Another look at elliptic homogenization

We consider the limit of sequences of normalized $(s,2)$-Gagliardo seminorms with an oscillating coefficient as $s\to 1$. In a seminal paper by Bourgain, Brezis and Mironescu (subsequently extended by Ponce) it is proven that if the coefficient is constant then this sequence $\mathrm{\Gamma }$-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by $\epsilon$ the scale of the oscillations and we assume that $1-s<\text{\hspace{0.17em}⁣}<{\epsilon }^2$, this sequence converges to the homogenized functional formally obtained by separating the effects of $s$ and $\epsilon$; that is, by the homogenization as $\epsilon \to 0$ of the Dirichlet integral with oscillating coefficient obtained by formally letting $s\to 1$ first.