Abstract |
The talk is devoted to the $\Gamma$-convergence analysis of the micromagnetic energy functional when the domain occupied by the nanomagnet is a thin shell generated by a bounded and convex smooth surface. Indeed, recently a significant interest to nanomagnets with curved shape has appeared. In particular, spherical shells are currently of great interest due to their capability to support skyrmion solutions which can be stabilized by curvature effects only, in contrast to the planar case where the intrinsic Dzyaloshinsky-Moriya interaction is required. It is well established that the effects of the demagnetizing field operator can be reduced to an effective easy-surface anisotropy for planar thin shells whose thickness is much less than the size of the system. More precisely, in G. Gioia and R. D. James [Micromagnetics of very thin films. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 453(1956):213 - 223, 1997] it is shown that the effects of the demagnetizing field operator can be reduced to an effective easy-surface anisotropy for planar thin shells. A generalization of this result can be found in G. Carbou [Thin layers in micromagnetism. Mathematical Models and Methods in Applied Sciences, 11(09):1529 - 1546, 2001] where, for thin shells generated by extruding surfaces whose closure is diffeomorphic to the closed unit disk of $\mathbb{R}^2$, the asymptotic behavior of the minimizers of the micromagnetic energy functional is investigated. In V. Slastikov [Micromagnetics of thin shells. Mathematical Models and Methods in Applied Sciences, 15(10):1469 - 1487, 2005] a $\Gamma$-convergence analysis is performed on pillow-like thin shells. In all the cited cases the investigation leaves out very interesting scenarios like the spherical one which cannot be easily recovered by simply gluing local patches. In this talk, we present a $\Gamma$-convergence analysis of the micromagnetic energy functional when the shell is generated, like in the case of a sphere, by a bounded and convex smooth surface. |