Inner-layer asymptotics in partially perforated domains: coupling across flat and oscillating interfaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914335668830208 |
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| author | Mel'nyk, Taras |
| author_facet | Mel'nyk, Taras |
| contents | The article examines a boundary-value problem in a domain consisting of perforated and imperforate regions, with Neumann conditions prescribed at
the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period $\varepsilon,$ we analyse the limit behavior of the problem as $\varepsilon \to 0.$ A crucial aspect of this study is deriving correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. For the flat interface, we construct and justify a complete asymptotic expansion of the solution in the $H^1$-Sobolev space. Furthermore, for the $\varepsilon$-periodically oscillating interface of amplitude $\mathcal{O}(\varepsilon),$ we provide an approximation to the solution and establish the corresponding asymptotic estimates in $H^1$-Sobolev spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02716 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inner-layer asymptotics in partially perforated domains: coupling across flat and oscillating interfaces Mel'nyk, Taras Analysis of PDEs 35B27, 35B40, 35B25, 35J25 The article examines a boundary-value problem in a domain consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period $\varepsilon,$ we analyse the limit behavior of the problem as $\varepsilon \to 0.$ A crucial aspect of this study is deriving correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. For the flat interface, we construct and justify a complete asymptotic expansion of the solution in the $H^1$-Sobolev space. Furthermore, for the $\varepsilon$-periodically oscillating interface of amplitude $\mathcal{O}(\varepsilon),$ we provide an approximation to the solution and establish the corresponding asymptotic estimates in $H^1$-Sobolev spaces. |
| title | Inner-layer asymptotics in partially perforated domains: coupling across flat and oscillating interfaces |
| topic | Analysis of PDEs 35B27, 35B40, 35B25, 35J25 |
| url | https://arxiv.org/abs/2504.02716 |