An Overdetermined Problem for an Elliptic Equation

Abstract

We consider the following overdetermined boundary value problem: Δ_
u_ + λ_u_ + 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 ⊂ ℝ2 is a smooth bounded convex open set. We first show that it may happen that the problem has no solution. Then we study the existence of solutions for a wide class of domains.