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

Given $p\ge 2$ 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: $$v\in W^{1,p}(B^n(0,1);{\mathbb{R}}^N)\mapsto {\int }_{B^n(0,1)}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}_g^{p}d{\mathrm{v}\mathrm{o}\mathrm{l}}_g={\int }_{B^n(0,1)}{(g^{\alpha \beta }(x)⟨{\mathrm{\partial }}_{\alpha }v(x),{\mathrm{\partial }}_{\beta }v(x)⟩)}^{\frac{p}{2}}\sqrt{\det g(x)}dx\mathrm{.}$$ 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 $\mathrm{\nabla }v$ is locally Hölder-continuous. As an application, we prove that any Hölder-continuous solution to $\mathrm{\mid }{\mathrm{\Delta }}_{g,p}u\mathrm{\mid }\lesssim \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}_g^p$ satisfies additional regularity properties depending on the regularity of $g$. In the case $p=n$, only the continuity is assumed $\mathit{\text{a}}\mathit{\text{priori}}$.