On the shape derivative of polygonal inclusions in the conductivity problem

TL;DR

This paper investigates the shape derivative of the electrostatic potential in a conductivity problem with polygonal inclusions, revealing that singularities at vertices prevent the derivative from being in H^1, unlike smooth cases.

Contribution

It extends the understanding of shape derivatives to polygonal inclusions, showing the failure of H^1 regularity due to vertex singularities.

Findings

Shape derivative characterized by an inhomogeneous transmission problem.

Singularities at polygon vertices prevent the shape derivative from being in H^1.

The known smooth case characterization extends to polygonal domains despite singularities.

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 $H^1$ in this case.