Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel–Zheng theory
Elisa Davoli, Leon Happ · Advances in Calculus of Variations · 2025
Abstract In this paper we prove a strong two-scale approximation result for sphere-valued maps in the space L 2 ( Ω ; W 0 1 , 2 ( Q 0 ; 𝕊 2 ) ) {L^{2}(\Omega;W^{1,2}_{0}(Q_{0};\mathbb{S}^{2}))} , where Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} is an open domain and Q 0 ⊂ Q {Q_{0}\subset Q} an open subset of the unit cube Q = ( 0 , 1 ) 3 {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.