Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition

TL;DR

This paper establishes geometric bounds for the Robin torsion and eigenvalues of the Laplacian on convex sets, generalizing classical inequalities and identifying optimal shapes as slabs.

Contribution

It introduces new bounds relating Robin torsion and eigenvalues to geometric quantities, extending Makai's inequality and identifying slab domains as optimizers.

Findings

Upper bound for Robin Torsion in terms of boundary distance norms.

Generalization of Makai inequality involving Robin Torsion, measure, and inradius.

Optimal shape for these functionals is a slab domain.

Abstract

In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the class of convex sets. Firstly, we prove an upper bound for the Robin Torsion in terms of the $L^1$ and $L^2$ norms of the distance function from the boundary, which allows us to prove a generalization of the Makai inequality involving the Robin Torsion, the Lebeasgue measure, and the inradius of a convex set. Subsequently, we prove quantitative estimates for the Robin Makai functional and for the Robin P\'olya functionals, which link the Lebesgue measure and the perimeter with the Robin Torsion and the Robin Eigenvalue respectively. In particular, we prove that the optimal values of all these shape functionals are achieved by slab domains.