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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