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

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Abstract

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) 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. © 2013 Elsevier Inc..

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Dipierro, S., & Pinamonti, A. (2013). A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. Journal of Differential Equations, 255(1), 85–119. https://doi.org/10.1016/j.jde.2013.04.001

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