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

Abstract

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

33 pages