Regularity for variational problems on Riemann surfaces

The interior (or epsilon-regularity) estimates for two-dimensional variational problems has a long history, ranging from the classical minimal/CMC surface equations to harmonic maps and prescribed mean-curvature equations and more recently Willmore surfaces, $W^{2,2}$ conformal immersions and Dirac-harmonic maps.

Here we will present some new estimates for a class of elliptic PDE with applications (in particular) to the free boundary problem for Dirac-harmonic maps, and global estimates for harmonic maps. This talk is comprised of separate joint works with Miaomiao Zhu and Tobias Lamm.