Shape sensitivity analysis of the heat equation and the Dirichlet-to-Neumann map
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911024272113664 |
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| author | Riva, Matteo Dalla Luzzini, Paolo Musolino, Paolo |
| author_facet | Riva, Matteo Dalla Luzzini, Paolo Musolino, Paolo |
| contents | We study a Dirichlet problem for the heat equation in a domain containing an interior hole. The domain has a fixed outer boundary and a variable inner boundary determined by a diffeomorphism $ϕ$. We analyze the maps that assign to the infinite-dimensional shape parameter $ϕ$ the corresponding solution and its normal derivative, and we prove that both are smooth. Motivated by an application to an inverse problem, we then compute the differential with respect to $ϕ$ of the normal derivative of the solution on the exterior boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shape sensitivity analysis of the heat equation and the Dirichlet-to-Neumann map Riva, Matteo Dalla Luzzini, Paolo Musolino, Paolo Analysis of PDEs 35K20, 31B10, 47H30, 45A05 We study a Dirichlet problem for the heat equation in a domain containing an interior hole. The domain has a fixed outer boundary and a variable inner boundary determined by a diffeomorphism $ϕ$. We analyze the maps that assign to the infinite-dimensional shape parameter $ϕ$ the corresponding solution and its normal derivative, and we prove that both are smooth. Motivated by an application to an inverse problem, we then compute the differential with respect to $ϕ$ of the normal derivative of the solution on the exterior boundary. |
| title | Shape sensitivity analysis of the heat equation and the Dirichlet-to-Neumann map |
| topic | Analysis of PDEs 35K20, 31B10, 47H30, 45A05 |
| url | https://arxiv.org/abs/2506.21196 |