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)sv=f(v)in Rn, where s∈(0,1)s∈(0,1) and the operator s(−Δ)(−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)=0on Rn×(0,+∞),−xαux=f(u)on Rn×{0}, where α∈(−1,1)α∈(−1,1), y∈Rny∈Rn, x∈(0,+∞)x∈(0,+∞) and u=u(y,x)u=u(y,x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Γα:u|∂R+n+1↦−xαux|∂R+n+1 is (−Δ)1−α2. More generally, we study the so-called boundary reaction equations given by{−div(μ(x)∇u)+g(x,u)=0on 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.