1.2. BASIC FACTS ABOUT LINEAR ELLIPTIC PDE' S 3

Besides, for every y G D, #£(•?/) is harmonic in D\{y}.

Let a denote Lebesgue measure on 3D. Then, for every u G

C2(D),

(1.2) u(x) = - - / u{y)^-(x,y)a(dy) - - gD(x,y)Au{y)dy, x € £.

(See [GiT], Section 2.4). Formula (1.2) is often referred to as Green's representation

formula.

Assume that / is bounded or that / is a nonnegative Borel function on D. Define

for every x G D

GD{f)(x)= / gD{x,y)f(y)dy.

JD

The operator GD is called the Green operator of D.

If / is bounded and belongs to CX(D) (f is locally Hlder-continuous), Gn(f) be-

longs to C2(D) and is the unique solution of the Dirichlet problem

(Au = - 2 /

[U\dD = 0.

Here the notation

"U\QJJ

= 0" means that

u(x) 0,

for every 7/ G 3D. (See for instance [Mi]).

Bounds for the Green function. The following global estimates on the Green

function and its derivatives are vital for our purposes. We use the notation intro-

duced in Section 1.1 and state a first classical lemma.

LEMMA 1.2 (see [GrW], Theorem 3.3). There exists a constant C = C(D)

2

such that for every (x,y) G D , x ^ y,

(i)

gD(x,y)^C\x-y\2-d.

(n)

9D(X,V)

^Cpix^x-yl1-4.

(in)

9D(X,V) ^ Cp{x)p(y)\x - y\~d.

(iv)

|V„ff

D

(:r,3/)|C|s-!/| 1 -' 1 .

(v)

\Vy9D(x,y)\^Cp(x)\x-y\-d.

Recall the properties of the function p' defined at the end of Section 1.1:

(V G C3(D),

| Vp' I = 1 in a neighborhood of 3D in D

[p7 = p in a neighborhood of 3D in D.

We state a lemma which provides more precise estimates on the gradient of the

Green function near the boundary. A detailed proof is given in the Appendix,

Section A.2.1.