Asymptotic analysis, in a thin multidomain, of minimizing maps with values in \( S^{2} \)

Abstract

We consider a thin multidomain of R 3 consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the second one with small thickness and given cross section. The first part of this paper is devoted to analyze, in this thin multidomain, a "static Landau-Lifshitz equation", when the volumes of the two cylinders vanish. We derive the limit problem, which decomposes into two uncoupled problems, well posed on the limit cylinders (with dimensions 1 and 2, respectively). We precise how the limit problem depends on limit of the ratio between the volumes of the two cylinders. In the second part of this paper, we study the asymptotic behavior of the two limit problems, when the exterior limit fields increase. We show that in some cases, contrary to the initial problem, the energies of the limit problems diverge and we find the order of these energies.Résumé. Nous considérons un multi-domaine mince de R 3 se composant de deux cylindres verticaux, superposés l'un sur l'autre : le premier possède une taille donnée et une petite section transversale, le second a une petiteépaisseur et une section transversale donnée. La première partie de cet article est consacréeà analyser, dans ce multi-domaine, uneéquation stationnaire de type Landau-Lifshitz, quand les volumes des deux cylindres tendent vers 0. Nous montrons que le problème limite, se décompose en deux probèmes découplés, bien posés sur le domaine limite. Ensuite, nous précisons comment le problème limite dépend de la limite du rapport des volumes des deux cylindres. Dans la deuxième partie de cet article, nousétudions le comportement asymptotique des deux problèmes limites, quand les champs extérieurs limites augmentent. Nous prouvons que dans certains cas, contrairement au problème initial, lesénergies des problèmes limites divergent et nous précisons l'ordre de cesénergies.

Ferromagnetic of nanowires of infinite length and infinite thin films

“…Therefore, converges (14) follow and these converges hold true for the whole sequence. Convergence (15) is a consequence of (14).…”

“…Reformulating the problem on a fixed domain through appropriate rescalings of the kind proposed by Ciarlet and Destuynder [5] and using the main ideas used by [3,4,8,12,13] and [14], we derive the limit problem in both cases. Specifically, in the case of nanowires, we prove that (see Theorem 1)…”

Asymptotic analysis for two joined thin slanting ferromagnetic films

“…[11][12][13][14]) who studied the junction of ferromagnetic bodies. In [13], they considered two orthogonal ferromagnetic thin films and they proved that the limit magnetization is coupled when the volumes of the two thin films vanish with the same rate.…”

“…In [13], they considered two orthogonal ferromagnetic thin films and they proved that the limit magnetization is coupled when the volumes of the two thin films vanish with the same rate. In [12] and [11], they developed an asymptotic analysis of minimizing maps with values in S 2 for the energy Ω n (|DM | 2 − 2F n M )dx, neglecting the term with non-local magnetostatic energy.…”

$$nD-pD$$ n D - p D Dimensional reduction of micromagnetic structures

“…We explicitly remark that formula (1.5) was obtained in [13] in the case n = 3 and p = 1. For junction 1D-2D we refer to [10] and [15]. For junction 1D-1D we refer to [11].…”