# A shape optimization problem on planar sets with prescribed topology

created by prinari on 17 Jan 2021
modified on 18 Jan 2021

[BibTeX]

Submitted Paper

Inserted: 17 jan 2021
Last Updated: 18 jan 2021

Year: 2021

ArXiv: 2101.02886 PDF

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 $k=1$.