A rigidity result for overdetermined elliptic problems in the plane

Abstract

Let $f:[0,+\infty )\to \mathbb{R}$ be a (locally) Lipschitz function and $\Omega \subset {\mathbb{R}}^2$ a $C^{1,\alpha }$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem $$ \left\{\begin{array} {ll} Δu + f(u) = 0 & \mbox{in }\; Ω \\ u= 0\, \, \, , \, \, \, \frac{\partial u}{\partial \vecν}=1 &\mbox{on }\; \partial Ω\end{array}\right. $$ we prove that $\Omega$ is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.

28 pages, 7 figures