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arXiv · 1211.2622

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

Serena Dipierro·Andrea Pinamonti
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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 $F\in C^{1,1}_{loc}(\R^2)$, $s_1,s_2\in (0,1)$ and the operator $(-Δ)^s$ 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.

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Serena Dipierro, Andrea Pinamonti. 2013-04-15. A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. https://arxiv.org/abs/1211.2622

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