(4) a ij (x) ~, ~j (a" u,,b = 0 (1) NOTE ON A LEMMA OF FINN AND GILBARG

(1)

N O T E O N A L E M M A O F F I N N A N D G I L B A R G

BY

L. E. P A Y N E and H. F. W E I N B E R G E R University of Maryland (1)

I n t h e preceding paper [1] it was shown b y F i n n a n d Gilbarg t h a t if u satisfics t h e elliptic differential e q u a t i o n

(a" u,,b = 0 (1)

in a n n-dimensional d o m a i n D c o n t a i n i n g t h e exterior St, of a sphere of radius r 0 a n d if t h e coefficients a ~i a n d t h e solution u b e h a v e suitably a t infinity, t h e n

A (r) = f a ~ J u , i u j d V (2)

S r

exists for r > r 0 a n d

where ~ = m i n [ ( n - 2), 2 I / n - 1].

A (r) ~< A (ro) (3)

I n t h e case of L a p l a c e ' s equation, it is easily SCell t h a t (3) holds with ~ replaced b y n - 2 , which is greater t h a n 2 for large n. Thus, (3) is n o t sharp.

W e shall replace (3) b y a sharp a s y m p t o t i c estimate u n d e r t h e a s s u m p t i o n t h a t a ij approaches t h e u n i t m a t r i x at infinity. As is p o i n t e d o u t in [1], if a ij approaches a n y positive definite matrix, one can m a k e this limit t h e u n i t m a t r i x b y means of a coordinate t r a n s f o r m a t i o n .

We define t h e t w o f u n c t i o n s

a ij (x) ~, ~j

a 0 ( r ) =

... ~

inf l,s

~: ~,~ (4)

Ixl_>r

(1) This research was supported by the United States Air Force through the Air Force office of Scientific Research of the Air Research and Development Command under contract No.

AF 18(600)-573.

(2)

298 L . E . P A Y N E A N D H . F . W E I N B E R G E R

and

a 1 (r) = ~ sup 1

~atJxtx t,

⁽⁵⁾

r I x l = r ~,t

where Ix[ is the length of the vector x.

We assume t h a t a tj approaches the unit m a t r i x in the sense t h a t % a n d a 1 approach 1 as r--> oo.

L e t DR (r) be the Diriehlet integral of u over the annular region between spheres of radii r and R > r. Clearly DR (r) increases with R, and b y (4)

D (r) = lim

DR (r) <~

A (r)/a o (r). (6)

We now write

R R

8(1) r r 8(1)

(7)

where S(1) is the surface of the unit sphere, d ~ its surface element, a n d g r a d a u the projection of the gradient of u on the unit sphere.

3 0 u 2

, grada u,'=r3[, grad u, - (~r) ] 9

⁽⁸⁾

B y Schwarz's and Wirtinger's inequalities / ' ~ u \3

s(1) f d~

R

r S(1) 8(1)

>~(n-2)rn-2f[u(R)-u(r)]2d~+(n-1)f[fu~d~-

s(1) r S(1) S(1)

(9)

where con is the area of S(1). Since b o t h integrals on the right are non-negative, each is separately bounded b y

Dn(r)

and hence uniformly in R b y

D(r).

B y consid- ering the first integral for r sufficiently large and R > r , we see t h a t

u(r)considered

as a function on the unit sphere converges in the m e a n square sense as r-->oo. This implies t h a t the limit of

f u(r)d~

exists. B y adding a suitable constant to u, we

8(1)

m a y m a k e this limit zero. We suppose this to have been done.

(3)

NOTE ON A LEMMA OF FINN AND GILBARG

299

T h e second integral on t h e right of (9) m u s t converge, a n d hence its i n t e g r a n d m u s t a p p r o a c h zero. T h u s we find

lim

f u2(R)d~=O.

⁽¹⁰⁾

R - ~ S(1)

Neglecting the second term in (9) and letting R-->c~ shows that

D(r))(n_2)r._ 2 f u2(r)da=

(n=2)r

f u'dS,

(11)

S(1) S(r)

where S(r) is the surface of the sphere of radius r.

I t is easily seen f r o m S c h w a r z ' s inequality t o g e t h e r with (10) a n d t h e Dirichlet integrability of u t h a t

lim

(~ ua~JuivjdS=O.

(12)

R-~Oo J 8( R) Hence, using S c h w a r z ' s inequality

, , O A (r)

[A(r)]2=[ f ua"u.,v, dS] <~al(r) f a"u,uadS f u'dS= -al(r)--a--~- f u~dS.

.~(r) S(r) Zcr)

^S(r)

Using (11) and (6) gives

[A (r)]~< -rax(r) OAA(r). (13)

( n - 2) a0 (r) ~ r This m e a n s t h a t for r > r o

r

j e a - h ~ ) ~0 9 (14)

ro

W e a k e n i n g this inequality, we find

.ao(Q)~

A (r) ~< A (ro) ~>,. t~,(~)~.

& g

If' we assume t h a t

ao(~)/ax(s

as Q - + c ~ , t h e e x p o n e n t in (15) is arbitrarily close t o n - 2 for sufficiently large r o.

I n general, (14) can also be w r i t t e n in t h e f o r m

/r ^~--2

r

A(r)<A(ro) t: ) exp{(n--2)f(1--ao/ax)de/e} ⁰⁶⁾

T$

which clearly displays the effect of the deviation of ao/a 1 from one.

R e f e r e n c e

[1]. R. FINN & D. GILBARG, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations.

Acta Math.,

98 (1957), 265--296.