Asymptotic shape of isolated magnetic domains

TL;DR

This paper analyzes the asymptotic shape of large isolated magnetic domains by deriving a limit energy functional and identifying the optimal domain configurations in the macroscopic regime.

Contribution

It introduces a $ ext{Gamma}$-convergence framework for the energy of magnetic domains and characterizes the asymptotic shapes as the domain size grows large.

Findings

Derived the $ ext{Gamma}$-limit of the energy functional for large domains.

Identified the asymptotic shape solutions for the limit problem.

Provided compactness results for the energy minimizers.

Abstract

We investigate the energy of an isolated magnetized domain $\Omega \subset \mathbb{R}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given by $$ \mathcal{E}(\Omega) \ = \ \int_{\mathbb{R}^n} |\nabla \chi_{\Omega}| \ dx + \int_{\mathbb{R}^n} |\nabla h_\Omega|^2 \ dx $$ penalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potential $h_\Omega$ is determined by $\Delta h_\Omega = \partial_1 \chi_\Omega$, corresponding to uniform magnetization within the domain. We consider the macroscopic regime $|\Omega| \rightarrow \infty$, in which we derive compactness and $\Gamma$-limit which is formulated in terms of the cross-sectional area of the anisotropically rescaled configuration. We then give the solutions for the limit problems.