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 the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>;</m:mo> <m:mrow> <m:msubsup> <m:mi>W</m:mi> <m:mn>0</m:mn> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>Q</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>;</m:mo> <m:msup> <m:mi>𝕊</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L²(Ω;W1,2₀(Q₀;S²))} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> {Ω³} is an open domain and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Q</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⊂</m:mo> <m:mi>Q</m:mi> </m:mrow> </m:math> {Q₀⊂ Q} an open subset of the unit cube <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Q</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> {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.