Regularity at the free boundary for Dirac-harmonic maps from surfaces

TL;DR

This paper develops a regularity theory for Dirac-harmonic maps from surfaces, proving smoothness at free boundaries and interior regions, advancing understanding of elliptic systems with anti-symmetric structures.

Contribution

It introduces a comprehensive regularity framework for Dirac-harmonic maps, including boundary and interior smoothness results, under inhomogeneous boundary conditions.

Findings

Full regularity at free boundaries for Dirac-harmonic maps

Interior epsilon-regularity and smoothness in all dimensions

Application of methods to elliptic systems with anti-symmetric structures

Abstract

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior $\epsilon$-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.