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.
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