Homogenization of Hamilton-Jacobi equations on networks
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929579469307904 |
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| author | Pozza, Marco Siconolfi, Antonio Sorrentino, Alfonso |
| author_facet | Pozza, Marco Siconolfi, Antonio Sorrentino, Alfonso |
| contents | We prove a homogenization result for a family of time-dependent Hamilton-Jacobi equations, rescaled by a parameter $\varepsilon$ tending to zero, posed on a periodic network, with a suitable notion of periodicity that will be defined. As $\varepsilon$ becomes infinitesimal, we derive a limiting Hamilton-Jacobi equation in a Euclidean space, whose dimension is determined by the topological complexity of the network and is independent of the ambient space in which the network is embedded. Among the key contributions of our analysis, we extend to the setting of networks and graphs Mather's result on the asymptotic behavior of the average minimal action functional, as time tends to infinity. Additionally, we establish the well-posedness of the approximating problems, representing a nontrivial generalization of existing results for finite networks to a non-compact setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homogenization of Hamilton-Jacobi equations on networks Pozza, Marco Siconolfi, Antonio Sorrentino, Alfonso Analysis of PDEs 35B27, 35R02, 35F21, 37J51, 49L25 We prove a homogenization result for a family of time-dependent Hamilton-Jacobi equations, rescaled by a parameter $\varepsilon$ tending to zero, posed on a periodic network, with a suitable notion of periodicity that will be defined. As $\varepsilon$ becomes infinitesimal, we derive a limiting Hamilton-Jacobi equation in a Euclidean space, whose dimension is determined by the topological complexity of the network and is independent of the ambient space in which the network is embedded. Among the key contributions of our analysis, we extend to the setting of networks and graphs Mather's result on the asymptotic behavior of the average minimal action functional, as time tends to infinity. Additionally, we establish the well-posedness of the approximating problems, representing a nontrivial generalization of existing results for finite networks to a non-compact setting. |
| title | Homogenization of Hamilton-Jacobi equations on networks |
| topic | Analysis of PDEs 35B27, 35R02, 35F21, 37J51, 49L25 |
| url | https://arxiv.org/abs/2411.03803 |