Regularity of unconstrained $p$-harmonic maps from curved domain and application to critical $n$-Laplace systems
Abstract
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: $$ v\in W^{1,p}\left(B^n(0,1);\mathbb{R}^N \right) \mapsto \int_{B^n(0,1)} |\nabla v|^p_g\, d\mathrm{vol}_g = \int_{B^n(0,1)} \left( g^{\alpha\beta}(x) \left\langle \partial_\alpha v(x), \partial_\beta v(x) \right\rangle \right)^{\frac{p}{2}}\, \sqrt{\det g(x)}\, dx. $$ 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 specialize to the case $p=n$ and we prove that any continuous solution to $|\Delta_{g,n}u|\lesssim |\nabla u|^n_g$ satisfies additional regularity properties depending on the regularity of $g$.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.13236
- arXiv:
- arXiv:2407.13236
- Bibcode:
- 2024arXiv240713236M
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 23 pages