# Overdetermined problems in unbounded domains with Lipschitz singularities

### Alberto Farina

Université de Picardie Jules Verne, Amiens, France### Enrico Valdinoci

Università di Roma Tor Vergata, Italy

## 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 $Ω$ is a locally Lipschitz epigraph, that is $C_{3}$ on $Γ⊆∂Ω$, with $∂Ω∖Γ$ consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets $Ω$ for which monotone solutions exist. In particular, if $C∈R_{n}$ is a cone and either $n=2$ or $n=3$ and $f≥0$, 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.

## Cite this article

Alberto Farina, Enrico Valdinoci, Overdetermined problems in unbounded domains with Lipschitz singularities. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 965–974

DOI 10.4171/RMI/623