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

Abstract

In the context of micromagnetics the partial differential equation

has to be solved in the entire space for a given magnetization $\mathbf{m}:\Omega \to {\mathbb{R}}^d$ and $\Omega \subseteq {\mathbb{R}}^d$. For an $L^p$ function $\mathbf{m}$ we show that the solution might fail to be in the classical Sobolev space $W^{1,p}({\mathbb{R}}^d)$ but has to be in a Beppo-Levi class $W_1^p({\mathbb{R}}^d)$. We prove unique solvability in $W_1^p({\mathbb{R}}^d)$ and provide a direct ansatz to obtain $u$ via a non-local integral operator ${\mathcal{L}}_p$ related to the Newtonian potential. A possible discretization to compute $\nabla ({\mathcal{L}}_2\mathbf{m})$ 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.