An Overdetermined Problem for an Elliptic Equation

Abstract

We consider the following overdetermined boundary value problem: ∆u + λu + µ = 0 in Ω, u = 0 on ∂Ω and ∂u/∂n = c on ∂Ω, where c = 0, λ and µ are real constants and Ω ⊂ R 2 is a smooth bounded convex open set. We first show that it may happen that the problem has no solution. Then we study the existence of solutions for a wide class of domains.2010 Mathematics Subject Classification: 35J05, 35R30. Keywords: overdetermined elliptic boundary value problem, Schiffer property, Schiffer conjecture. §1. IntroductionLet Ω ⊂ R 2 be a smooth bounded simply-connected open set. We consider solutions of the following overdetermined elliptic boundary value problem:where λ, µ and c are real constants and ∂/∂n is the outward normal derivative.If c = 0 and µ = 0 (or equivalently µ = 1) we get as a special case Schiffer's problem (Yau [18, p. 688, problem 80]). If µ = 0 and c = 0 the problem was posed by Berenstein [1].In 1981 Williams [16] proved that if ∂Ω is Lipschitz and (1.1)-(1.3) has a solution for c = 0 and µ = 1, then ∂Ω is real analytic.