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

In this paper we consider the so-called procedure of {\it Continuous Steiner Symmetrization}, introduced by Brock in~\cite{bro95,bro00}. It transforms every domain $\Omega\comp\R^d$ 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 $t\mapsto\O_t$, it can be slightly modified so to obtain the $\gamma$-continuity for a $\gamma$-dense class of domains $\O$, namely, the class of polyedral sets in $\R^d$. This allows to obtain a sharp characterization of the Blaschke-Santaló diagram of torsion and eigenvalue.

Comparison results for solutions of elliptic Neumann problems with lower-order terms via Steiner symmetrization

In this paper, we consider a Neumann problem for a linear elliptic equation with lower-order terms. A comparison result for solutions of the problem is proved by using Steiner symmetrization.