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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.18430 |
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| _version_ | 1866912849692983296 |
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| author | Arrieta, José M. Domínguez-de-Tena, Joaquín |
| author_facet | Arrieta, José M. Domínguez-de-Tena, Joaquín |
| contents | This work analyses the homogenization of a linear elliptic equation with Neumann boundary conditions in a comb/brush domain, composed of a fixed base and a family of thin teeth. The teeth are defined as rescalings of order less than or equal to $\varepsilon$ of a model tooth of arbitrary shape. Periodicity in their distribution is not assumed; instead, the existence of an asymptotic limit density $θ$, which may vanish in certain regions, is assumed. The convergence analysis is performed using an adaptation of the unfolding operator method to a non-periodic framework. Finally, it is shown that, under certain conditions on the geometry of the teeth, the resulting limit problem can be interpreted as a differential equation on a graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18430 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A brush problem. Homogenization involving thin domains and PDEs in graphs Arrieta, José M. Domínguez-de-Tena, Joaquín Analysis of PDEs 35B27, 35J25, 35R02 This work analyses the homogenization of a linear elliptic equation with Neumann boundary conditions in a comb/brush domain, composed of a fixed base and a family of thin teeth. The teeth are defined as rescalings of order less than or equal to $\varepsilon$ of a model tooth of arbitrary shape. Periodicity in their distribution is not assumed; instead, the existence of an asymptotic limit density $θ$, which may vanish in certain regions, is assumed. The convergence analysis is performed using an adaptation of the unfolding operator method to a non-periodic framework. Finally, it is shown that, under certain conditions on the geometry of the teeth, the resulting limit problem can be interpreted as a differential equation on a graph. |
| title | A brush problem. Homogenization involving thin domains and PDEs in graphs |
| topic | Analysis of PDEs 35B27, 35J25, 35R02 |
| url | https://arxiv.org/abs/2601.18430 |