On the relations between principal eigenvalue and torsional rigidity

TL;DR

This paper investigates the extremal relationships between the first Dirichlet eigenvalue and torsional rigidity of domains, providing exact results in one dimension and bounds in higher dimensions across various domain classes.

Contribution

It characterizes the full Blaschke-Santaló diagram for these quantities in one dimension and offers bounds for higher dimensions, advancing understanding of their geometric interplay.

Findings

Full diagram in 1D for $rac{ ext{eigenvalue}}{ ext{rigidity}}$

Bounds for higher dimensions

Analysis across all, convex, and thin domains

Abstract

We consider the problem of minimising or maximising the quantity $\lambda(\O)T^q(\O)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $\lambda(\O)$ denotes the first eigenvalue of the Dirichlet Laplacian on $H^1_0(\O)$, while $T(\O)$ is the torsional rigidity of $\O$. The optimisation problem above is considered in the class of {\it all domains} $\O$, in the class of {\it convex domains} $\O$, and in the class of {\it thin domains}. The full Blaschke-Santal\'o diagram for $\lambda(\O)$ and $T(\O)$ is obtained in dimension one, while for higher dimensions we provide some bounds.