On two functionals involving the maximum of the torsion function

Henrot, Antoine; Lucardesi, Ilaria; Philippin, Gérard

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1585-1604.

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T\left(\Omega \right)∕\left(M\left(\Omega \right)\left|\Omega \right|\right)$ and $M\left(\Omega \right){\lambda }_1\left(\Omega \right)$, where $\Omega$ is a bounded open set of ${ℝ}^d$ with finite Lebesgue measure $\left|\Omega \right|,M\left(\Omega \right)$ denotes the maximum of the torsion function, $T\left(\Omega \right)$ the torsion, and ${\lambda }_1\left(\Omega \right)$ the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

Classification : 35P15, 49R05, 35J25, 35B27, 49Q10
Mots clés : Torsional rigidity, first Dirichlet eigenvalue, shape optimization

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