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)\qquad {in $\R^n$,}$ where $s \in (0,1)$ 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^\a \nabla u)=0 \qquad {on $\R^n\times(0,+\infty)$} -x^\a u_x = f(u) \qquad {on $\R^n\times\{0\}$} $ where $\a \in (-1,1)$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\Gamma_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-\Delta)^{\frac{1-\a}{2}}$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\Gamma_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-\Delta)^{\frac{1-\a}{2}}$. More generally, we study the so-called boundary reaction equations given by ${-div (\mu(x) \nabla u)+g(x,u)=0 {on $\R^n\times(0,+\infty)$} - \mu(x) u_x = f(u) {on $\R^n\times{0}$}$ under some natural assumptions on the diffusion coefficient $\mu$ and on the nonlinearities $f$ and $g$. We prove a geometric formula of Poincar\'e-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.