# Revista Matemática Iberoamericana

Volume 26, Issue 3, 2010, pp. 965–974
DOI: 10.4171/RMI/623

Published online: 2010-12-31

Overdetermined problems in unbounded domains with Lipschitz singularities

Alberto Farina[1] and Enrico Valdinoci[2]

(1) Université de Picardie Jules Verne, Amiens, France
(2) Università di Roma Tor Vergata, Italy

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.

Keywords: Elliptic partial differential equations, rigidity results, nonexistence of solutions

Farina Alberto, Valdinoci Enrico: Overdetermined problems in unbounded domains with Lipschitz singularities. Rev. Mat. Iberoamericana 26 (2010), 965-974. doi: 10.4171/RMI/623