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arXiv · 2309.02408

Wiener type regularity for non-linear integro-differential equations

Shaoguang Shi·Guanglan Wang·Zhichun Zhai
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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, $(\alpha,p)-$barriers, and $(\alpha,p)-$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 $(\alpha,p)-$potentials and $(\alpha,p)-$Perron solutions, and the existence of a capacitary function for an arbitrary condenser.

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Shaoguang Shi, Guanglan Wang, Zhichun Zhai. 2023-09-05. Wiener type regularity for non-linear integro-differential equations. https://arxiv.org/abs/2309.02408

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