Removable set for Hölder continuous solutions of $\mathscr{A}$-harmonic functions on Finsler manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912234026827776 |
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| author | Apaza, Juan Pablo Alcon |
| author_facet | Apaza, Juan Pablo Alcon |
| contents | 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 $μ=\operatorname{div}( \mathscr{A} (\cdot, \nabla u))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Removable set for Hölder continuous solutions of $\mathscr{A}$-harmonic functions on Finsler manifolds Apaza, Juan Pablo Alcon Analysis of PDEs 35J60, 35J70, 35J92, 58J05 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 $μ=\operatorname{div}( \mathscr{A} (\cdot, \nabla u))$. |
| title | Removable set for Hölder continuous solutions of $\mathscr{A}$-harmonic functions on Finsler manifolds |
| topic | Analysis of PDEs 35J60, 35J70, 35J92, 58J05 |
| url | https://arxiv.org/abs/2502.11922 |