Overdetermined problems with fractional laplacian
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 924-938.

Let N1 and s(0,1). In the present work we characterize bounded open sets Ω with C 2 boundary (not necessarily connected) for which the following overdetermined problem

(-Δ) s u=f(u)inΩ;u=0in N Ω;( η ) s u= Const .onΩ
has a nonnegative and nontrivial solution, where η is the outer unit normal vectorfield along Ω and for x 0 Ω
η s u(x 0 )=-lim t0 u(x 0 -tη(x 0 )) t s .
Under mild assumptions on f, we prove that Ω must be a ball. In the special case f1, 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.

Reçu le :
DOI : 10.1051/cocv/2014048
Classification : 35B50, 35N25
Mots clés : Fractional Laplacian, maximum principles, Hopf’s Lemma, overdetermined problems
Fall, Mouhamed Moustapha 1 ; Jarohs, Sven 2

1 African Institute for Mathematical Sciences of Senegal. Km 2, Route de Joal. BP 1418 Mbour, Senegal
2 Goethe-Universität Frankfurt, Institut für Mathematik. Robert-Mayer-Str. 10, 60054 Frankfurt, Germany
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Fall, Mouhamed Moustapha; Jarohs, Sven. Overdetermined problems with fractional laplacian. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 924-938. doi : 10.1051/cocv/2014048. http://archive.numdam.org/articles/10.1051/cocv/2014048/

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