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

Abstract

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

arXiv comment23 pages. Added in v2: Typos corrected, second part generalized to $p\ne n$. Added in v3: Typos corrected, final version