Nonlocal Variational problems with applications to Micromagnetics

Abstract

In this thesis, we consider multiscale variational problems characterized by nonlocal interaction energies, with applications in the theory of Micromagnetics.In Chapter 1, we present a general overview of nonlocal exchange models, with particular emphasis on nonlocal functionals that mimic gradient-type contributions, and on the properties of the interaction kernels. We next formally introduce the micromagnetic setting, which constitutes the main application of this work.Chapter 2 concerns the nonlocal-to-local analysis of a family of exchange energy functionals, extending the well-known Bourgain-Brezis-Mironescu formula to encompass the scenarios that also include antisymmetric contributions. The key points are a pointwise convergence result and a Gamma-convergence argument, yielding as byproduct a rigorous justification of the so-called Dzyaloshinskii-Moriya interaction term.In Chapter 3, we investigate the existence and qualitative properties of minimizers for a class of nonlocal micromagnetic energy functionals. In particular, we study the competition between a nonlocal symmetric exchange interaction, which penalizes spatial variations in magnetization and is modulated by a Lévy-type kernel, and a magnetostatic self-energy term accounting for long-range dipolar interactions. For spherical domains, we generalize the so-called Brown’s Fundamental Theorem by identifying critical radii that separate distinct energetic regimes: in the small-body regime, uniform magnetizations are energetically preferred, whereas in the large-body regime, non-uniform magnetization configurations dominate.Building upon a tailored notion of two-scale convergence, Chapter 4 is devoted to the ho- mogenization of nonlocal micromagnetic functionals for composite ferromagnetic materials, incorporating both symmetric and antisymmetric exchange contributions. Assuming that the nonlocal interaction range and the scale of the heterogeneities vanish simultaneously, we charac- terize the resulting asymptotic behavior, which leads to an effective local functional expressed through a tangentially constrained nonlocal cell problem. In addition, a nonlocal analogue of the classical limit decomposition result for gradient fields is established.Finally, Chapter 5 investigates the asymptotic behavior of nonlocal energy functionals associated with nonnegative measurable kernels that recovers the Lp-norm of the function, thereby generalizing the so-called Maz’ya–Shaposhnikova formula. By formulating the problem in differ- ent functional spaces, we characterize the necessary and sufficient conditions on the family of interaction kernels that guarantee convergence. The results are established in the smooth setting and then extended, via density arguments preserving the kernel conditions, to integer-order and fractional Sobolev spaces.