Calculus of Variations and Geometric Measure Theory

G. Buttazzo - A. Pratelli

An application of the continuous Steiner symmetrization to Blaschke-Santalò diagrams

created by pratelli on 17 Nov 2020
modified on 20 Jan 2022


Published Paper

Inserted: 17 nov 2020
Last Updated: 20 jan 2022

Journal: ESAIM Control Optim. Calc. Var.
Year: 2021


In this paper we consider the so-called procedure of {\it Continuous Steiner Symmetrization}, introduced by Brock in Brock95,Brock00. It transforms every domain $\Omega\subset\subset\mathbb R^d$ into the ball keeping the volume fixed and letting the first eigenvalue and the torsion respectively decrease and increase. While this does not provide, in general, a $\gamma$-continuous map $t\mapsto\Omega_t$, it can be slightly modified so to obtain the $\gamma$-continuity for a $\gamma$-dense class of domains $\Omega$, namely, the class of polyedral sets in $\mathbb R^d$. This allows to obtain a sharp characterization of the Blaschke-Santalò diagram of torsion and eigenvalue.