Asymptotic analysis, in a thin multidomain, of minimizing maps with values in \(S^2\). (English) Zbl 1156.78300
Summary: We consider a thin multidomain of \(\mathbb R^3\) 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.
MSC:
| 78A25 | Electromagnetic theory (general) |
| 74K05 | Strings |
| 74K30 | Junctions |
| 74K35 | Thin films |
| 35B25 | Singular perturbations in context of PDEs |
Keywords:
maps with values in \(S^{2}\); thin multidomains; dimension reduction; singular perturbationsReferences:
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