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D. Praetorius:
"Analysis of the Operator $\Laplace^{-1}\div$ Arising in Magnetic Models";
Zeitschrift für Analysis und ihre Anwendungen (ZAA), 23 (2004), 589 - 605.

In the context of micromagnetics the partial differential equation

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

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.

http://aleph.ub.tuwien.ac.at/F?base=tuw01&func=find-c&ccl_term=AC04969719

http://www.anum.tuwien.ac.at/~dirk/download/preprint/potential.pdf