Overdetermined problems in unbounded domains with Lipschitz singularities

Abstract

We study the overdetermined problem $$ \left{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\Omega$,} \ u = 0 & \mbox{ on $\partial\Omega$,} \ \partial\_\nu u = c & \mbox{ on $\Gamma$,} \end{array} \right. $$ where $\Omega$ is a locally Lipschitz epigraph, that is $C^3$ on $\Gamma\subseteq\partial\Omega$, with $\partial\Omega\setminus\Gamma$ consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets $\Omega$ for which monotone solutions exist. In particular, if $\mathcal{C} \in \mathbb{R}^n$ is a cone and either $n=2$ or $n=3$ and $f \ge 0$, then there exists no solution of $$ \left{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\mathcal{C}$,} \ u > 0 & \mbox{ in $\mathcal{C}$,} \ u = 0 & \mbox{ on $\partial\mathcal{C}$,} \ \partial\_\nu u = c & \mbox{ on $\partial\mathcal{C} \setminus {0}$.} \end{array} \right. $$ This answers a question raised by Juan Luis Vázquez.