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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 $\lambda(\O)T^q(\O)$ 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.

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Michiel van den Berg, Giuseppe Buttazzo, Aldo Pratelli. 2019-10-31. On the relations between principal eigenvalue and torsional rigidity. https://arxiv.org/abs/1910.14593

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