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[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