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

In this paper we prove a strong two-scale approximation result for sphere-valued maps in $L^2(\mathrm{\Omega };W_0^{1,2}(Q_0;{\mathbb{S}}^2))$, where $\mathrm{\Omega }\subset {\mathbb{R}}^3$ 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.