F. Prinari | Searcharxiv

arXiv2021

A shape optimization problem on planar sets with prescribed topology

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 $k=1$.

math.OC↗

arXiv2020

Some inequalities involving perimeter and torsional rigidity

We consider shape functionals of the form $F_q(\Omega)=P(\Omega)T^q(\Omega)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(\Omega)$ denotes the perimeter of $\Omega$ and $T(\Omega)$ is the torsional rigidity of $\Omega$. The minimization and maximization of $F_q(\Omega)$ is considered on various classes of admissible domains $\Omega$: in the class $\mathcal{A}_{all}$ of all domains, in the class $\mathcal{A}_{convex}$ of convex domains, and in the class $\mathcal{A}_{thin}$ of thin domains.

math.AP↗