Differentiability of the Dirichlet to Neumann map under movements of polygonal inclusions with an application to shape optimization

TL;DR

This paper rigorously derives the derivatives of the Dirichlet to Neumann and Neumann to Dirichlet maps for the conductivity equation as polygonal inclusions move, enabling shape optimization applications.

Contribution

It introduces a novel shape derivative approach for polygonal inclusions, advancing the mathematical understanding of boundary map sensitivities.

Findings

Derived explicit formulas for shape derivatives of boundary maps.

Applied derivatives to optimize shapes of polygonal inclusions.

Enhanced methods for shape optimization in conductivity problems.

Abstract

In this paper we derive rigorously the derivative of the Dirichlet to Neumann map and of the Neumann to Dirichlet map of the conductivity equation with respect to movements of vertices of triangular conductivity inclusions. We apply this result to formulate an optimization problem based on a shape derivative approach.