Analysis of the operator Δ-1div arising in magnetic models

Abstract

In the context of micromagnetics the partial differential equation div(-∇u + m) = 0 in ℝd has to be solved in the entire space for a given magnetization m : Ω → ℝd and Ω ⊆ ℝd. For an Lp function m we show that the solution might fail to be in the classical Sobolev space W 1,p(ℝd) but has to be in a Beppo-Levi class W 1p(ℝd). We prove unique solvability in W1p(ℝd) and provide a direct ansatz to obtain u via a non-local integral operator Lp related to the Newtonian potential. A possible discretization to compute ∇(L 2m.) 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.