On the shape derivative of polygonal inclusions in the conductivity problem

Institut für Mathematik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany

^* Corresponding author: hanke@mathematik.uni-mainz.de

Received: 10 September 2024
Accepted: 8 June 2025

Abstract

We consider the conductivity problem for a homogeneous body with an inclusion of a different, but known, conductivity. Our interest concerns the associated shape derivative, i.e., the derivative of the corresponding electrostatic potential with respect to the shape of the inclusion. For a smooth inclusion it is known that the shape derivative is the solution of a specific inhomogeneous transmission problem. We show that this characterization of the shape derivative is also valid when the inclusion is a polygonal domain, but due to singularities at the vertices of the polygon, the shape derivative fails to belong to H1 in this case.