Some Results Related to Schiffer's Problem

Abstract

We consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain Ω with analytic boundary ∂Ω having at least one bounded connected component {−Δu=g(u)in Ω, ∂u/∂ν=0 and u=c on ∂Ω, where c is a constant. When g(c)=0 the constant solution u≡c is the unique solution. For g(c)≠0, we show that the boundary is a circle if and only if the problem admits a solution that has constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.