Another look at elliptic homogenization
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arXiv
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| Format: | Preprint |
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2023
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| author | Braides, Andrea Brusca, Giuseppe Cosma Donati, Davide |
| author_facet | Braides, Andrea Brusca, Giuseppe Cosma Donati, Davide |
| contents | 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 $Γ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12325 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Another look at elliptic homogenization Braides, Andrea Brusca, Giuseppe Cosma Donati, Davide Analysis of PDEs 49J45, 35B27, 35R11 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 $Γ$-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. |
| title | Another look at elliptic homogenization |
| topic | Analysis of PDEs 49J45, 35B27, 35R11 |
| url | https://arxiv.org/abs/2306.12325 |