# Boundary regularity for conformally invariant variational problems with Neumann data

## TL;DR

This paper investigates boundary regularity of conformally invariant variational maps with Neumann boundary conditions, introducing a nonlocal boundary potential and establishing regularity results for such systems.

## Contribution

It introduces a novel boundary system with a nonlocal antisymmetric potential for conformally invariant variational problems and proves boundary regularity for solutions.

## Key findings

- Boundary systems with nonlocal antisymmetric potentials are derived.
- Boundary regularity is established for solutions to these systems.
- The approach connects interior potentials with boundary conditions to ensure regularity.

## Abstract

We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support manifold. For example, harmonic maps, or $H$-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Riviere, that these maps satisfy a system with an antisymmetric potential, from which one can derive regularity of the solution. We show that these maps satisfy along the boundary a system with a nonlocal antisymmetric boundary potential which contains information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems.