A rigidity result for overdetermined elliptic problems in the plane

Let $f:[0,+\mathrm{\infty })\to \mathbb{R}$ be a (locally) Lipschitz function and $\mathrm{\Omega }\subset {\mathbb{R}}^2$ a $C^{1,\alpha }$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem $$\{\begin{array}{ll}\mathrm{\Delta }u+f(u)=0& in\mathrm{\Omega }\\ u=0,\frac{\mathrm{\partial }u}{\mathrm{\partial }\vec{\nu }}=1& on\mathrm{\partial }\mathrm{\Omega }\end{array}$$ we prove that $\mathrm{\Omega }$ 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.