On the shape derivative of polygonal inclusions in the conductivity problem
Analysis of PDEs
2025-06-10 v2
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 in this case.
Keywords
Cite
@article{arxiv.2402.02793,
title = {On the shape derivative of polygonal inclusions in the conductivity problem},
author = {Martin Hanke},
journal= {arXiv preprint arXiv:2402.02793},
year = {2025}
}