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# Revista Matemática Iberoamericana

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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 and Enrico Valdinoci

(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. Iberoam. 26 (2010), 965-974. doi: 10.4171/RMI/623