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September, 2010 Overdetermined problems in unbounded domains with Lipschitz singularities
Alberto Farina , Enrico Valdinoci
Rev. Mat. Iberoamericana 26(3): 965-974 (September, 2010).

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 $\Omega$ 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.

Citation

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Alberto Farina . Enrico Valdinoci . "Overdetermined problems in unbounded domains with Lipschitz singularities." Rev. Mat. Iberoamericana 26 (3) 965 - 974, September, 2010.

Information

Published: September, 2010
First available in Project Euclid: 27 August 2010

zbMATH: 1211.35206
MathSciNet: MR2789372

Subjects:
Primary: 35B65, 35J20, 35J25

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.26 • No. 3 • September, 2010
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