Fine regularity of fractional harmonic maps and applications

TL;DR

This paper establishes new regularity results for fractional harmonic maps into spheres, extending previous work and applying advanced nonlocal PDE techniques to improve understanding of their smoothness and related flows.

Contribution

It introduces novel regularity results for fractional harmonic maps, especially without classical monotonicity formulas, and applies these to free boundary problems and fractional harmonic map heat flows.

Findings

Proved small energy regularity results for fractional harmonic maps.

Improved regularity results even in cases with available monotonicity formulas.

Derived higher differentiability for fractional harmonic map heat flow.

Abstract

In this paper, we derive several regularity results for harmonic mappings into Euclidean spheres associated with rather general energies related to fractional Sobolev spaces. These maps generalize families of maps introduced by Da Lio, Rivi\`ere and Schikorra and are related to harmonic maps with free boundaries. In our context, there is in general no monotonicity formula, which prevents the use of some classical methods. Despite this limitation, under natural assumptions on a Gagliardo-type energy, we succeed in proving a variety of small energy regularity results and improve on known results, even in the isotropic case for which some monotonicity formula is available. To this end, we exploit recent developments in the regularity theory of nonlocal equations and as a by-product, we explain how these results apply to classes of harmonic maps with free boundary and lead to new potential-theoretic estimates. As another application, we obtain higher differentiability results for the fractional harmonic map heat flow.