Search
Talk

Optimal shape and regularity of magnetic needle domains

  • Florian Nolte (Universität Heidelberg)
A3 01 (Sophus-Lie room)

Abstract

We study the nonlocal energy \begin{align*} \mathcal{E}(\Omega) = \mathcal{P}(\Omega) + \int_{\mathbb{R}^n} \left|\nabla \Phi_\Omega\right|^2 \text{\rm d} x \end{align*} where $\mathcal{P}(\Omega)$ denotes the perimeter of a set $\Omega \subset \mathbb{R}^n$ with prescribed volume $|\Omega|=V >0$ and $\Phi_\Omega \in \dot H^1(\mathbb{R}^n)$ is the weak solution of $\Delta \Phi_\Omega = \partial_1 \chi_\Omega$. This energy is related to the study of long and slender "needle" domains in uniaxial ferromagnetic materials where the second term models dipole interactions. We prove existence of minimizers and show that they are ($\mathcal{L}^n$ equivalent to) bounded, connected sets with smooth boundary. In particular we show that $\nabla \Phi_\Omega \in L^\infty$ for local minimizers. For the physically important case $n=3$, we furthermore establish a scaling law for the minimal energy in terms of the prescribed mass $V$.

Links

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

Upcoming Events of This Seminar

  • Mar 11, 2024 tba with Carlos Román Parra
  • Mar 15, 2024 tba with Esther Bou Dagher
  • Mar 27, 2024 tba with Christian Wagner
  • May 21, 2024 tba with Immanuel Zachhuber