Removable Sets for H\"older Continuous p(x)-Harmonic Functions

TL;DR

This paper characterizes removable sets for Hölder continuous p(x)-harmonic functions in terms of Hausdorff measure, extending understanding of singularities and removability in variable exponent harmonic analysis.

Contribution

It provides a new criterion for removability of sets based on Hausdorff measure and variable exponent p(x), generalizing classical results to Hölder continuous p(x)-harmonic functions.

Findings

Removability depends on Hausdorff measure of the set.

Established a measure-theoretic criterion for removability.

Extended classical harmonic analysis results to variable exponent case.

Abstract

We establish that a closed set $E$ is removable for $C^{0,\alpha}$ H\"{o}lder continuous $p(x)$-harmonic functions in a bounded open domain $\Omega$ of $\mathbb{R}^n$, $n\geq 2$, provided that for each compact subset $K$ of $E$, the $(n-p_K+\alpha(p_K-1))$-Hausdorff measure of $K$ is zero, where $p_K=\max_{x\in K} p(x)$.