a i n e d in 0o~j, a n d v coincides w i t h u in co 1. B y T h e o r e m s 33.1 a n d 18.2 in I-Iervd [4], t h e L * - p o t c n t i a l s c a n b e r e p r e s e n t e d as in (2.1). I n t h i s case we o b t a i n

] 5 8 P E T E R S J O G R E ~

[ .

v(y) = J G~(x, y)d#(x), (3.5)

for some positive m e a s u r e # w i t h s u p p o r t c o n t a i n e d in 0%.

T h e r e exists a p o s i t i v e L * - p o t e n t i a l P * in [2 a n d t h u s also a p o s i t i v e L * - h a r m o n i c f u n c t i o n in a n e i g h b o u r h o o d of r An L * - h a r m o n i c u of a r b i t r a r y sign is t h e r e f o r e in % a difference b e t w e e n t w o p o s i t i v e L * - h a r m o n i c functions. H e n c e , we o b t a i n a r e p r e s e n t a t i o n similar t o (3.5) for a n y u, b u t w h e r e # n e e d n o t be positive.

B e c a u s e of (2.3), t h e f u n c t i o n ~p satisfies

y2(x) = -- f G~ y)Ly~(y)dy. (3.6)

i "'

N o w (3.5--6) a n d F u b m s t h e o r e m i m p l y t h a t

f ^uLFdx

^{-- --}

^{f = o,}

a n d t h u s L *u ~ O.

Conversely, suppose u is locally i n t e g r a b l e in ~) a n d satisfies L*u = 0. W e t a k e a c o m p l e t e l y d e t e r m i n i n g ~o 1 w i t h ~51 C ~o a n d a p o i n t y E co 1. P u t

f(x) = f P~(x)d(e,(z) --

where e x is t h e m e a s u r e consisting of a u n i t mass at y. F r o m (2.4) it follows t h a t f(x) = 0 for x e ~ , since / ~ . . . . Px in t 9 ~ . N o w d e f i n e

a n d

[ .

s =

J Pz(x)(gdz) - h~(z))dz.

Since P~(x) is a f u n d a m e n t a l solution, we see t h a t f~ is L - h a r m o n i c o u t s i d e t h e s u p p o r t s o f g~ a n d h~ a n d t h a t f ~ - - ~ f = 0 in ~ 9 ~ 5 1 as e - + 0, u n i f o r m l y on c o m p a c t subsets of ~Q~51. F r o m T h e o r e m 35, I V in M i r a n d a [6], it follows t h a t t h e first- a n d s e c o n d - o r d e r d e r i v a t i v e s of f~ t e n d to 0 in ~ r as e --> 0, uni- f o r m l y on c o m p a c t subsets. T a k e ~ E ~(eg) e q u a l to 1 in a n e i g h b o u r h o o d U o f

~51. T h e n

in U, a n d L(9)f~ ) ---> 0 u n i f o r m l y in o ~ U as e --> 0.

Since (?f~ is of class C (2), it is clear t h a t f uL(q~f~)dx = 0

O]N- T t t E A D J O I N T O F A~'q : E L L I P T I C LLNEAI:~ ] ) I F F ~ ] R E ~ T I A L O r E R A T O I ~ 1 5 9

or e q u i v a l e n t l y ,

f uL(~fJx-- f ug~dx § f uh~dx=O.

B y L e m m a 1 we c a n a s s u m e t h a t u is continuous. L e t t i n g e - ~ 0, we get u(y) =

f ^{%d(y; )~,}

a n d t h e f i r s t p a r t o f T h e o r e m 1 is p r o v e d .

T h e p r o o f t h a t L * u ~_ 0 for a n L * - s u p e r h a r m o n i c f u n c t i o n u is quite similar t o t h e c o r r e s p o n d i n g p r o o f for L * - h u r m o n i e f u n c t i o n s a n d is o m i t t e d .

Conversely, suppose t h a t L * u is a n e g a t i v e m e a s u r e - - / ~ . T a k e a b o u n d e d (o~ of class C 0'~) a n d such t h a t ~1 c (o. B e c a u s e of (2.3), a n y yJ C 95(0~1) satisfies

f uL~c~x= f f r176

w h e r e t h e d o u b l e i n t e g r a l is a b s o l u t e l y c o n v e r g e n t . T h e f i r s t p a r t of T h e o r e m 1 n o w shows t h a t t h e f u n c t i o n v d e f i n e d b y

is e q u a l to un L * - h a r m o n i c f u n c t i o n ~.e. in (o~. T h e i n t e g r a l in (3.7) r e p r e s e n t s a n L * - p o t e n t i ~ l , so u coincides w i t h an L * - s u p e r h a r m o n i c f u n c t i o n a.e. in (o~

a n d t h u s also in (o. T h e p r o o f of T h e o r e m i is complete.

4. Regularity of the L*-harmonie lunetions

THEOREM 2. I f U iS L*-harmonic in (o C t~, then u C C}~162

Proof. S u p p o s e n ~ 2, t a k e a c o m p a c t set K C ~o, a n d let ~ C 95((o) be 1 i n a n e i g h b o u r h o o d U of K . I f y E K , we h a v e

f = f _ - -

for a n y ~0 E 95(~o). As in t h e p r o o f of L e m m a 1 we f i n d t h a t

+ u(x)(a~J(x) f - a'J(y))~ ^(~(z)g(~, y))dx + (4.1)

f

§ u(x)(b~(x) ~x~ (q~(x)g(x, y)) § c(x)?(x)H(x, y))dx.

1 6 0 !~E T E 1~ S J 0 GI%EN

,Now t a k e y a n d z E K w i t h ~ = [ z - - y l so small t h a t B - - {x: Ix -- yI _< 2e} c V, a n d consider (4.1) a n d t h e c o r r e s p o n d i n g f o r m u l a for u(z).

of t h e a ~i it follows t h a t

a n d

F r o m t h e r e g u l a r i t y

H ( x , y) - - H ( x , z) = O ( ~ I x -- yl 2-n + ~lx - - yl~-"), H~(x, y) Hx~(x, z) ~-- O(~alx - - y[1-~ ~_ ~l x _ y]-~),

H~,~(x, y) - - Hx~,~(x, z) = O ( ~ l x - - yl -'~ + ~Ix--

yl-~-'~),

if x ~ B. Since u is c o n t i n u o u s , t h e s e e s t i m a t e s easily i m p l y t h a t t h e d i f f e r e n c e b e t w e e n t h e f i r s t t e r m s in t h e f o r m u l a s for u(y) a n d u(z) is 0(~ ~) as Q --> 0, a n d t h e same is t r u e for t h e s e c o n d a n d f o u r t h t e r m s .

T h e t h i r d t e r m in (4.1) we split as f B + f u \ , + f ( o \ V ' a n d t h e i n t e g r a l s o v e r B in this e x p r e s s i o n a n d in t h e c o r r e s p o n d i n g e x p r e s s i o n for u(z) are 0 ( ~ ) . T h e difference b e t w e e n t h e integrals o v e r ( o ~ U is also 0 ( ~ ) . T h e r e m a i n i n g difference can be w r i t t e n as

f u(x)(a'J(z) -

a~J(y))Hxi,~(x,

z)dx +

U~B (4.2)

-l- f u(x)(a'i(x) - - a ;i ))(Hx~ (x (Y"" , 7' ' y) -- Hx,v(x' z))dx.

U \ B

H e r e t h e second t e r m is 0 ( ~ ) , a n d t h e f i r s t t e r m is O(e ~ log 1/~). H e n c e , u E ~(0,~)~1or for some fi ~ 0. B u t t h e n we can i m p r o v e t h e last e s t i m a t e . Since

f Hxi5.(x, z)dSx = 0

Ix-=l = r

for all r, t h e fh'st t e r m in (4.2) equals

f (u(x) - u(z))(a~J(z) alJ(u))Hx <x z)ax + 0(~),

U \ B

w h i c h is b o u n d e d b y

o+=>. f z< + o(+> o(+>.

U \ B

T h e case n ~ 2 n o w follows as in t h e p r o o f o f L e m m a 1, a n d T h e o r e m 2 is p r o v e d .

O N TIIE ADffOII~T OF A N E L L I P T I C LII~EAI~ D I F F E I ~ E N T I A L OI~EI~ATOI~ ] 6 1

Remark. I t is clear t h a t t h e e x p o n e n t cr is best possible. I n a similar w a y , o n e c a n p r o v e r e g u l a r i t y p r o p e r t i e s of solutions of n o n h o m o g e n e o u s e q u a t i o n s L * u -~ f.

F o r e x a m p l e , i f f E LF~or t h e n u e C~~ ~) i f p = n/(2 - - a), a n d u is c o n t i n u o u s if p ~ n/2.

5. A formula for the L*-harmonie measure

W e shall a p p r o x i m a t e t h e coefficients of 25 w i t h m o r e r e g u l a r f u n c t i o n s , a n d s t a r t b y e x a m i n i n g h o w t h e G r e e n ' s f u n c t i o n varies w i t h t h e coefficients. F o r s --> 0, a s s u m e t h a t

25~u q i

bsui

=- a~ uq ~- + c~u

is a n o p e r a t o r w i t h coefficients in C(~ for some ~ ~ 9 . L e t G : ~ be t h e c o r r e s p o n d i n g G r e e n ' s f u n c t i o n , w h e n e v e r it exists.

L E M ~ A 2. A s s u m e that r is a bounded C (~'~) domain with ~ C ~ . Let the C(~162 norms of the a~ be bounded for small s, and let a~ j ~ a ~j u n i f o r m l y i n r as s --> O, and analogously for b~ and c~. T h e n G~(x, y) ~ G~ y) u n i f o r m l y on a n y compact subset of o)• ~ which is disjoint with the diagonal.

Proof. Since t h e r e is a p o s i t i v e L - p o t e n t i a l in tg, t h e r e exists a C (2'~) f u n c t i o n v in ~5 w h i c h is p o s i t i v e a n d satisfies L v < 0 in ~. Since t h e n also L v ~ 0 if e is small e n o u g h , GO ~ exists for such e.

I n B o b o c a n d M u s t a t ~ [1, Chap. 4], t h e r e is a c o n s t r u c t i o n o f t h e G r e e n ' s f u n c t i o n for L in t h e f o r m o f a n L - p o t e n t i a l w h o s e s u p p o r t is t h e p o i n t y. F r o m this con- s t r u c t i o n it c a n b e seen t h a t G:~(x, y) is u n i f o r m l y c o n t i n u o u s in y w h e n x a n d y s t a y w i t h i n d i s j o i n t c o m p a c t subsets o f w, a n d this c o n t i n u i t y is u n i f o r m in e for small e. B o b o c a n d M u s t a t ~ a s s u m e t h a t t h e c o e f f i c i e n t c is n o n p o s i t i v e , b u t f this is n o t t h e case, we can a p p l y t h e i r p r o o f t o t h e o p e r a t o r Me: u --> v-lL~(vu), n w h i c h t h e c o e f f i c i e n t of u is n e g a t i v e . T h e n t h e G r e e n ' s f u n c t i o n o f L~ is g i v e n b y

v(x),

i G~'(x, y) -~ G ~ ( x , y ) ~ ( ~

a n d t h e same e q u i c o n t i n u i t y follows.

F o r f C C(~ a n d e small, let u~ be t h e s o l u t i o n of t h e p r o b l e m L ~ u ~ - ~ f in 04 u ~ = 0 on ~o.

I f we d e f i n e u similarly b y m e a n s o f L, t h e n t h e C(2)(~o) n o r m s of u a n d u~

are u n i f o r m l y b o u n d e d for small e, as follows f r o m M i r a n d a [6, T h e o r e m 35, IV].

:Now L ( u - - u~) = (L~ - - L)u~, so supo [L(u - - u~)[ - ~ 0 as s --> 0. F r o m T h e o r e m 35, I X in [6], we t h e n conclude t h a t u~ --> u as e --> 0, u n i f o r m l y in co. Since

162 ~ETE~ SJOGRE~

[ ,

u(x) -- u,(x) = -- J (G~~ y) -- G:'(x, y))f(y)dy,

t h e l e m m a t h e n follows if we choose f s u i t a b l y a n d use t h e e q u i e o n t i n u i t y o f G:?.

Suppose t h a t (o is a b o u n d e d C (''~) d o m a i n w i t h o5 c s T h e n ~ is a c o m p a c t (n - - 1)-manifold, i m b e d d e d in R ~, a n d for t o p o l o g i c a l reasons e a c h of its c o m p o n e n t s separates R". I t follows t h a t &5 ---- &o, w h i c h m e a n s t h a t a t e a c h p o i n t o f this m a n i f o l d , o) lies on one side a n d R " ~ o 5 on t h e o t h e r . I t is easily s h o w n t h a t o) is L - c o m p l e t e l y d e t e r m i n i n g (see t h e p r o o f of this f a c t for o p e n balls in H e r v 6 [4, p. 565]).

I f x E aco, we let n~ ~ (cos ~1,. 9 cos ~,~) be t h e e x t e r i o r u n i t n o r m a l o f &o a t x. T h e c o n o r m a l d e r i v a t i v e a t x is d e f i n e d b y a/av ~ aq(x) cos ~jO/Ox~. T h e area m e a s u r e of 0~o is d e n o t e d dS.

TI-IEOI~E~ 3. I f y is a point in the C (~'~) domain ~o described above, then the L*-harmonic measure ~ is absolutely continuous with respect to dS, and

d~; aG~(x, y)

dS - - o ~ (5.D

for x E &o. T h i s density of o~' is or continuous and positive on 0o~.

_Proof. Since a~ ~ is i n d e p e n d e n t of ~Q D ~5, we can a s s u m e t h a t ,(2 is b o u n d e d a n d o f class C (~';~), a n d t h a t t h e coefficients o f L are d e f i n e d in a s l i g h t l y l a r g e r d o m a i n , so t h a t L e m m a 2 applies to f2. T h e n we can w r i t e Py(x) = G(x, y). P u t

u = ~ ? , ~ , (5.2)

w h i c h is a n L - h a r m o n i c f u n c t i o n in ~Q~ a(o. Since all t h e p o i n t s o f &o are regular, u equals Py in Y2~o9 a n d H ~ in ~o, a n d u is c o n t i n u o u s in ~ a n d zero on 0tg. Py D u e to T h e o r e m 3.1 in W i d m a n [7], g r a d u is ~-I-ISlder c o n t i n u o u s in ~5 a n d in

~ o , b u t its b o u n d a r y values on a~o n e e d n o t coincide. W e w r i t e au'/Ov a n d au"/3v for t h e c o n o r m a l d e r i v a t i v e s o b t a i n e d f r o m t h e v a l u e s o f g r a d u in r a n d t g ~ , resp. F u r t h e r , we p u t Aau/av ~ 3u'/Ov -- Ou"/Ov.

N o w d e f i n e

a~! = a ' J , w ~

a n d a n a l o g o u s l y for bl, a n d c,, for s > O, a n d w r i t e L , anti G, = G~ as before.

B y O/Ov, we shall m e a n t h e c o n o r m a l d e r i v a t i v e on ao~ associated w i t h L,, a n d A Ou/av~ is d e f i n e d analogously. W e also n e e d t h e a u x i l i a r y f u n c t i o n

aa~(x)\,

d e f i n e d for x e 309.

O N T H E A D J O I N T O ~ A l ~ E L L I P T I C L I l ~ E A R D I F F E R E N T I A L O P E R A T O R 163

F r o m L e m m a 3.3 in W i d m a n [7], it follows t h a t each second derivative

u~ i

^is

integrable in tg. Therefore, we can a p p l y Green's a n d Stokes's formulas, a n d for x E o) we o b t a i n

au"(z)

o = -

f a~(x, z)L~u(z)az -- f G~(x, z)--~- dSz +

.Q \ o) 0~o

Oa, Ow

a n d

au'(z) u(x) -~ -- f G~(x, z)L u(z)c~z + f G~(x, z ) ~ dS, --

(o 0(.o

_

f aG~(x, z)

d 3%, u(z)dS~ + fb(z)G~(x, z)u(z)clS~.

0o, &o

(Cf. M i r a n d a [6, pp. 12--20]). Adding, we get

s s au(z)

u(x) = - . ] G~(x, z)L~u(z)dz -~ . ] G~(x, z)A - ~ - v dSz,

(5.3)

OoJ

a n d in a similar w a y , this f o r m u l a can be p r o v e d for x C ~ 5 . N o w let s --> 0.

F r o m t h e construction of t h e Green's f u n c t i o n in Boboc a n d ~ u s t a t ~ [1], we conclude t h a t

G~(x, z) -~ O(H(x, z))

in ~ • 9 , u n i f o r m l y in s. Hence, it follows f r o m L e m m a 2 t h a t t h e first integral in (5.3) t e n d s to

f G(x, z)Lu(z)dz -~ O,

a n d so

f Ou(z)

u(x) = G(x, z)~ ~ czs~. (5.4)

0~o

F r o m (2.2) a n d (5.2) we see t h a t

au(z) aG~(z, y)

A _av

^--

_av~ ~o

for z r aco, so (2.4) a n d (5.4) i m p l y (5.1). I t follows f r o m T h e o r e m 3.1 in W i d m a n [7] t h a t

da~/dS C C(~

To prove t h a t

aG~(x, y)/O~,~

is n e g a t i v e on ao~, we n o t e t h a t

G~(x,

y), con- sidered as a f u n c t i o n of x, is L - h a r m o n i c a n d positive in ~o~{y} a n d zero on aco, L e t co 1 be a d o m a i n o b t a i n e d b y t a k i n g a w a y f r o m co a small ball centred a t y.

I f t h e coefficient c is nonpositive, t h e result follows d i r e c t l y f r o m T h e o r e m 3,IV in M i r a n d a [6], applied in co 1. F o r a r b i t r a r y c, we t a k e v as in t h e p r o o f o f L e m m a 2 a n d a p p l y t h e same t h e o r e m to

G/v

a n d t h e operator

u---> v-l.L(vu).

T h e o r e m 3 is proved.

164 P E T E R S J O G R E N

6. The Diriehlet problems for L and L*

For subdomains of t9 0 the Dirichlet problems for L and L* need not be uniquely solvable, but we have the following Fredholm type theorem.

THEOREM 4. Let ~ be a bounded C (1'~) domain with (5 c Y2 o.

problems

and

Consider the

L u = f in (o, u : O on ~(o, (6.1)

L*v = g in the sense of ~'(~0), v =q~ on a~o, (6.2) where f E CI~162 and f is continuous in (o, g E Lp(~o) for some p > n/2, and q~ is continuous on ~o). Then either (6.1) and (6,2) are both uniquely solvable, or else the corresponding homogeneous problems have the same finite number of linearly independent solutions us and v, i ~ 1 , . . . , m . I n the second case (6.1)is solvable if and only if

f f v ~ d x = O , i = 1, ^{@ @} m,

o )

and (6.2) if and only if

f

^{gu~dx ~-} q~ ~ dS = O, i = 1, . . ., m.

e o 0 o )

Proof. I f we let M be the operator L -- y and choose the constant y large enough, there exists a Green's function G of M in ~o. Then u is a solution of (6.1) if and only if M n = f - - ~ u in ~o and u = 0 on 0~o, which is equivalent to u(x) = y f S(x, y)u(y)dy -- f S(x, y)f(y)dy. (6.3)

J J

o )

Similarly, v solves (6.2) if and only if v ( y ) ~ - , f a ( x , y ) v ( x ) d x - - f a(x,

( o o )

f a G ( x , y) q)(x)dS~. (6.4) y)g(x)dx -- 0~

Ore

To this pair of integral equations, Fredholm's t h e o r y is applicable (see Miranda [6]).

Hence, the equations are either both uniquely solvable, or else the corresponding homogeneous equations have the same finite number of linearly independent solutions us and v~, i = 1 , . . . , m. These functions are t h e n also solutions of the homogeneous problems (6.1) and (6.2). Moreover, in the second case (6.3) is solvable if and only if

O N T H E A D J O I N T O F A N E L L I P T I C L I N E A R D I F F E R E N T I A L O P E R A T O R 165

f v,(x)dx f.(x,

y ) f ( y ) d y = O, i =- 1 , . . . , m , w h i c h is e q u i v a l e n t t o

f fv~dx = O, i = 1 . . . . , m .

t~or (6.4) t h e c o n d i t i o n o f s o l v a b i l i t y is

f f Ix, l Ixl x +

o) o~

f s OG(x, y)

o) 0 ( o

(6.5)

U s i n g t h e m e t h o d s o f W i d m a n [7 o r 8], o n e s h o w s t h a t OG(x, y ) / ~ v , = O(Ix - - y 11-~) f o r x E 0(o, y C o9. I t f o l l o w s t h a t (6.5) is e q u i v a l e n t t o

f

gu~dx + j ~ - ~ v d S = O, i == 1, . . . , m ,

o) 0 o )

a n d t h e p r o o f o f T h e o r e m 4 is c o m p l e t e .

References

1. BoBoc, N., & MUSTATA, P., Espaces harmoniques associds aux opgrateurs d,'ffgrentiels tindaires du second ordre de type elliptique. Lecture Notes in Mathematics 68. Springer-Verlag, Berlin-Heidelberg-New York, 1968.

2. BRELOT, M., Une a x i o m a t i q u e g@n@rale du probl~me de DiricMet dans les espaces localement compacts. Sdminaire de Thdorie du potentiel, Ire a n n i e (1957).

3. BnELOT, M., A x i o m a t i q u e des fonctions harmoniques et surharmoniques dans un espace localement compact. Sdminaire de Thdorie du potentiel, 2e a n n i e (1958).

4. HERV~, R.-M., Reeherches axiomatiques sur la thdorie des fonctions surbarmoniqucs et du potentiel. Ann. Inst. Fourier, 12 (1962), 415 -571.

5. LITTMA~, W., Generalized subharmonic functions: monotonic approximations and an i m p r o v e d m a x i m u m principle. A n n . Sc. Norm. Sup. Pisa, 17 (1963), 207--222.

6. MIRANDA, C., Partial Differential Equations of Elliptic Type. Second R e v i s e d Edition.

Springer-Verlag, Berlin-Heidelberg-New York, 1970.

7. WID)IA~, K.-O., Inequalities for the Green function a n d b o u n d a r y c o n t i n u i t y of the gradient of solutions of elliptic differentiM equations. Math. Scand. 21 (1967), 17--37.

8. --~-- Inequalities for Green functions of second order elliptic operators. R e p o r t 8 (1972).

Link6ping University, Dept. of Math.

Received March 17, 1972 Peter Sj6gren

D e p a r t m e n t of Mathematics University of G6teborg F a e k

S-402 20 G6teborg 5 Sweden