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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