Boundary regularity via Uhlenbeck-Rivi\`ere decomposition

TL;DR

This paper proves boundary regularity for weak solutions of systems with skew-symmetric structure, extending previous results and confirming a conjecture by Heinz using advanced decomposition techniques.

Contribution

It establishes boundary continuity for solutions with continuous boundary trace in systems with skew-symmetric structure, utilizing Rivi e's decomposition method.

Findings

Weak solutions with continuous boundary trace are continuous up to the boundary.

Extension of previous boundary regularity results to broader systems.

Application of Rivi e's decomposition to boundary regularity problems.

Abstract

We prove that weak solutions of systems with skew-symmetric structure, which possess a continuous boundary trace, have to be continuous up to the boundary. This applies, e.g., to the H-surface system with bounded H and thus extends an earlier result by P. Strzelecki and proves the natural counterpart of a conjecture by E. Heinz. Methodically, we use estimates below natural exponents of integrability and a recent decomposition result by T.Rivi\`ere.