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(- Δ)s v = f (v) in Rn, where s ∈ (0, 1) and the operator (- Δ)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, on Rn × (0, + ∞),; - xα ux = 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 |∂ R+n + 1 {mapping} - xα ux |∂ R+n + 1 is (- Δ)frac(1 - α, 2). More generally, we study the so-called boundary reaction equations given by{(- div (μ (x) ∇ u) + g (x, u) = 0, on Rn × (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.