Overdetermined problems with fractional laplacian

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 (Equation Presented) has a nonnegative and nontrivial solution, where η is the outer unit normal vectorfield along ∂Ω and for x0 ∈ ∂Ω (∂η)s u(x0) = - limt→0 u(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.