On the relations between principal eigenvalue and torsional rigidity

Abstract

We consider the problem of minimising or maximising the quantity $\lambda (Ø)T^q(Ø)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $\lambda (Ø)$ denotes the first eigenvalue of the Dirichlet Laplacian on $H_0^1(Ø)$, while $T(Ø)$ is the torsional rigidity of $Ø$. The optimisation problem above is considered in the class of {\it all domains} $Ø$, in the class of {\it convex domains} $Ø$, and in the class of {\it thin domains}. The full Blaschke-Santaló diagram for $\lambda (Ø)$ and $T(Ø)$ is obtained in dimension one, while for higher dimensions we provide some bounds.

23 pages, 2 figures