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arXiv · 2406.11365

Shape perturbation of a nonlinear mixed problem for the heat equation

Matteo Dalla Riva·Paolo Luzzini·Riccardo Molinarolo·Paolo Musolino
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Abstract

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.

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Matteo Dalla Riva, Paolo Luzzini, Riccardo Molinarolo, Paolo Musolino. 2024-12-12. Shape perturbation of a nonlinear mixed problem for the heat equation. https://arxiv.org/abs/2406.11365

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