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

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