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arXiv · 1910.14593

On the relations between principal eigenvalue and torsional rigidity

Michiel van den Berg·Giuseppe Buttazzo·Aldo Pratelli
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Abstract

We consider the problem of minimising or maximising the quantity $λ(Ø)T^q(Ø)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $λ(Ø)$ denotes the first eigenvalue of the Dirichlet Laplacian on $H^1_0(Ø)$, 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 $λ(Ø)$ and $T(Ø)$ is obtained in dimension one, while for higher dimensions we provide some bounds.

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Michiel van den Berg, Giuseppe Buttazzo, Aldo Pratelli. 2019-11-14. On the relations between principal eigenvalue and torsional rigidity. https://arxiv.org/abs/1910.14593

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