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

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.