Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel-Zheng theory

TL;DR

This paper extends the Bethuel-Zheng theory to a two-scale setting for sphere-valued Sobolev maps, providing a strong approximation result and applying it to high-contrast micromagnetic variational problems.

Contribution

It generalizes the Bethuel-Zheng two-scale approximation to high-contrast sphere-valued Sobolev maps, with applications in micromagnetics.

Findings

Established a two-scale approximation for sphere-valued Sobolev maps.

Extended Bethuel-Zheng argument to a high-contrast setting.

Applied the theory to a micromagnetic variational problem.

Abstract

In this paper we prove a strong two-scale approximation result for sphere-valued maps in $L^2(\Omega;W^{1,2}_0(Q_0;\mathbb{S}^2))$, where $\Omega\subset \mathbb{R}^3$ is an open domain and $Q_0\subset Q$ an open subset of the unit cube $Q=(0,1)^3$. The proof relies on a generalization of the seminal argument by F. Bethuel and X.M. Zheng to the two-scale setting. We then present an application to a variational problem in high-contrast micromagnetics.