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Zeitschrift für Analysis und ihre Anwendungen

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Volume 23, Issue 3, 2004, pp. 589–605
DOI: 10.4171/ZAA/1212

Published online: 2004-09-30

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

Dirk Praetorius[1]

(1) Technische Universität Wien, Austria

\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 $\Omega\subseteq\R^d$. 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 $W_1^p(\R^d)$. We prove unique solvability in $W_1^p(\R^d)$ 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.

Keywords: Laplace equation, integral representation, Calderón-Zygmund kernel, micromagnetics, magnetic potential, panel clustering, hierarchical matrices

Praetorius Dirk: Analysis of the Operator Δ^-1div Arising in Magnetic Models. Z. Anal. Anwend. 23 (2004), 589-605. doi: 10.4171/ZAA/1212