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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.13236 |
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| _version_ | 1866910878381637632 |
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| author | Martino, Dorian |
| author_facet | Martino, Dorian |
| contents | Given $p\geq 2$ and a map $g : B^n(0,1)\to S_n^{++}$, where $S_n^{++}$ is the group of positively definite matrices, we study critical points of the following functional: $$ v\in W^{1,p}\left(B^n(0,1);\mathbb{R}^N \right) \mapsto \int_{B^n(0,1)} |\nabla v|^p_g\, d\mathrm{vol}_g = \int_{B^n(0,1)} \left( g^{αβ}(x) \left\langle \partial_αv(x), \partial_βv(x) \right\rangle \right)^{\frac{p}{2}}\, \sqrt{\det g(x)}\, dx. $$ We show that if $g$ is uniformly close to a constant matrix, then $v$ is locally Hölder-continuous. If $g$ is Hölder-continuous, we show that $\nabla v$ is locally Hölder-continuous. As an application, we prove that any Hölder-continuous solution to $|Δ_{g,p}u|\lesssim |\nabla u|^p_g$ satisfies additional regularity properties depending on the regularity of $g$. In the case $p=n$, only the continuity is assumed \textit{a priori}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13236 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity of unconstrained $p$-harmonic maps from curved domain and application to critical $p$-Laplace systems Martino, Dorian Analysis of PDEs Given $p\geq 2$ and a map $g : B^n(0,1)\to S_n^{++}$, where $S_n^{++}$ is the group of positively definite matrices, we study critical points of the following functional: $$ v\in W^{1,p}\left(B^n(0,1);\mathbb{R}^N \right) \mapsto \int_{B^n(0,1)} |\nabla v|^p_g\, d\mathrm{vol}_g = \int_{B^n(0,1)} \left( g^{αβ}(x) \left\langle \partial_αv(x), \partial_βv(x) \right\rangle \right)^{\frac{p}{2}}\, \sqrt{\det g(x)}\, dx. $$ We show that if $g$ is uniformly close to a constant matrix, then $v$ is locally Hölder-continuous. If $g$ is Hölder-continuous, we show that $\nabla v$ is locally Hölder-continuous. As an application, we prove that any Hölder-continuous solution to $|Δ_{g,p}u|\lesssim |\nabla u|^p_g$ satisfies additional regularity properties depending on the regularity of $g$. In the case $p=n$, only the continuity is assumed \textit{a priori}. |
| title | Regularity of unconstrained $p$-harmonic maps from curved domain and application to critical $p$-Laplace systems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.13236 |