Wiener type regularity for non-linear integro-differential equations

2023/09/05 by Shi, Shaoguang, Wang, Guanglan, Zhai, Zhichun
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.02408

Abstract

The primary purpose of this paper is to study the Wiener-type regularity criteria for non-linear equations driven by integro-differential operators, whose model is the fractional p-Laplace equation. In doing so, with the help of tools from potential analysis, such as fractional relative Sobolev capacities, Wiener type integrals, Wolff potentials, (α,p)-barriers, and (α,p)-balayages, we first prove the characterizations of the fractional thinness and the Perron boundary regularity. Then, we establish a Wiener test and a generalized fractional Wiener criterion. Furthermore, we also prove the continuity of the fractional superharmonic function, the fractional resolutivity, a connection between (α,p)-potentials and (α,p)-Perron solutions, and the existence of a capacitary function for an arbitrary condenser.

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