Shape sensitivity analysis of the heat equation and the Dirichlet-to-Neumann map

Matteo Dalla Riva; Paolo Luzzini; Paolo Musolino

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arXiv · 2506.21196

Shape sensitivity analysis of the heat equation and the Dirichlet-to-Neumann map

Matteo Dalla Riva·Paolo Luzzini·Paolo Musolino

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Abstract

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 $\phi$. We analyze the maps that assign to the infinite-dimensional shape parameter $\phi$ 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 $\phi$ of the normal derivative of the solution on the exterior boundary.

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Matteo Dalla Riva, Paolo Luzzini, Paolo Musolino. 2025-06-26. Shape sensitivity analysis of the heat equation and the Dirichlet-to-Neumann map. https://arxiv.org/abs/2506.21196

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