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

TL;DR

This paper investigates symmetry properties of solutions to elliptic systems involving fractional Laplacians, deriving geometric inequalities and applying Poincaré formulas to establish symmetry for stable and monotone solutions.

Contribution

It introduces new Poincaré-type formulas for fractional harmonic extensions and proves symmetry results for solutions of fractional elliptic systems, extending classical symmetry results to nonlocal operators.

Findings

Established symmetry for stable solutions

Proved symmetry for monotone solutions

Derived geometric inequalities for fractional systems

Abstract

We study the symmetry properties for solutions of elliptic systems of the type (-\Delta)^{s_1} u = F_1(u, v), (-\Delta)^{s_2} v= F_2(u, v), where $F\in C^{1,1}_{loc}(\R^2)$, $s_1,s_2\in (0,1)$ and the operator $(-\Delta)^s$ is the so-called fractional Laplacian. We obtain some Poincar\'e-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.