Mathematics

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, Brezis and Mironescu (subsequently extended by Ponce) 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 \ll \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.This is a joint work with Professor Andrea Braides and Giuseppe Cosma Brusca.