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

We deal with symmetry properties for solutions of nonlocal equations of the type (− )sv = f (v) in Rn, where 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 −div xα∇u =0 onRn ×(0,+∞), −xαux = f (u) on Rn × {0}, where α ∈ (−1, 1), y ∈ Rn, x ∈ (0,+∞) and u = u(y, x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Γα : u| ∂Rn+1 + → −xαux | ∂Rn+1 + is (− ) 1−α 2 . More generally, we study the so-called boundary reaction equations given by −div μ(x)∇u + g(x,u) =0 onRn ×(0,+∞), −μ(x)ux = f (u) on Rn × {0}under 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. © 2009 Elsevier Inc. All rights reserved