Regularity of unconstrained $p$-harmonic maps from curved domain and application to critical $p$-Laplace systems
Dorian Martino
TL;DR
This work extends regularity theory for unconstrained $p$-harmonic maps to domains equipped with low-regularity, curved metrics $g$ by developing a reflection-avoidant approach that compares solutions to corresponding constant-metric problems. The authors prove local $C^{0,\gamma}$ regularity of $v$ when $g$ is uniformly close to a constant metric and upgrade to $C^{0,\gamma}$-regularity of $\nabla v$ when $g$ is Hölder continuous; further, for $C^1$ metrics they obtain higher-order integrability for $\nabla v$ with respect to $g$. As an application, they study systems $|\Delta_{g,p}u|\lesssim |\nabla u|^p$ and show Hölder-continuity in the critical case $p=n$, as well as $L^{\infty}$-bounds and $C^{1,\alpha}$-regularity in the noncritical regime, with all estimates depending on the regularity and proximity of $g$ to constants. The results provide a robust framework for regularity of critical $p$-Laplace systems on curved domains with minimal metric regularity, extending known flat-domain theory to curved, low-regularity settings.
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: $$ v\in W^{1,p}\left(B^n(0,1);\mathbb{R}^N \right) \mapsto \int_{B^n(0,1)} |\nabla v|^p_g\, d\mathrm{vol}_g = \int_{B^n(0,1)} \left( g^{αβ}(x) \left\langle \partial_αv(x), \partial_βv(x) \right\rangle \right)^{\frac{p}{2}}\, \sqrt{\det g(x)}\, dx. $$ We show that if $g$ is uniformly close to a constant matrix, then $v$ is locally 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 $|Δ_{g,p}u|\lesssim |\nabla u|^p_g$ satisfies additional regularity properties depending on the regularity of $g$. In the case $p=n$, only the continuity is assumed \textit{a priori}.