Sacks-Uhlenbeck type regularity for subcritical generalized $p$-harmonic maps into Homogeneous targets

TL;DR

This paper establishes uniform regularity results for generalized p-harmonic maps into homogeneous targets in the subcritical range, extending previous methods to a broader class of solutions not in the energy space.

Contribution

It introduces a new notion of solutions for generalized p-harmonic maps into homogeneous targets and proves uniform regularity as p approaches the domain dimension, extending prior work.

Findings

Proves uniform $C^{1,eta}$-regularity for generalized p-harmonic maps in the subcritical range.

Extends regularity results to solutions not belonging to the classical energy space.

Adapts and verifies estimates from previous methods for a broader class of solutions.

Abstract

Adapting \cite{strz3}, we define generalized $p$-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range $p$ greater than the domain dimension $n$, we show a uniform $C^{1,\alpha}$-regularity result for a sequence of such maps in the limit $p \searrow n$, assuming a uniform $n$-energy bound on its elements. The method of the proof follows the exact same lines as in \cite{strz3} but we need to check uniformity of estimates not previously considered there.