Homogenization of non-local energies on disconnected sets
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Braides, Andrea Scalabrino, Sergio Trifone, Chiara |
| author_facet | Braides, Andrea Scalabrino, Sergio Trifone, Chiara |
| contents | We consider the problem of the homogenization of non-local quadratic energies defined on $δ$-periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale $\varepsilon$. For kernels with unbounded support we show that we may have three regimes: (i) $\varepsilon<\!<δ$, for which the $Γ$-limit even in the strong topology of $L^2$ is $0$; (ii) $\frac\varepsilonδ\toκ$, in which the energies are coercive with respect to a convergence of interpolated functions, and the limit is governed by a non-local homogenization formula parameterized by $κ$; (iii) $δ<\!<\varepsilon$, for which the $Γ$-limit is computed with respect to a coarse-grained convergence and exhibits a separation-of-scales effect; namely, it is the same as the one obtained by formally first letting $δ\to 0$ (which turns out to be a pointwise weak limit, thanks to an iterated use of Jensen's inequality), and then, noting that the outcome is a nonlocal energy studied by Bourgain, Brezis and Mironescu, letting $\varepsilon\to0$. A slightly more complex description is necessary for case (ii) if the kernel is compactly supported. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09677 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homogenization of non-local energies on disconnected sets Braides, Andrea Scalabrino, Sergio Trifone, Chiara Analysis of PDEs Optimization and Control 35B27, 74Q05, 49J45, 26A33, 74A70 We consider the problem of the homogenization of non-local quadratic energies defined on $δ$-periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale $\varepsilon$. For kernels with unbounded support we show that we may have three regimes: (i) $\varepsilon<\!<δ$, for which the $Γ$-limit even in the strong topology of $L^2$ is $0$; (ii) $\frac\varepsilonδ\toκ$, in which the energies are coercive with respect to a convergence of interpolated functions, and the limit is governed by a non-local homogenization formula parameterized by $κ$; (iii) $δ<\!<\varepsilon$, for which the $Γ$-limit is computed with respect to a coarse-grained convergence and exhibits a separation-of-scales effect; namely, it is the same as the one obtained by formally first letting $δ\to 0$ (which turns out to be a pointwise weak limit, thanks to an iterated use of Jensen's inequality), and then, noting that the outcome is a nonlocal energy studied by Bourgain, Brezis and Mironescu, letting $\varepsilon\to0$. A slightly more complex description is necessary for case (ii) if the kernel is compactly supported. |
| title | Homogenization of non-local energies on disconnected sets |
| topic | Analysis of PDEs Optimization and Control 35B27, 74Q05, 49J45, 26A33, 74A70 |
| url | https://arxiv.org/abs/2405.09677 |