Variational analysis of discrete Dirichlet problems in periodically perforated domains
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912266911219712 |
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| author | Fusco, Giuliana |
| author_facet | Fusco, Giuliana |
| contents | In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, $\varepsilon>0$, tends to zero. We consider pairwise interaction energies satisfying $p$-growth conditions, $p<d$, $d$ being the dimension of the reference configuration, defined on discrete functions subject to Dirichlet conditions on a $δ$-periodic array of small squares of side $r_δ\sim δ^{d/d-p}$. Our analysis is performed in the framework of $Γ$-convergence and we prove that, in the regime $\varepsilon=o(r_δ)$, the discrete energy and their continuum counterpart share the same $Γ$-limit and the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational analysis of discrete Dirichlet problems in periodically perforated domains Fusco, Giuliana Analysis of PDEs 49J45, 49M25, 74Q05, 82B20 In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, $\varepsilon>0$, tends to zero. We consider pairwise interaction energies satisfying $p$-growth conditions, $p<d$, $d$ being the dimension of the reference configuration, defined on discrete functions subject to Dirichlet conditions on a $δ$-periodic array of small squares of side $r_δ\sim δ^{d/d-p}$. Our analysis is performed in the framework of $Γ$-convergence and we prove that, in the regime $\varepsilon=o(r_δ)$, the discrete energy and their continuum counterpart share the same $Γ$-limit and the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals. |
| title | Variational analysis of discrete Dirichlet problems in periodically perforated domains |
| topic | Analysis of PDEs 49J45, 49M25, 74Q05, 82B20 |
| url | https://arxiv.org/abs/2503.06723 |