Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result

Abstract

We deal with symmetry properties for solutions of nonlocal equations of the typewhere s ∈ (0, 1) and the operator (− ) s is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation u(y, x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Γ α : u| ∂R n+1More generally, we study the so-called boundary reaction equations given byunder some natural assumptions on the diffusion coefficient μ and on the nonlinearities f and g. We prove a geometric formula of Poincaré-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.

Stable Solutions of Elliptic Equations on Riemannian Manifolds

“…The use of such a type of formula in the Euclidean setting was started in [SZ98a,SZ98b] and its importance for symmetry results was explained in [Far02]. Further applications to PDEs have been given in [FSV08,SV08,FV08]. We now give two additional results in the spirit of Theorem 1, under a sign assumption on the nonlinearity and on the growth of the volume of the geodesic balls.…”

Fractional Kirchhoff equation with a general critical nonlinearity

“…where N > 2s with 0 < s < 1, a, b are positive constants and (−∆) s u is the fractional Laplacian which arises in the description of various phenomena in the applied science, such as the phase transition [19], Markov processes [1] and fractional quantum mechanics [15]. When a = 1 and b = 0, (1.1) becomes the fractional Schrödinger equations which have been studied by many authors.…”

Nonlocal phase transitions in homogeneous and periodic media

“…In this case, a positive answer to this problem was given in [CSM05] when n = 2 and s = 1/2, in [SV09,CS15] when n = 2 and s ∈ (0, 1), in [CC10] when n = 3 and s = 1/2 and in [CC14] when n = 3 and s ∈ (1/2, 1) (see also [SV13] and [BV16] for different proofs, also related to nonlocal minimal surfaces).…”