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

TL;DR

This paper develops variational methods to analyze a singular stochastic PDE modeling magnetization ripples in ferromagnetic films, establishing universality and existence of regular minimizers through $$-convergence and renormalization.

Contribution

It introduces a novel variational framework for a singular SPDE, proving universality of the magnetization ripple and existence of minimizers with optimal regularity.

Findings

Universality of the magnetization ripple law independent of noise approximation.

Existence of minimizers with optimal regularity.

Sharp stochastic estimates under spectral gap inequality.

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.