Shape perturbation of a nonlinear mixed problem for the heat equation
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910741472215040 |
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| author | Riva, Matteo Dalla Luzzini, Paolo Molinarolo, Riccardo Musolino, Paolo |
| author_facet | Riva, Matteo Dalla Luzzini, Paolo Molinarolo, Riccardo Musolino, Paolo |
| contents | We consider the heat equation in a domain that has a hole in its interior. We impose a Neumann condition on the exterior boundary and a nonlinear Robin condition on the boundary of the hole. The shape of the hole is determined by a suitable diffeomorphism $ϕ$ defined on the boundary of a reference domain. Assuming that the problem has a solution $u_0$ when $ϕ$ is the identity map, we demonstrate that a solution $u_ϕ$ continues to exist for $ϕ$ close to the identity map and that the "domain-to-solution" map $ϕ\mapsto u_ϕ$ is of class $C^\infty$. Moreover, we show that the family of solutions $\{u_ϕ\}_ϕ$ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks a the linear case complete the paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_11365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shape perturbation of a nonlinear mixed problem for the heat equation Riva, Matteo Dalla Luzzini, Paolo Molinarolo, Riccardo Musolino, Paolo Analysis of PDEs 35K20, 31B10, 47H30, 45A05 We consider the heat equation in a domain that has a hole in its interior. We impose a Neumann condition on the exterior boundary and a nonlinear Robin condition on the boundary of the hole. The shape of the hole is determined by a suitable diffeomorphism $ϕ$ defined on the boundary of a reference domain. Assuming that the problem has a solution $u_0$ when $ϕ$ is the identity map, we demonstrate that a solution $u_ϕ$ continues to exist for $ϕ$ close to the identity map and that the "domain-to-solution" map $ϕ\mapsto u_ϕ$ is of class $C^\infty$. Moreover, we show that the family of solutions $\{u_ϕ\}_ϕ$ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks a the linear case complete the paper. |
| title | Shape perturbation of a nonlinear mixed problem for the heat equation |
| topic | Analysis of PDEs 35K20, 31B10, 47H30, 45A05 |
| url | https://arxiv.org/abs/2406.11365 |