Harmonic maps on domains with piecewise Lipschitz continuous metrics

TL;DR

This paper studies harmonic maps from domains with piecewise Lipschitz metrics to compact manifolds, extending regularity results and introducing generalized stationary harmonic maps.

Contribution

It generalizes the concept of stationary harmonic maps and establishes partial regularity results for domains with piecewise Lipschitz metrics.

Findings

Proves partial regularity of harmonic maps in this setting

Establishes global Lipschitz regularity under certain conditions

Shows piecewise $C^{1,eta}$ regularity for harmonic maps

Abstract

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Lipschitz and piecewise $C^{1,\alpha}$-regularity of harmonic maps from $(\Omega, g)$ manifolds that support convex distance functions.