Magnetization switching on small ferromagnetic ellipsoidal samples

Abstract

Abstract. The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.Mathematics Subject Classification.

Existence of Regular Solutions of the Landau--Lifshitz--Gilbert Equation in 3D with Natural Boundary Conditions

“…For the 3D torus (with periodic boundary conditions) [8] proves H 2 -regularity local in time for the coupled system of LLG and Maxwell-equations. The work [3] proves global existence of strong solutions for small initial energies on small ellipsoids. The survey article [21] summarizes results in the context of the evolution of harmonic maps (which however does not cover the LLG equation).…”

Hysteresis for ferromagnetism: asymptotics of some two-scale Landau–Lifshitz model

“…Global solutions and equilibria. In [1,Th. 4.3], in the case of ellipsoidal domains Ω ⊂ R 3 and under a smallness assumption (on h ext L ∞ and ∆m 0 L 2 ), Alouges and Beauchard construct global smooth solutions to (2.1).…”

“…Standard energy estimates ensure local-in-time existence and uniqueness of solutions continuous in time, with values in H 2 (Ω)) (with an existence time depending on ε): see for example [1] or [4]. By the usual continuation argument, we simply need to bound the H 2 norm of m ε to ensure existence up to time T .…”

The Newtonian potential and the demagnetizing factors of the general ellipsoid

“…Despite the well-knowness of (1.2) in the mathematical and physical community, and its importance in theoretical and applied studies [20][21][22][23][24][25][26][27][28][29], rigorous proofs of that result are not readily available in the literature: to the best knowledge of the author, relative modern treatments of the interior problem can be found in [13,14], and more recently in [30,31] where also the the exterior problem is investigated. However, in all the cited references, the solution of the problem is always based on the use of ellipsoidal coordinates which tends to focus the attention on the computational details of the question rather than on its geometric counterpart.…”