Overdetermined problems in unbounded domains with Lipschitz singularities

Abstract

We study the overdetermined problem

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

This answers a question raised by Juan Luis Vázquez.