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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2406.16040 |
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| _version_ | 1866909230026457088 |
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| author | Alicandro, Roberto Gelli, Maria Stella Leone, Chiara |
| author_facet | Alicandro, Roberto Gelli, Maria Stella Leone, Chiara |
| contents | In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $Γ$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the asymptotic behaviour of the functionals when the order parameter is subject to Dirichlet conditions on a periodically perforated domains, given by a periodic array of small balls of radius $r_δ$ centered on a $δ$--periodic lattice, being $δ> 0$ an additional small parameter and $r_δ=o(δ)$. We highlight differences and analogies with the local case, according to the interplay between the three scales $\varepsilon$, $δ$ and $r_δ$. A fundamental tool in our analysis turns out to be a non local variant of the classical Gagliardo-Nirenberg-Sobolev inequality in Sobolev spaces which may be of independent interest and useful for other applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variational analysis of nonlocal Dirichlet problems in periodically perforated domains Alicandro, Roberto Gelli, Maria Stella Leone, Chiara Analysis of PDEs In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $Γ$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the asymptotic behaviour of the functionals when the order parameter is subject to Dirichlet conditions on a periodically perforated domains, given by a periodic array of small balls of radius $r_δ$ centered on a $δ$--periodic lattice, being $δ> 0$ an additional small parameter and $r_δ=o(δ)$. We highlight differences and analogies with the local case, according to the interplay between the three scales $\varepsilon$, $δ$ and $r_δ$. A fundamental tool in our analysis turns out to be a non local variant of the classical Gagliardo-Nirenberg-Sobolev inequality in Sobolev spaces which may be of independent interest and useful for other applications. |
| title | Variational analysis of nonlocal Dirichlet problems in periodically perforated domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.16040 |