Another look at elliptic homogenization
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Date
2023-06-21
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Abstract
We consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms
with an oscillating coefficient as s → 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 Γ-converges to a multiple of the Dirichlet integral. Here we prove that,
if we denote by ε the scale of the oscillations and we assume that 1−s << ε2, this sequence converges to the homogenized functional formally obtained by separating the effects of s and ε; that is, by the homogenization as ε → 0 of the Dirichlet integral with oscillating coefficient obtained by formally letting s → 1 first.
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Keywords
Γ-convergence, non-local functionals, fractional Sobolev spaces, homogenization