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

We study the symmetry properties for solutions of elliptic systems of the type{(−Δ)s1u=F1(u,v),(−Δ)s2v=F2(u,v), where F∈Cloc1,1(R2), s1,s2∈(0,1)s1,s2∈(0,1) and the operator (−Δ)s(−Δ)s is the so-called fractional Laplacian. We obtain some Poincaré-type formulas for the α-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions.