A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian

Abstract

We study the symmetry properties for solutions of elliptic systems of the type (-Δ)^{s_1} u = F_1(u, v), (-Δ)^{s_2} v= F_2(u, v), where $F\in C_{loc}^{1,1}({\mathbb{R}}^2)$, $s_1,s_2\in (0,1)$ and the operator $(-\Delta {)}^s$ is the so-called fractional Laplacian. We obtain some Poincaré-type formulas for the $\alpha$-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions.