Variational methods for a singular SPDE yielding the universality of the magnetization ripple |
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Authors: | Radu Ignat Felix Otto Tobias Ried Pavlos Tsatsoulis |
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Affiliation: | 1. Institut de Mathématiques de Toulouse & Institut Universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, Toulouse, Cedex 9 France;2. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22, Leipzig, Germany |
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Abstract: | The magnetization ripple is a microstructure formed in thin ferromagnetic films. It can be described by minimizers of a nonconvex energy functional leading to a nonlocal and nonlinear elliptic SPDE in two dimensions driven by white noise, which is singular. We address the universal character of the magnetization ripple using variational methods based on Γ-convergence. Due to the infinite energy of the system, the (random) energy functional has to be renormalized. Using the topology of Γ-convergence, we give a sense to the law of the renormalized functional that is independent of the way white noise is approximated. More precisely, this universality holds in the class of (not necessarily Gaussian) approximations to white noise satisfying the spectral gap inequality, which allows us to obtain sharp stochastic estimates. As a corollary, we obtain the existence of minimizers with optimal regularity. |
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