Wiener type regularity for non-linear integro-differential equations

TL;DR

This paper develops Wiener-type regularity criteria for non-linear integro-differential equations, specifically fractional p-Laplace equations, using potential analysis tools to characterize boundary regularity and solution properties.

Contribution

It introduces new Wiener-type criteria and characterizations for boundary regularity of fractional p-Laplace equations using advanced potential analysis techniques.

Findings

Characterization of fractional thinness and Perron boundary regularity

Establishment of a Wiener test and generalized fractional Wiener criterion

Proof of continuity and resolutivity of fractional superharmonic functions

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.