Overdetermined problems with fractional laplacian
African Institute for Mathematical Sciences of Senegal. Km
2, Route de Joal.
2 Goethe-Universität Frankfurt, Institut für Mathematik. Robert-Mayer-Str. 10, 60054 Frankfurt, Germany
Revised: 29 August 2014
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 has a nonnegative and nontrivial solution, where η is the outer unit normal vectorfield along ∂Ω and for x0 ∈ ∂Ω 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.
Mathematics Subject Classification: 35B50 / 35N25
Key words: Fractional Laplacian / maximum principles / Hopf’s Lemma / overdetermined problems
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