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 $(-\mathrm{\Delta }{)}^s$ 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 ${\mathbb{R}}^n\times \{0\}$} $where$\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 ${\mathbb{R}}^n\times 0$}$undersomenaturalassumptionsonthediffusioncoefficient$\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.