CHAPTER 1 An analytic approach to the equation Au = u2 1.1. Parametrization near the boundary of the domain We consider PDE's in a smooth and bounded domain D in Rd. Unless otherwise specified, we will always assume that d ^ 3. For technical reasons, we restrict ourselves to domains of class C4, even though many results hold under more general assumptions. The reason for this restriction is that we need a sufficiently smooth parametrization near the boundary for which the distance to the boundary is one of the coordinates. Let us describe this parametrization, whose existence is proved in the Appendix, Section A.l. It is inspired by the one used by Le Gall in [LG95], Section 3.2. Let D be a bounded domain of class C4 in Rd. We denote by B(x,r) the ball of radius r centered at x. For every x £ D, we put p(x) = dist(x,dD). There exist e 0, yi,...,yn G dD and n mappings pi : B(yi,2e) Rd,...,^n : B(yn, 2e) Rd such that the following properties hold: ( l ) a D C % , e ) U . . . U % , £ ) . (2) For every 1 ^ i ^ n, there exists a neighborhood Oi of 0 in Rd such that ipi can be extended to a C3-diffeomorphism from a neighborhood of J5(^,2e) onto a neighborhood of 0{. Moreover, ifi{B(yil2e)) = Oi and ^ ( D n % , 2 £ ) ) = ffnOi, where H stands for the half-space H = {(r,s) r 0, s £ R ^ 1 } . (3) For every 1 ^ i ^ n and every x £ B(yi, 2e), let us set ifi(x) = (ri(x),si,i(a ),...,Sd-i,i(^)). If x £ DnB(yi,2e), then Ti(x) = p(x). In addition, for every x £ B(yi, 2e), |Vr^(x)| = 1 and Vri(x).Vsj^(x) 0, for every 1 ^ j ^ d 1. Let us examine some consequences of these properties. For every 1 ^ i ^ n, put r\i ip~x. If / is a function of class C1 on D fl 5(i/j, 2e), then or/i(r,s) = (V/.Vp)or/2(r,s) = (r,s), l
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