Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

TL;DR

This paper introduces a new heat flow for half-harmonic maps with free boundary, establishes weak solutions, and proves their partial regularity, extending previous work on harmonic maps and half-harmonic maps.

Contribution

It develops a novel heat flow for half-harmonic maps with free boundary and proves partial regularity using Ginzburg-Landau approximation, expanding understanding of these maps.

Findings

Constructed weak solutions for the new heat flow.

Proved partial regularity in space and time.

Extended previous studies on harmonic and half-harmonic maps.

Abstract

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivi\`ere. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.