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}.
Categories: math.AP