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On the shape derivative of polygonal inclusions in the conductivity problem

Authors :
Hanke, Martin
Publication Year :
2024

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.

Subjects

Subjects :
Mathematics - Analysis of PDEs

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2402.02793
Document Type :
Working Paper