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arXiv · 1605.00906

Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations

Janne Korvenpaa·Tuomo Kuusi·Giampiero Palatucci
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Abstract

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of $(s,p)$-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method in nonlinear Potential Theory.

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Janne Korvenpaa, Tuomo Kuusi, Giampiero Palatucci. 2016-05-03. Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations. https://arxiv.org/abs/1605.00906

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