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 (-Delta)(s) v = f(v) in R-n, where s is an element of (0, 1) and the operator (-Delta)(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(x(alpha)del u) = 0 on R-n x (0, +infinity), -x(alpha)u(x) = f (u) on R-n x {0}, where alpha is an element of (-1, 1), y is an element of R-n, x is an element of (0, +infinity) and u = u(y, x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Gamma(alpha) : u vertical bar(partial derivative R+n+1) bar right arrow -x(alpha)u(x)vertical bar(partial derivative R+n+1) is (-Delta)(1-alpha/2). More generally, we study the so-called boundary reaction equations given by {-div(mu(x)del u) + g(x, u) = 0 on R-n x (0, +infinity), -mu (x)u(x) = f(u) on R-n x {0} under some natural assumptions on the diffusion coefficient mu and on the nonlinearities f and g. We prove a geometric formula of Poincare-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi. (C) 2009 Elsevier Inc. All rights reserved.