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 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.
Cite this article
Robert Dalmasso, An Overdetermined Problem for an Elliptic Equation. Publ. Res. Inst. Math. Sci. 46 (2010), no. 3, pp. 591–606DOI 10.2977/PRIMS/19