2013/11/29 by Fall, Mouhamed Moustapha, Jarohs, Sven · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1311.7549
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 ( -Δ)s u = f(u) in Ω, u=0 in ℝN∖ Ω, (∂η)s u=Const. on ∂ Ω has a nonnegative and nontrivial solution, where η is the outer unit normal vectorfield along ∂Ω and for x0∈∂Ω (∂η)su(x0)=-limt→ 0\fracu(x0-tη(x0))ts. 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.