A shape optimization problem on planar sets with prescribed topology

TL;DR

This paper investigates shape optimization problems for planar sets with fixed topology, analyzing the existence of optimal domains based on a parameter q and providing explicit results for specific cases.

Contribution

It introduces a relaxed formulation for topology-constrained shape optimization problems and characterizes the existence of solutions depending on the parameter q.

Findings

Optimal relaxed domains exist for q<1/2.

The problem is ill-posed for q>1/2.

Explicit infimum values are given for q=1/2 with k=0 and k=1.

Abstract

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $P(\Omega)T^q(\Omega)|\Omega|^{-2q-1/2}$ and the class of admissible domains consists of two-dimensional open sets $\Omega$ satisfying the topological constraints of having a prescribed number $k$ of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem and we show that when $q<1/2$ an optimal relaxed domain exists. When $q>1/2$ the problem is ill-posed and for $q=1/2$ the explicit value of the infimum is provided in the cases $k=0$ and $k=1$.