Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

TL;DR

This paper investigates the regularity properties of harmonic and quasi-harmonic discs in metric spaces that satisfy a quadratic isoperimetric inequality, establishing their continuity and Hölder continuity up to the boundary.

Contribution

It extends regularity results for harmonic discs to a broad class of metric spaces including Lipschitz manifolds and Alexandrov spaces, under quadratic isoperimetric conditions.

Findings

Harmonic and quasi-harmonic discs are locally Hölder continuous.

Continuity up to the boundary is established for these discs.

The results apply to various spaces with curvature bounds and Lipschitz structures.

Abstract

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hoelder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.