# Analysis of the Operator Δ^-1div Arising in Magnetic Models

### Dirk Praetorius

Technische Universität Wien, Austria

## Abstract

\newcommand{\m}{\text{\bf m}} \newcommand{\R}{\mathbb R} \newcommand{\LL}{{\cal L}} In the context of micromagnetics the partial differential equation

\[ \text{div}(-\nabla u+\m)=0\text{ in }\R^d \]has to be solved in the entire space for a given magnetization \( \m:\Omega\to\R^d \) and $Ω⊆R_{d}$. For an $L_{p}$ function \( \m \) we show that the solution might fail to be in the classical Sobolev space $W_{1,p}(R_{d})$ but has to be in a Beppo-Levi class $W_{1}(R_{d})$. We prove unique solvability in $W_{1}(R_{d})$ and provide a direct ansatz to obtain $u$ via a non-local integral operator \( \LL_p \) related to the Newtonian potential. A possible discretization to compute \( \nabla(\LL_2\m) \) is mentioned, and it is shown how recently established matrix compression techniques using hierarchical matrices can be applied to the full matrix obtained from the discrete operator.

## Cite this article

Dirk Praetorius, Analysis of the Operator Δ^-1div Arising in Magnetic Models. Z. Anal. Anwend. 23 (2004), no. 3, pp. 589–605

DOI 10.4171/ZAA/1212