Uniform small energy regularity for fractional geometric problems

TL;DR

This paper establishes small energy regularity results for fractional geometric problems, including a parabolic boundary reaction Ginzburg-Landau problem and fractional harmonic maps, uniformly across the fractional parameter range.

Contribution

It provides the first uniform small energy regularity results for fractional geometric problems as the fractional parameter approaches 1.

Findings

Proved small energy regularity for fractional Ginzburg-Landau problem in (0,1)

Established similar regularity for fractional harmonic maps to spheres

Results are uniform as the fractional parameter s approaches 1

Abstract

We prove small energy regularity for a parabolic boundary reaction Ginzburg-Landau problem in the full range $s\in (0,1)$, answering a question posed by Hyder, Segatti, Sire and Wang. We also obtain a similar small energy regularity result for fractional harmonic maps to spheres. Both results are uniform as $s\to 1$.