2025/02/08 by Giovanni Di Fratta, Di Fratta, Giovanni, Rossella Giorgio +3 · 1 voice
Computer Science · Mathematics · Physics and Astronomy · #35A15 #35R11 #49J45 #49S05 #82D40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Magnetic properties of thin films #Numerical methods in inverse problems #math.AP
paper · pdf · doi:10.48550/arxiv.2502.05532
openalex publication_date 2025/02/08 · arxiv published 2025/02/08 · arxiv updated 2025/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2(Ω;\mathbbS2) 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 ≤ R^*, the uniform magnetization state is energetically preferable (small-body regime), whereas for R ≥ 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.