On the adjoint of an elliptic linear differential operator
and its potential theory
PETER SJOGREN
University of G6teborg, Sweden
1. Introduction
In her thesis [4], R.-M. Herr6 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 that of L*. In 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 Littman [5]. He shows that
they 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 type theorem for the I)Mchlet problems for
15 and L* in a given domain.
The author wishes to thank 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. Preliminaries
Suppose we are given a domain ~20 C R n, n > 2, a n d a differential operator
L u = aiJuij -~- b~u~ + cu,
154
PETER SJOGREN
defined in D 0. W e assume t h a t a (i ~ a jl, t h a t L is elliptic in Y20, and t h a t t h e
coefficients are locally ~-HSlder continuous, for some ~ w i t h 0 < ~ < 1. As
I-Ierv6 shows, we can let the C (2) functions u satisfying Lu = 0 be t h e h a r m o n i c
functions in Brelot'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 Brelot [2, 3]. I n this
w a y we obtain a p o t e n t i a l t h e o r y satisfying Brelot's Axioms 1, 2, and 3' (see I-Ierv6
[4]). W e write ~>L-harmonic~>, ~>L-potentiab>, etc. when we refer to concepts of this
theory.
To m a k e possible the c o n s t r u c t i o n of a n adjoint t h e o r y , we m u s t limit ourselves
to a d o m a i n D C z90 where a positive L - p o t e n t i a l exists. D e p e n d i n g on the coefficient c of the operator, Y2 m a y be chosen in the following way, as shown b y
l-[erv6 [4, p. 562].
1. I f c < 0
and 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 which is a n admissible
D.
F r o m now on we fix such an .(2. I n the sequel o), 0~1, 9 9 9 will always be subdomains
of z9 or z90.
W e follow the n o t a t i o n of Brelot and K e r v d and write / ~ for the b a l a y a g e d
f u n c t i o n of 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 function v a n d a set E c ~9. I f 05
is c o m p a c t and contained in $2, a n d f is defined a n d continuous on ~9, t h e n the
solution of the Dirichlet p r o b l e m for L in ~o with b o u n d a r y values f is d e n o t e d
b y H y . A point x 0 E ~ is called L-regular for ~o if Hy(x)-+f(Xo) as x ~ Xo,
x C o9, for a n y continuous f. As shown b y Hervd, x 0 is L-regular if a n d only if it
is regular 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] constructs an L - p o t e n t i a l Py in D with s u p p o r t {y}
and such t h a t the m a p p i n g (x, y)--~ Pz(x) is continuous for x, y C D, x # y.
The s u p p o r t of a p o t e n t i a l P is d e f i n e d as the c o m p l e m e n t of the largest open set
in which P is harmonic. The 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 represented as
P(x) =~f P:~(x)d#(y)
(2.1)
for a u n i q u e positive measure # in s The s u p p o r t of # coincides with the s u p p o r t
of the 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 the Green's function
is given b y
G+(x, y) = _P,(x) -- H77(x).
The f u n c t i o n G ~ can be used to solve a b o u n d a r y value problem, as follows. I f f
is continuous in 05 a n d locally KSlder continuous in ~, the u n i q u e solution of t h e
problem
Lu=f
in ~,
u=0
o~1 0oJ
ON
THE
ADJOINT
OF
AN
ELLIPTIC
LINEAR
DIF]~IERENTIAL
OPERATOR
155
is given by
u(x) - _ f Go(x, y)f(y)dy.
(2.3)
(o
For this see Miranda [6].
To construct the adjoint potentials, Kerv~ uses the concept of completelydetermining 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. In the adjoint theory 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. From 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 any 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
f %(x)dx = 1.
We let H(x, y) be the fundamental solution of the operator aq (y ) a2/ Oxiaxi
defined by
]56
PETE~ SJOG~EN
1
H(x, y) --
/ ~ : w . . log (~a~i(y)(xi -- yi)(xi -- yj))-'/' if n = 2,
2~ Y A ( y )
(n
1
2-~onV A ( y ) (~aq(y)(x, -- y,)(xj -- yi)) (2-n)/2 if n > 2.
H e r e ~o= is t h e area of t h e u n i t sphere in R", and A(y) a n d
d e t e r m i n a n t a n d the inverse, resp., of the 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 with 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 of the distribution solutions of
L*u=O
in a d o m a i n coCs
L]~MM~, 1. I f u E L~oo@) satisfies L*u : 0 in the sense of %'@),
coincides a.e. in co with a continuous function.
Let
then u
Proof. Assume 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.
U be a r e l a t i v e l y c o m p a c t open subset of co, and pick y E U so t h a t
f
lu(x) -- u(y)idx = o(~ n)
(3.1)
B~o
as e --> 0, where ~ , is the ball {x: Ix - - y[ < ~}. Choose ~0 E ~(eo~/~el2) equal
to 1 in U ~ B ~ . I n B(~B(,/2 we let the derivatives of ~0 satisfy ~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 of ~ a n d y.
Since x - + 7(x)F(x, y) is a C (2) function, we conclude t h a t
f
:But L~F(x,y)----O
~,
u(x)L,~(q~(x)F(x, y))dx = O.
for x 4 : y ,
so
B~
(3.2)
f .
y)dx + ]
Be
We know that
y))dx = o.
eo~U
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. Miranda [6, pp. 18--20]). H e n c e , (3.1) implies t h a t the t h i r d t e r m in (3.2) is
o(1) as e - + O, a n d the first t e r m equals
ON THE
ADJOINT
OF
AN
ELLIPTIC
LINEAR
DIFFERENTIAL
OPERATOR
fu(y)aq(y)qaii(x)H(
, y)dx + o(1).
157
(3.3)
B9
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 second t e r m in (3.2) equals
t h e s a m e expression, e x c e p t for a f a c t o r - - 2. ]~ut H is a f u n d a m e n t a l solution
of a~J(y)O~/Ox~Oxi, a n d ~ 0 - 1 c a n be considered 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 the i n t e g r a l in (3.3) equals u(y). L e t t i n g e--> 0, we get
u(y) =
f
y))d
o,i
(3.4)
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 new 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 of ~ ' ( e o •
since for ~ C ~(o~•
we h a v e
f
f f
+f
o)
f
xjx=o
(o
T h u s we can m a k e v a n d hence 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 complete.
Remark. As we shall see later, u is in f a c t I~51der continuous, a n d t h e r e f o r e
t h e p r o o f of (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 =
in the sense of ~'(a~) i f and only i f u coincides a.e. in a~ with a function which
Z*-harmonic in co. Similarly, u is locally integrable in co and satisfies L*u ~
in the sense of ~5'(co) i f and only if u coincides a.e. in a~ with a function which
L*-superharmonic in o~.
0
is
0
is
_Proof. S u p p o s e u is L * - h a r m o n i c in co, a n d let ~0 E ~(eo). T a k e ~o1 a n d o~2
such t h a t
supp~oC~IC~IC~2C~2Co~,
and let (02 be bounded
and of class C 0'~). If u ~ O, define
v =
(R*~'~/(o s
x--u
which is the balayaged function of u in the Z*-potential theory in 0) 2. Then v
is a n L * - p o t e n t i a l in co2 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 co1. 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 can be r e p r e s e n t e d
as in (2.1). I n this case we o b t a i n
]58
PETER SJOGRE~
[.
v(y) = J G~(x, y)d#(x),
(3.5)
for some positive measure # with s u p p o r t contained in 0%.
T h e r e exists a positive L * - p o t e n t i a l P * in [2 and t h u s also a positive 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 therefore in % a difference b e t w e e n two positive 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 to (3.5) for a n y u, b u t where # need not be
positive.
Because of (2.3), the f u n c t i o n ~p satisfies
y2(x) = -- f G~
y)Ly~(y)dy.
(3.6)
Now (3.5--6) and F u b i m"' s t h e o r e m i m p l y t h a t
f uLFdx
-
-
-
-
f
= o,
and thus L *u ~ O.
Conversely, suppose u is locally integrable in ~) and satisfies L*u = 0. W e
take a c o m p l e t e l y d e t e r m i n i n g ~o1 w i t h ~51 C ~o a n d a p o i n t y E co1. P u t
f(x) = f P~(x)d(e,(z) -where ex is the measure 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 define
and
[.
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 outside
the supports of g~ a n d h~ and t h a t f ~ - - ~ f = 0 in ~9~51 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, IV in Miranda [6], it follows t h a t
the first- and second-order derivatives of f~ t e n d to 0 in ~ r
as e --> 0, unif o r m l y on c o m p a c t subsets. T a k e ~ E ~(eg) equal to 1 in a n e i g h b o u r h o o d U of
~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
ADJOINT
O F A~'q : E L L I P T I C LLNEAI:~ ] ) I F F ~ ] R E ~ T I A L
OrERATOI~
159
or equivalently,
f uL(~fJx-- f ug~dx§ f uh~dx=O.
B y L e m m a 1 we can assume t h a t
u is continuous. L e t t i n g
u(y) =
e - ~ 0, we get
f %d(y;)~,
a n d the first p a r t of T h e o r e m 1 is proved.
The p r o o f t h a t L*u ~_ 0 for an L * - s u p e r h a r m o n i c function u is quite similar
to the corresponding p r o o f for L * - h u r m o n i e functions a n d is omitted.
Conversely, suppose t h a t L*u is a negative measure --/~. 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. Because of (2.3), a n y yJ C 95(0~1) satisfies
f uL~c~x=f f r176
where the double integral is a b s o l u t e l y convergent. The first p a r t of T h e o r e m 1
now shows t h a t the f u n c t i o n v defined b y
is equal to un L * - h a r m o n i c function ~.e. in (o~. T h e integral in (3.7) represents
a n L*-potenti~l, so u coincides with 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. The 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. Suppose 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 the 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)- a'J(y))~ (~(z)g(~, y))dx+
§
f
f
u(x)(b~(x) ~x~ (q~(x)g(x, y)) § c(x)?(x)H(x, y))dx.
(4.1)
160
!~E T E 1~ S J 0 GI%EN
,Now t a k e y a n d z E K
with ~=
[z--yl
so small t h a t
B -- {x: Ix -- yI _< 2e} c V,
a n d consider (4.1) and the corresponding f o r m u l a for u(z).
of the a ~i it follows t h a t
F r o m the 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]-~),
and
H~,~(x, y) -- Hx~,~(x, z) = O ( ~ l x -- yl -'~ + ~Ix--
yl-~-'~),
if x ~ B. Since u is continuous, these estimates easily i m p l y t h a t the difference
b e t w e e n the first t e r m s in t h e formulas for u(y) a n d u(z) is 0(~ ~) as Q --> 0,
and the same is t r u e for t h e second a n d f o u r t h terms.
The 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 the integrals
over B in this expression a n d in t h e corresponding expression for u(z) are 0 ( ~ ) .
The difference b e t w e e n t h e integrals over ( 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 +
(4.2)
U~B
-l- f
u(x)(a'i(x) -- a ;i(Y""
))(Hx~, 7'(x ' y) -- Hx,v(x' z))dx.
U\B
H e r e the second t e r m is 0 ( ~ ) , and the first t e r m is O(e ~ log 1/~). Hence, 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 estimate. 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
which is b o u n d e d b y
o+=>.
f
z<
+ o(+> o(+>.
U\B
T h e case n ~ 2 now follows as in the p r o o f of L e m m a 1, a n d T h e o r e m 2 is proved.
ON
TIIE ADffOII~T
OF
AN
ELLIPTIC
LII~EAI~ D I F F E I ~ E N T I A L
OI~EI~ATOI~
]61
Remark. I t is clear t h a t the e x p o n e n t cr is best possible. I n a similar way, o n e
can p r o v e r e g u l a r i t y properties of solutions of n o n h o m o g e n e o u s equations L * u -~ f.
F o r example, if f E LF~or t h e n u e C~~~) if 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 more regular functions, a n d
s t a r t b y e x a m i n i n g how the Green's f u n c t i o n varies with the coefficients. F o r
s --> 0, assume t h a t
25~u =- a~quq ~- bsui
i + c~u
is an o p e r a t o r w i t h coefficients in C(~
for some ~ ~ 9 .
corresponding Green's function, w h e n e v e r it exists.
L e t G :~ be t h e
LEM~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 uniformly i n
r as s --> O, and analogously for b~ and c~. Then G~(x, y) ~ G~
y) uniformly
on any compact subset of o)• ~ which is disjoint with the diagonal.
Proof. Since t h e r e is a positive 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 which is positive a n d satisfies L v < 0 in ~. Since t h e n also L v ~ 0
if e is small enough, GO~ exists for such e.
I n B o b o c a n d Mustat~ [1, Chap. 4], t h e r e is a c o n s t r u c t i o n of the Green's f u n c t i o n
for L in the f o r m o f an L - p o t e n t i a l whose s u p p o r t is t h e p o i n t y. F r o m this cons t r u c t i o n it can be 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 disjoint c o m p a c t subsets of w, and 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 and Mustat~ assume t h a t the coefficient c is nonpositive, b u t
f this is not the case, we can a p p l y t h e i r p r o o f to the o p e r a t o r Me: u --> v-lL~(vu),
n which the coefficient of u is negative. T h e n the Green's f u n c t i o n of L~ is given b y
i
v(x),
G~'(x, y) -~ G ~ ( x , y ) ~ ( ~
a n d the 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 the solution of the p r o b l e m
L~u~-~f
in 04 u ~ = 0
on
~o.
I f we define u similarly b y means of L, t h e n the 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 Miranda [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,
the l e m m a t h e n follows if we choose f suitably and use the e q u i e o n t i n u i t y of 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
Then ~
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 topological reasons each of
its c o m p o n e n t s separates R". I t follows t h a t &5 ---- &o, which m e a n s t h a t at each
point of this manifold, o) lies on one side and R"~o5 on the other. I t is easily
shown 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 the p r o o f of this fact for open
balls in H e r v 6 [4, p. 565]).
I f x E aco, we let n~ ~ (cos ~1,. 9 cos ~,~) be the exterior unit n o r m a l of &o
a t x. T h e conormal d e r i v a t i v e at x is d e f i n e d b y a/av ~ aq(x) cos ~jO/Ox~. The
area measure 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~;
dS
aG~(x, y)
--
(5.D
o~
for x E &o. This 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 assume t h a t ,(2 is b o u n d e d
and of class C (~';~), a n d t h a t the coefficients of L are defined in a slightly larger
domain, so t h a t L e m m a 2 applies to f2. T h e n we can write Py(x) = G(x, y). P u t
u = ~?,~,
(5.2)
which is an L - h a r m o n i c function in ~Q~ a(o. Since all the points of &o are regular,
u equals Py in Y2~o9 a n d H ~Py in ~o, a n d u is continuous in ~ a n d zero on 0tg.
Due 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 continuous in ~5 and in
~o,
b u t its b o u n d a r y values on a~o need not coincide. W e write au'/Ov a n d
au"/3v for the conormal derivatives o b t a i n e d f r o m t h e values of grad u in r
and tg~,
resp. F u r t h e r , we p u t Aau/av ~ 3u'/Ov -- Ou"/Ov.
Now define
a~! = a'J,w~
a n d analogously for bl, and c,, for s > O, a n d write L, anti G, = G~ as before.
B y O/Ov, we shall m e a n the conormal d e r i v a t i v e on ao~ associated w i t h L,, and
A Ou/av~ is defined analogously. W e also need the a u x i l i a r y f u n c t i o n
aa~(x)\,
defined for x e 309.
ON T H E
ADJOINT
O ~ Al~ 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
OPERATOR
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 apply Green's a n d Stokes's formulas, a n d for
x E o) we obtain
au"(z)
f a~(x,z)L~u(z)az-- f G~(x,z)--~- dSz +
o=-
.Q \ o)
0~o
Oa,
Ow
and
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. Miranda [6, pp. 12--20]). Adding, we get
u(x) = -
s
s
au(z)
. ] 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 formula can be proved for x C ~ 5 .
Now let s --> 0.
F r o m the construction of the Green's function 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 from L e m m a
2 t h a t the first integral in (5.3) tends to f G(x, z)Lu(z)dz -~ O, a n d so
f
u(x) =
G(x, z)~ ~Ou(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)
A av
-
-
aG~(z, y)
av~
~o
for z r aco, so (2.4) and (5.4) i m p l y (5.1). I t follows from Theorem 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 negative on ao~, we note t h a t G~(x, y), considered as a function of x, is L-harmonic a n d positive in ~o~{y} and zero on aco,
L e t co1 be a d o m a i n obtained b y t a k i n g a w a y from co a small ball centred at y.
I f the coefficient c is nonpositive, the result follows directly from Theorem 3,IV
in Miranda [6], applied in co1. F o r a r b i t r a r y c, we t a k e v as in the p r o o f o f L e m m a 2
a n d a p p l y the same theorem to G/v a n d the operator u--->v-l.L(vu).
Theorem 3 is proved.
164
P E T E R SJOGREN
6. The Diriehlet problems for L and L*
For subdomains of t90 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 Y2o. Consider the
problems
Lu=f
in (o, u : O
on ~(o,
(6.1)
and
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
ffv~dx=O,
i=
1,
@
@
m,
o)
and (6.2) if and only if
f
gu~dx ~-
eo
q~~ dS = O, i = 1, . . ., m.
0o)
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.
J
J
(6.3)
o)
Similarly, v solves (6.2) if and only if
v(y)~-, f
(o
, y) q)(x)dS~.
a ( x , y ) v ( x ) d x - - f a(x, y)g(x)dx -- f a G ( x0~
o)
(6.4)
Ore
To this pair of integral equations, Fredholm's theory 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 then also solutions of
the homogeneous problems (6.1) and (6.2). Moreover, in the second case (6.3) is
solvable if and only if
ON
THE
ADJOINT
OF
AN
ELLIPTIC
f v,(x)dxf.(x,
y)f(y)dy
LINEAR
=
O,
DIFFERENTIAL
i =- 1 , . . . ,
OPERATOR
165
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)
(6.5)
o~
f
s OG(x, y)
o)
0(o
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
o)
and the proof of Theorem
~ - ~ v d S = O, i == 1, . . . , m,
0o)
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 axiomatique 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., Axiomatique 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
improved m a x i m u m principle. Ann. Sc. Norm. Sup. Pisa, 17 (1963), 207--222.
6. MIRANDA, C., Partial Differential Equations of Elliptic Type. Second Revised Edition.
Springer-Verlag, Berlin-Heidelberg-New York, 1970.
7. WID)IA~, K.-O., Inequalities for the Green function and boundary continuity 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. Report 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
Faek
S-402 20 G6teborg 5
Sweden