Another Look at Elliptic Homogenization

Abstract

We consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms with an oscillating coefficient as \(s\rightarrow 1\). In a seminal paper by Bourgain et al. (Another look at Sobolev spaces. In: Optimal control and partial differential equations. IOS, Amsterdam, pp 439–455, 2001) it is proven that if the coefficient is constant then this sequence \(\Gamma \)-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by \(\varepsilon \) the scale of the oscillations and we assume that \(1-s<\!<\varepsilon ^2\), this sequence converges to the homogenized functional formally obtained by separating the effects of s and \(\varepsilon \); that is, by the homogenization as \(\varepsilon \rightarrow 0\) of the Dirichlet integral with oscillating coefficient obtained by formally letting \(s\rightarrow 1\) first.