Lp-gradient harmonic maps into spheres and SO(N)

TL;DR

This paper studies fractional gradient energy minimizers mapping into spheres or SO(N), proving their Hölder continuity and providing a new proof of regularity for classical harmonic maps.

Contribution

It introduces a fractional gradient energy framework for harmonic maps and establishes regularity results, including a novel proof for classical cases.

Findings

Critical points are Hölder continuous.

Provides a new proof for regularity of harmonic maps into spheres.

Extends regularity results to fractional gradient energies.

Abstract

We consider critical points of the energy $E(v) := \int_{\mathbb{R}^n} |\nabla^s v|^{\frac{n}{s}}$, where $v$ maps locally into the sphere or $SO(N)$, and $\nabla^s = (\partial_1^s,\ldots,\partial_n^s)$ is the formal fractional gradient, i.e. $\partial_\alpha^s$ is a composition of the fractional laplacian with the $\alpha$-th Riesz transform. We show that critical points of this energy are H\"older continuous. As a special case, for $s = 1$, we obtain a new, more stable proof of Fuchs and Strzelecki's regularity result of $n$-harmonic maps into the sphere, which is interesting on its own.