Variational analysis of nonlocal Dirichlet problems in periodically perforated domains

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter \(\varepsilon>0\) and \(\Gamma\)-converging to local functionals defined on Sobolev spaces as \(\varepsilon\to 0\). We study the asymptotic behaviour of the functionals when the order pa...

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Bibliographic Details
Published in:arXiv.org 2024-06
Main Authors: Alicandro, Roberto, Gelli, Maria Stella, Leone, Chiara
Format: Article
Language:English
Subjects:
Asymptotic properties
Convergence
Dirichlet problem
Order parameters
Sobolev space
Online Access:Get full text
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Summary:In this paper we consider a family of non local functionals of convolution-type depending on a small parameter \(\varepsilon>0\) and \(\Gamma\)-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 \(r_\delta\) centered on a \(\delta\)--periodic lattice, being \(\delta > 0\) an additional small parameter and \(r_\delta=o(\delta)\). We highlight differences and analogies with the local case, according to the interplay between the three scales \(\varepsilon\), \(\delta\) and \(r_\delta\). 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.
ISSN:2331-8422