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 Höldercontinuous. If $g$ is Höldercontinuous, we show that $\nabla v$ is locally Höldercontinuous. 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:

arXiv eprints
 Pub Date:
 July 2024
 DOI:
 10.48550/arXiv.2407.13236
 arXiv:
 arXiv:2407.13236
 Bibcode:
 2024arXiv240713236M
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 23 pages