Nonlocal Micromagnetics: Compactness Criteria, Existence of Minimizers, and Brown's Fundamental Theorem

TL;DR

This paper extends classical micromagnetic theory to nonlocal models, establishing existence of minimizers and identifying size-dependent regimes for uniform and non-uniform magnetization states in bounded domains.

Contribution

It develops a rigorous variational framework for nonlocal micromagnetic energy functionals, generalizes Brown's fundamental theorem, and identifies critical radii that determine magnetic configurations.

Findings

Existence of minimizers under mild assumptions on the interaction kernel.

Identification of critical radii R* and R** for uniform and non-uniform states.

Analysis of energetic regimes using Poincaré inequalities and explicit energy comparisons.

Abstract

This paper investigates the existence and qualitative properties of minimizers for a class of nonlocal micromagnetic energy functionals defined on bounded domains. The considered energy functional consists of a symmetric exchange interaction, which penalizes spatial variations in magnetization, and a magnetostatic self-energy term that accounts for long-range dipolar interactions. Motivated by the extension of Brown's fundamental theorem on fine ferromagnetic particles to nonlocal settings, we develop a rigorous variational framework in $L^2(\Omega;\mathbb{S}^2)$ under mild assumptions on the interaction kernel $j$, including symmetry, L\'evy-type integrability, and prescribed singular behavior. For spherical domains, we generalize Browns fundamental results by identifying critical radii $R^*$ and $R^{**}$ that delineate distinct energetic regimes: for $ R \leq R^*$, the uniform magnetization state is energetically preferable (small-body regime), whereas for $R \geq R^{**}$, non-uniform magnetization configurations become dominant (large-body regime). These transitions are analyzed through Poincar\'e-type inequalities and explicit energy comparisons between uniform and vortex-like magnetization states. Our results directly connect classical micromagnetic theory and contemporary nonlocal models, providing new insights into domain structure formation in nanoscale magnetism. Furthermore, the mathematical framework developed in this work contributes to advancing theoretical foundations for applications in spintronics and data storage technologies.