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(- \u394)s v = f (v) in Rn, where s 08 (0, 1) and the operator (- \u394)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\u3b1 07 u) = 0, on Rn 7 (0, + 1e),; - x\u3b1 ux = f (u), on Rn 7 {0},) where \u3b1 08 (- 1, 1), y 08 Rn, x 08 (0, + 1e) and u = u (y, x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator \u393\u3b1 : u | 02 R+n + 1 {mapping} - x\u3b1 ux | 02 R+n + 1 is (- \u394)frac(1 - \u3b1, 2). More generally, we study the so-called boundary reaction equations given by{(- div (\u3bc (x) 07 u) + g (x, u) = 0, on Rn 7 (0, + 1e),; - \u3bc (x) ux = f (u), on Rn 7 {0}) under some natural assumptions on the diffusion coefficient \u3bc and on the nonlinearities f and g. We prove a geometric formula of Poincar\ue9-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi