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 $\Gamma$-convergence. Due to the infinite energy of the system, the (random) energy functional has to be renormalized. Using the topology of $\Gamma$-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.

arXiv comment63 pages, 1 figure

References in corpus (7)

• A theory of regularity structures
• The dynamic ${\Phi }_3^4$ model comes down from infinity
• An analytic BPHZ theorem for regularity structures
• Global solutions to elliptic and parabolic ${\Phi }^4$ models in Euclidean space
• The Formation and Coarsening of the Concertina Pattern
• Interaction energy of domain walls in a nonlocal Ginzburg-Landau type model from micromagnetics
• A priori bounds for the ${\Phi }^4$ equation in the full sub-critical regime

Cited by in corpus (2)

• A diagram-free approach to the stochastic estimates in regularity structures
• Flow equation approach to singular stochastic PDEs