Optimal Shape of Isolated Ferromagnetic Domains
Hans Knüpfer, Florian Nolte · SIAM Journal on Mathematical Analysis · 2018
We investigate the optimal shape and the scaling of the minimal energy of an isolated magnetic domain. We study the nonlocal energy ${\mathcal E}(\Omega) = \text{Perimeter}(\Omega) + \int_{\mathbb{R}^n} \left| abla \Phi_\Omega\right|^2 \, {d} x$ for sets $\Omega \subset \mathbb{R}^n$ with prescribed volume $V := |\Omega|$ and finite perimeter. The magnetostatic potential $\Phi_\Omega \in \dot H^1(\mathbb{R}^n)$ is defined as the solution of $\Delta \Phi_\Omega = \partial_1 \chi_\Omega$. This energy appears in the nucleation theory for magnetization reversal in uniaxial materials. It can also be used as a basic model to describe the shape of ferrofluidic droplets in an isolating environment. We show existence of minimizers for all volumes $V>0$. Furthermore, we derive a scaling law for the minimal energy in the physical case $n=3$. Moreover, we show further properties on regularity and topology for local minimizers and the potential. For the analysis, we do not use any assumptions on the shape of the domain.