1.2. BASIC FACTS ABOUT LINEAR ELLIPTIC PDE'S 5 Here Ny(= Vp(y)) denotes the inward-pointing normal to dD at y. Note that hv+v hv + hv. Besides, z/ $J 2/ if and only if hv ^ / v . If vn G A/- and z^n | v G jV, then hVn | /i^. 1.2.2. Convergence properties of solutions. Let us recall that p(x) = dist(x, dD), for every x G D. Let i/, G C2(D) such that Aw = f in D. Then, it is classical that (1.3) sup{p\Vu\) ^ C(D)(sup \u\ + sup(p 2 |/|)). (see [GiT]) D D D Let us state a proposition P R O P O S I T I O N 1.5. Let (un)n^o be a sequence of functions in C2(D). Put fn Aun. If the sequences (un)n^o and (fn)n^o are uniformly bounded on the compact subsets of D, then there exists a subsequence of (un)n^o which converges uniformly on the compact subsets of D. Let u be the limit function of one such subsequence. If, in addition, (fn)n^o converges pointwise to a function f in CX(D), then Au = / . PROOF. The proof is very classical. It is left to the reqder who cqn refer to [GiT] or [Msl], Chapter 1. 1.2.3. A n auxiliary function. For every x G D, let us define p(x) = GD(l)(x) = / gD(x,y)dy. JD The following lemma is proved in the Appendix, Section A.3.1. L E M M A 1.6. The function p belongs to C4(D). Moreover, there exists a constant C = C(D) 0 such that, for every x G D, (1.4) C-1p(x)^p(x)^Cp(x). Besides, Ap = - 2 ^ 0. We state another useful lemma concerning p which "compares" the gradients of p' and p near the boundary. We recall that p' G C3(D) and that the properties p' p and |V//| = 1 hold in a neighborhood of dD. L E M M A 1.7. For every x G D, let us set r,{x) = V(p)(x).Vp'(x) and let T(x) be defined by Vp(x) = rj(x)Vpf(x) + T(x). There exists C = C(D) such that, for every x G D, \q(x)\ ^C and \T(x)\ ^Cp(x). (See the Appendix, Section A.3.2, for the proof). We use the auxiliary function p to prove a technical lemma. L E M MA 1.8. Assume that Au ^ 0 in a domain D\ C D. Assume also that there exists C\ 0 such that limsup u(x) ^ C\p(y) x—±y, x£Di
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