Variational methods for a singular SPDE yielding the universality of the magnetization ripple

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.

Characterizing models in regularity structures: a quasilinear case

“…We want to emphasize that such situations (in particular for non-Gaussian approximations) naturally appear in applications, e.g. in thin ferromagnetic films whose idealized physical model is triggered by white noise, but the actual physical approximation is not Gaussian due to the polycrystallinity of the material [14,15]. • Lifting symmetries.…”

“…A first universality result in the context of a spectral gap assumption was obtained in [15] for a mildly singular SPDE; it relies on a characterization of the limiting model by suitable commutator estimates instead of the Malliavin derivative. Although such commutator estimates could also serve for a characterization of the model associated to (1.1), we believe that the approach via the Malliavin derivative pursued here is more suitable for extensions to other equations.…”