2 1. ANALYTIC APPROAC H

for every (r, s) G H HOi. Besides, note that

{x G D ; p(x) c} C Bfoi, 2c) U ... U B(yn, 2c).

In particular, we see that p belongs to

C3

({x e D ; p(x) ^ e}), |Vp(x)| = 1 if

p(x) ^ e and, for every 0 s ^ c, the subdomain Z)s of D defined by

£s = {x G £ ; p(x) s}

is of class

C3.

From now on, we fix a function p' G C3(D) such that p'(#) = p(x) if p(x) ^ c.

Unless otherwise specified, we always use the notation and parametrization in the

domain D as described above.

1.2. Basic facts about linear elliptic PDE's

We use the following version of the maximum principle.

THEOREM

1.1 (Maximum principle, see [GiT], Theorem 3.5). Let u G

C2(D)

and let a be a nonnegative function in D such that

Au ^ au in D.

If, for every y G dD,

then

limsup u(x) ^ 0,

^y, x e D

u ^ 0 in D.

Let u, v G

C2(D)

and let -&- denote the outer normal derivative on dD. Then,

by virtue of Green's formula,

(1.1) jj?*u-u*v) = jBD(v%-u%)

where we integrate with respect to the Lebesgue measure on D and dD respectively.

The Green function g (of Brownian motion) of

Rd

is defined on

Rd

x

Rd

by

g(x,y) =

( d

_

2

) 7 T d / 2 \ x ~ y \ 2 ~ d [ i x ^ y

K

g(x,x) = +oo

where T stands for the Euler-function. For every y G

Md,

g(-,y) is harmonic in

Rd\{y}.

There exists a unique continuous function

gD:DxD—*[0,oc}

such that for every y G D, the function hy defined on D\{y} by hy(x) = g(x,y) —

grj(x,y) can be extended to a nonnegative harmonic function belonging to

CX{D)

and satisfies

hy(x) — g(x,y) for every x G dD.

The function go is called the Green function of D.

It is symmetric in the sense that gjj{x,y) = goiyj^x) for every x,y G D. For every

x G D and every z G dD, we have

gD(x,z) = 0.