Shape differentials and sensitivity of the heat equation

Abstract

We consider Dirichlet and Neumann problems 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 $\varphi$. We analyze the maps that assign to the infinite-dimensional shape parameter $\varphi$ the corresponding solutions, and we prove that they are smooth. Motivated by applications to inverse problems, we then compute the differentials with respect to $\varphi$ of the Dirichlet-to-Neumann and Neumann-to-Dirichlet maps.