35S15 Boundary value problems for pseudodifferential operators
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We present a symmetry result to solutions of equations involving the fractional Laplacian in a domain with at least two perpendicular symmetries. We show that if the solution is continuous, bounded, and odd in one direction such that it has a fixed sign on one side, then it will be symmetric in the perpendicular direction. Moreover, the solution will be monotonic in the part where it is of fixed sign. In addition, we present also a class of examples in which our result can be applied.
We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.
We derive a shape derivative formula for the family of principal Dirichlet eigenvalues λs(Ω) of the fractional Laplacian (−Δ)s associated with bounded open sets Ω⊂RN of class C1,1. This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case s=12. As an application, we consider the maximization problem for λs(Ω) among annular-shaped domains of fixed volume of the type B∖B¯¯¯¯′, where B is a fixed ball and B′ is ball whose position is varied within B. We prove that λs(B∖B¯¯¯¯′) is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.