The Hopf-Laplace equation: harmonicity and regularity

TL;DR

This paper investigates the Hopf-Laplace equation, showing that solutions are harmonic outside a small singular set where they are Lipschitz but not necessarily smooth, advancing understanding of harmonic maps and energy minimizers.

Contribution

It proves that solutions to the Hopf-Laplace equation are harmonic outside a small singular set, clarifying their regularity and structure beyond classical harmonic maps.

Findings

Solutions are harmonic outside a small singular set.

On the singular set, solutions are locally Lipschitz.

Solutions may not be differentiable on the singular set.

Abstract

The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the Dirichlet energy among homeomorphisms often leads to mappings which are neither harmonic nor homeomorphisms. We prove that such mappings are harmonic outside of a singular set with small image. On the singular set they are locally Lipschitz, but not necessarily differentiable.