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

In the context of micromagnetics the partial differential equation div(−∇u + m) = 0 in R 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 p 1 (Rd). We prove unique solvability in W p 1 (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 ∇(L2m) 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.