An Overdetermined Problem for an Elliptic Equation
Robert Dalmasso · Publications of the Research Institute for Mathematical Sciences · 2010
We consider the following overdetermined boundary value problem: Δu + 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.