JournalszaaVol. 23, No. 3pp. 589–605

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

  • Dirk Praetorius

    Technische Universität Wien, Austria
Analysis of the Operator Δ^-1div Arising in Magnetic Models cover
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Abstract

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

div(u+\m)=0 in Rd\text{div}(-\nabla u+\m)=0\text{ in }\R^d

has to be solved in the entire space for a given magnetization \m:ΩRd\m:\Omega\to\R^d and ΩRd\Omega\subseteq\R^d. For an LpL^p function \m\m we show that the solution might fail to be in the classical Sobolev space W1,p(Rd)W^{1,p}(\R^d) but has to be in a Beppo-Levi class W1p(Rd)W_1^p(\R^d). We prove unique solvability in W1p(Rd)W_1^p(\R^d) and provide a direct ansatz to obtain uu via a non-local integral operator \LLp\LL_p related to the Newtonian potential. A possible discretization to compute (\LL2\m)\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