Removable Sets for Hölder Continuous p(x)-Harmonic Functions

Abstract

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