Removable set for Hölder continuous solutions of $\mathscr{A}$-harmonic functions on Finsler manifolds

TL;DR

The paper addresses removability questions for Hölder continuous $\mathscr{A}$-harmonic functions on reversible Finsler manifolds with variable exponent growth. It establishes a removability criterion based on vanishing Hausdorff measure of a dimension tied to $n_1$, $p^+$ and the Hölder exponent $\alpha$, and develops an obstacle-problem–driven framework to obtain quantitative flux estimates via $\mu_\ell$ and the limit measure $\mu$. The approach extends capacity and measure-theoretic methods from Euclidean settings to the non-Euclidean Finsler setting with nonstandard growth, utilizing energy estimates, curvature-like volume control, and covering arguments. The results advance the understanding of singularity removal in nonlinear PDEs on Finsler spaces and provide tools potentially applicable to geometric analysis and nonlinear potential theory on manifolds with asymmetric distance.

Abstract

We establish that a closed set $\mathcal{S}$ is removable for $α$-Hölder continuous $\mathscr{A}$-harmonic functions in a reversible Finsler manifold $(Ω, F, \mathtt{V})$ of dimension $n \geq 2$, provided that (under certain conditions on $(Ω, F, \mathtt{V})$ and the variable exponent $p$ ) for each compact subset $K$ of $\mathcal{S}$, the $\mathrm{n}_1-p_K^{+}+α\left(p_K^{+}-1\right)$-Hausdorff measure of $K$ is zero. Here, $p_K^{+}=\sup _K p$ and $\mathrm{n}_1$ is chosen so that $\mathtt{V}(B(x, r)) \leq \mathtt{K} r^{\mathrm{n}_1}$ for every ball. The estimates used to remove the singularities will focus on a family $\left\{u_{\ell}\right\}_{\ell \in \mathcal{J}} \subset W_{\mathrm{loc}}^{1, p(x)}(Ω; \mathtt{V})$ that converges to $u$ in a certain sense. As a second main result of this article, we will also obtain an estimate (when $\lim _{d\left(x, 0_Ω\right) \rightarrow \infty} p=1$ ) for $$ μ_{\ell}(B(x, r)):=\sup \left\{\int_{B(x, r)} \mathscr{A} \left(\cdot, \nabla u_{\ell}\right) \bullet \mathcal{D} ζ\mathrm{dV} \mid 0 \leq ζ\leq 1 \text { and } ζ\in C_0^{\infty}(B(x, r))\right\}, $$ which is related to the measure $μ=\operatorname{div}( \mathscr{A} (\cdot, \nabla u))$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Lemma 2.7
• proof