Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime

TL;DR

This paper investigates the existence and symmetry properties of energy-minimizing harmonic maps with axial symmetry between surfaces of revolution, providing conditions for symmetry and reducing the problem to a one-dimensional profile analysis.

Contribution

It establishes the existence of axially symmetric minimizers and characterizes when these minimizers are unique and symmetric, advancing the understanding of variational problems in micromagnetism models.

Findings

Axially symmetric minimizers always exist.

If target surface T is not flat, minimizers must have line symmetry.

The problem reduces to computing an optimal one-dimensional profile.

Abstract

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields $H^1(S,T)$, where $S$ and $T$ are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface $T$ is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.