Boundary regularity and Wiener-type criteria at infinity for nonlinear elliptic equations of $p$-Laplace type

TL;DR

This paper establishes Wiener-type criteria for boundary regularity at infinity for nonlinear p-Laplace equations, linking geometric boundary properties to solution behavior in unbounded domains.

Contribution

It introduces a novel approach using circular inversion to relate Wiener criteria at infinity to those at the origin, providing new regularity conditions for p-Laplace type equations.

Findings

For p>n, infinity is regular iff the boundary is unbounded.

For p=n, the regularity criterion differs, with counterexamples.

The criteria extend to p-harmonic functions in metric measure spaces.

Abstract

We study boundary regularity at the infinity point $\boldsymbol{\infty}$ for nonlinear elliptic equations of $p$-Laplace type in unbounded open sets $\Omega \subset \mathbf{R}^n$. We consider the case $p \ge n \ge 2$ and characterize the regularity at $\boldsymbol{\infty}$ by means of Wiener-type integrals. Our approach uses circular inversion, which maps $\boldsymbol{\infty}$ to the origin and the original nonlinear equation to a similar weighted equation. The Wiener criterion at the origin for such equations is then transformed back to provide Wiener-type criteria at $\boldsymbol{\infty}$. When $p>n$, the criteria simplify so that $\boldsymbol{\infty}$ is regular if and only if the boundary $\partial\Omega$ is unbounded. For $p=n$ this is not true, as shown by an example. This simplified criterion is also proved for $p$-harmonic functions in unbounded open subsets of Ahlfors $Q$-regular metric measure spaces with $Q<p$, supporting a Poincar\'e inequality.