An Overdetermined Problem for an Elliptic Equation
Résumé
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.