Calculus of Variations and Geometric Measure Theory

A. Braides - G. C. Brusca - D. Donati

Another look at elliptic homogenization

created by braidesa on 21 Jun 2023
modified on 04 Dec 2023


Accepted Paper

Inserted: 21 jun 2023
Last Updated: 4 dec 2023

Journal: Milan J. Math
Year: 2023
Doi: 10.1007/s00032-023-00389-y
Links: view-only version


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 $\Gamma$-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 $\varepsilon\to 0$ of the Dirichlet integral with oscillating coefficient obtained by formally letting $s\to 1$ first.