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

TL;DR

This paper investigates how small changes in the shape of a domain affect solutions to the heat equation and the Dirichlet-to-Neumann map, with implications for inverse problems.

Contribution

It proves the smoothness of the solution and normal derivative maps with respect to domain shape and computes their derivatives for inverse problem applications.

Findings

Both maps are smooth functions of the shape parameter.

Explicit formulas for the derivatives with respect to shape are derived.

Results facilitate shape optimization and inverse problem solutions.

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.