Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Removable set for Hölder continuous solutions of $\mathscr{A}$-harmonic functions on Finsler manifolds (2502.11922v1)

Published 17 Feb 2025 in math.AP

Abstract: We establish that a closed set $\mathcal{S}$ is removable for $\alpha$-H\"older continuous $\mathscr{A}$-harmonic functions in a reversible Finsler manifold $(\Omega, F, \mathtt{V})$ of dimension $n \geq 2$, 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 $\mathtt{V}(B(x, r)) \leq \mathtt{K} r{\mathrm{n}_1}$ for every ball. The estimates used to remove the singularities will focus on a family $\left{u{\ell}\right}{\ell \in \mathcal{J}} \subset W{\mathrm{loc}}{1, p(x)}(\Omega ; \mathtt{V})$ that converges to $u$ in a certain sense. As a second main result of this article, we will also obtain an estimate (when $\lim {d\left(x, 0{\Omega}\right) \rightarrow \infty} p=1$ ) for $$ \mu_{\ell}(B(x, r)):=\sup \left{\int_{B(x, r)} \mathscr{A} \left(\cdot, \nabla u_{\ell}\right) \bullet \mathcal{D} \zeta \mathrm{dV} \mid 0 \leq \zeta \leq 1 \text { and } \zeta \in C_0{\infty}(B(x, r))\right}, $$ which is related to the measure $\mu=\operatorname{div}( \mathscr{A} (\cdot, \nabla u))$.

Summary

We haven't generated a summary for this paper yet.