Overdetermined problems with fractional laplacian

¹ African Institute for Mathematical Sciences of Senegal. Km 2, Route de Joal. BP 1418 Mbour, Senegal
mouhamed.m.fall@aims-senegal.org
² Goethe-Universität Frankfurt, Institut für Mathematik. Robert-Mayer-Str. 10, 60054 Frankfurt, Germany
jarohs@math.uni-frankfurt.de.

Received: 12 May 2014
Revised: 29 August 2014

Abstract

Let N ≥ 1 and s ∈ (0,1). In the present work we characterize bounded open sets Ω with C² boundary (not necessarily connected) for which the following overdetermined problem $\begin{array}{ccc}\{\begin{array}{c}\begin{array}{c}\end{array}\mathrm{(}\mathrm{-}\mathit{\Delta }{\mathrm{)}}^{\mathit{s}}\mathit{u}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{u}\mathrm{\right)}\begin{array}{c}\end{array}\mathrm{in}\mathit{\Omega }\mathrm{;}\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\mathit{u}\mathrm{=}\mathrm{0}\begin{array}{c}\end{array}\mathrm{in}{R}^{\mathit{N}}\mathrm{\backslash }\mathit{\Omega }\mathrm{;}\\ \begin{array}{c}\end{array}\mathrm{(}{\mathit{\partial }}_{\mathit{\eta }}{\mathrm{)}}_{\mathit{s}}\mathit{u}\mathrm{=}\mathrm{Const}\mathit{.}\begin{array}{c}\end{array}\mathrm{on}\mathit{\partial \Omega }\end{array}& & \end{array}$ has a nonnegative and nontrivial solution, where η is the outer unit normal vectorfield along ∂Ω and for x₀ ∈ ∂Ω $\begin{array}{ccc}{\mathrm{(}{\mathit{\partial }}_{\mathit{\eta }}\mathrm{)}}_{\mathit{s}}\mathit{u}\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}\mathrm{\right)}\mathrm{=}\mathrm{-}\underset{\mathit{t}\mathrm{\to }\mathrm{0}}{\lim }\frac{\mathit{u}\mathrm{(}{\mathit{x}}_{\mathrm{0}}\mathrm{-}\mathit{t\eta }\mathrm{(}{\mathit{x}}_{\mathrm{0}}\mathrm{\left)}\mathrm{\right)}}{{\mathit{t}}^{\mathit{s}}}\mathit{.}& & \end{array}$ Under mild assumptions on f, we prove that Ω must be a ball. In the special case f ≡ 1, we obtain an extension of Serrin’s result in 1971. The fact that Ω is not assumed to be connected is related to the nonlocal property of the fractional Laplacian. The main ingredients in our proof are maximum principles and the method of moving planes.