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

TL;DR

This work addresses the existence and qualitative structure of minimizers for a nonlocal micromagnetic energy $\mathcal{E}_{\Omega}(m)=\mathcal{J}_{\Omega}(m)+\mathcal{W}_{\Omega}(m)$ defined on bounded domains, where $\mathcal{J}_{\Omega}$ encodes a symmetric nonlocal exchange via a Lévy-type kernel $j$ and $\mathcal{W}_{\Omega}$ represents magnetostatic self-energy. The authors develop a robust variational framework in $L^2(\Omega;\mathbb{S}^2)$ under symmetry, Lévy-type integrability, and singular behavior assumptions on $j$, and extend Brown's fundamental theorem to the nonlocal setting on spherical domains by identifying critical radii $R^*$ and $R^{**}$ that separate uniform from vortex-like minimizers via Poincaré-type inequalities and energy comparisons. They prove a general compactness theorem ensuring strong $L^2$-convergence of minimizing sequences and existence of minimizers for a broad class of functionals, then specialize to micromagnetics in 3D to show: (i) small-body regime ($R\le R^*$) implies uniform minimizers, (ii) large-body regime ($R\ge R^{**}$) implies non-uniform minimizers, typically vortex-like; (iii) precise energy comparisons using vortex constructions and exchange estimates. The results bridge classical micromagnetics with nonlocal energy models, offering rigorous insight into nanoscale domain formation relevant for spintronics and data storage.

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 $L^2(Ω;\mathbb{S}^2)$ under mild assumptions on the interaction kernel $j$, including symmetry, Lévy-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 Poincaré-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.

Figures (1)

• Figure 1: On the left, the rescaled configuration $m_{\bullet}$ restricted to the equatorial plane. On the right, a vector plot of $m_{\bullet}$ over the slice coordinate planes stacked along the $z$-axis.

Theorems & Definitions (34)

• Theorem 2.1: Compactness
• Remark 2.2: On the necessity of Assumption \ref{['J3']}
• Remark 2.3: About Assumption \ref{['J2']}
• Remark 2.4: Motivation for the function $\psi$
• Theorem 2.5: Existence of minimizers
• Remark 2.6
• Remark 2.7
• Lemma 2.8
• proof
• Theorem 2.9: Existence of magnetic equilibrium configurations