Poisson-type problems with transmission conditions at boundaries of infinite metric trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909932754829312 |
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| author | Kachanovska, Maryna Naderi, Kiyan Pankrashkin, Konstantin |
| author_facet | Kachanovska, Maryna Naderi, Kiyan Pankrashkin, Konstantin |
| contents | The paper introduces a Poisson-type problem on a mixed-dimensional structure combining a Euclidean domain and a lower-dimensional self-similar component touching a compact surface (interface). The lower-dimensional piece is a so-called infinite metric tree (one-dimensional branching structure), and the key ingredient of the study is a rigorous definition of the gluing conditions between the two components. These constructions are based on the recent concept of embedded trace maps and some abstract machineries derived from a suitable Green-type formula. The problem is then reduced to the study of Fredholm properties of a linear combination of Dirichlet-to-Neumann maps for the tree and the Euclidean domain, which yields desired existence and uniqueness results. One also shows that finite sections of tree can be used for an efficient approximation of the solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poisson-type problems with transmission conditions at boundaries of infinite metric trees Kachanovska, Maryna Naderi, Kiyan Pankrashkin, Konstantin Analysis of PDEs Mathematical Physics Functional Analysis Spectral Theory The paper introduces a Poisson-type problem on a mixed-dimensional structure combining a Euclidean domain and a lower-dimensional self-similar component touching a compact surface (interface). The lower-dimensional piece is a so-called infinite metric tree (one-dimensional branching structure), and the key ingredient of the study is a rigorous definition of the gluing conditions between the two components. These constructions are based on the recent concept of embedded trace maps and some abstract machineries derived from a suitable Green-type formula. The problem is then reduced to the study of Fredholm properties of a linear combination of Dirichlet-to-Neumann maps for the tree and the Euclidean domain, which yields desired existence and uniqueness results. One also shows that finite sections of tree can be used for an efficient approximation of the solutions. |
| title | Poisson-type problems with transmission conditions at boundaries of infinite metric trees |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis Spectral Theory |
| url | https://arxiv.org/abs/2506.11218 |