Calculus of Variations and Geometric Measure Theory

L. Briani - G. Buttazzo - F. Prinari

A shape optimization problem on planar sets with prescribed topology

created by prinari on 17 Jan 2021
modified on 19 Nov 2021


Published Paper

Inserted: 17 jan 2021
Last Updated: 19 nov 2021

Journal: Jota
Year: 2021

ArXiv: 2101.02886 PDF


We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $P(\Omega)T^q(\Omega)
^{-2q-1/2}$ 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 $k=1$.