Calculus of Variations and Geometric Measure Theory

M. van den Berg - G. Buttazzo - A. Pratelli

On the relations between principal eigenvalue and torsional rigidity

created by buttazzo on 08 Nov 2019
modified by pratelli on 20 Jan 2022


Published Paper

Inserted: 8 nov 2019
Last Updated: 20 jan 2022

Journal: Comm. Contemp. Math.
Year: 2021


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

Keywords: Torsional rigidity, Convex domains, shape optimisation, principal eigenvalue