Regularity of unconstrained $p$-harmonic maps from curved domain and application to critical $p$-Laplace systems

TL;DR

This paper investigates the regularity of critical points of a $p$-harmonic functional on curved domains, showing local Hölder continuity under certain conditions and applying results to critical $p$-Laplace systems.

Contribution

It establishes regularity results for $p$-harmonic maps on curved domains with variable metrics, extending known results to non-flat geometries and applying to critical $p$-Laplace systems.

Findings

Critical points are locally Hölder continuous when the metric is close to constant.

Gradient of solutions is Hölder continuous if the metric is Hölder continuous.

Solutions to certain $p$-Laplace inequalities exhibit enhanced regularity depending on metric regularity.

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 H\"older-continuous. If $g$ is H\"older-continuous, we show that $\nabla v$ is locally H\"older-continuous. As an application, we prove that any H\"older-continuous solution to $|\Delta_{g,p}u|\lesssim |\nabla u|^p_g$ satisfies additional regularity properties depending on the regularity of $g$. In the case $p=n$, only the continuity is assumed \textit{a priori}.