An overdetermined problem for an elliptic equation (Q710657)

Summary: We consider the following overdetermined boundary value problem: \(\Delta u + \lambda u + \mu = 0\) in \(\Omega , u = 0\) on \(\partial \Omega \) and \(\partial u/\partial n = c\) on \(\partial \Omega \), where \(c \neq 0, \lambda \) and \(\mu \) are real constants and \(\Omega \subset \mathbb R^{2}\) 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.