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.
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