Boundary regularity for minimizers of the micromagnetic energy functional

TL;DR

This paper studies the boundary regularity of minimizers of the micromagnetic energy functional, showing they are regular for small particle volumes using reflection techniques and harmonic map regularity theory.

Contribution

It establishes boundary regularity results for micromagnetic energy minimizers, extending harmonic map techniques to the micromagnetic context with small volume conditions.

Findings

Minimizers are regular at the boundary for sufficiently small particles.

A reflection construction at the boundary is used to analyze regularity.

The approach adapts harmonic map regularity theory to micromagnetic energy minimizers.

Abstract

Motivated by the construction of time-periodic solutions for the three-dimensional Landau-Lifshitz-Gilbert equation in the case of soft and small ferromagnetic particles, we investigate the regularity properties of minimizers of the micromagnetic energy functional at the boundary. In particular, we show that minimizers are regular provided the volume of the particle is sufficiently small. The approach uses a reflection construction at the boundary and an adaption of the well-known regularity theory for minimizing harmonic maps into spheres.