On the Γ-convergence of the energies and the convergence of almost minimizers in infinite magnetic cylinders

Abstract

In this thesis we study static 180 degree domain walls in infinite thin magnetic wires with either a rectangular or a centrally symmetric Lipschitz cross section. We explore the magnetization energy minimization problem by finding an approximation for the magnetostatic energy. Two different pattern formations of the magnetization have been observed. In dependence of the thickness of the wire, different pattern formations of the magnetization vector are observed. We prove an existence of global minimizers(even for Lipschitz cross sections). We prove a Γ-convergence result for both types of thin wires. For rectangular cross sections we distinguish two different regimes and establish the minimal energy scaling in terms of the cross section edge's lengths. For a centrally symmetric cross section we establish as well the minimal energy scaling in terms of the diameter of the cross section and some geometric parameters relating to it. We prove as well a rate of convergence for the minimal energies for all cases. For thick wires with a rectangular cross section we prove an upper bound and give a reference for a lower bound on the minimal energy. For thin wires a Néel wall occurs and for thick wires a vortex wall is expected to occur.