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

Abstract

In this paper we prove a strong two-scale approximation result for sphere-valued maps in the space 𝐿²⁢(Ω;𝑊¹,² 0⁢(𝑄0;𝕊²)) , where Ω⊂ℝ³ is an open domain and 𝑄₀⊂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.