JournalsrmiVol. 26, No. 3pp. 965–974

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
Overdetermined problems in unbounded domains with Lipschitz singularities cover
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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\Omega is a locally Lipschitz epigraph, that is C3C^3 on ΓΩ\Gamma\subseteq\partial\Omega, with ΩΓ\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 Ω\Omega for which monotone solutions exist. In particular, if CRn\mathcal{C} \in \mathbb{R}^n is a cone and either n=2n=2 or n=3n=3 and f0f \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.

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