Stationary Configuration of some Optimal Shaping

TL;DR

This paper investigates the optimal placement of Dirichlet regions within a domain to minimize a functional, focusing on configurations of either one-dimensional connected sets or finite point sets, and provides necessary optimality conditions.

Contribution

It introduces a novel analysis of optimal Dirichlet region configurations, considering both connected sets and finite point sets, with derived necessary optimality conditions.

Findings

Necessary conditions for optimality of Dirichlet regions.

Extension of shape optimization to lower-dimensional and point configurations.

Framework applicable to various control variable classes in shape optimization.

Abstract

We consider the problem of optimal location of a Dirichlet region in a $d$-dimensional domain $\Omega$ subjected to a given right-hand side $f$ in order to minimize some given functional of the configuration. While in the literature the Dirichlet region is usually taken $d-1$ dimensional, in this shape optimization problems, we consider two classes of control variables, namely the class of one dimensional closed connected sets of finite one dimensional Hausdorff measure and the class of sets of points of finite cardinality, and we give a necessary condition of optimality.