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