On a conjecture of a P\'olya functional for triangles and rectangles

TL;DR

This paper proves a longstanding conjecture about bounds of a scale-invariant functional involving the first Dirichlet eigenvalue and torsional rigidity for triangles and rectangles, confirming specific bounds for these shapes.

Contribution

The paper establishes the conjectured bounds for the functional specifically for triangles and rectangles, advancing understanding of shape optimization in spectral geometry.

Findings

Proved the conjecture for triangles.

Confirmed the bounds for rectangles with a monotonicity property.

Extended the upper bound to tangential quadrilaterals.

Abstract

We consider the functional given by the product of the first Dirichlet eigenvalue and the torsional rigidity of planar domains normalized by the area. This scale invariant functional was studied by P\'olya and Szeg\H{o} in 1951 who showed that it is bounded above by 1 for all domains. It has been conjectured that within the class of bounded convex planar domains the functional is bounded below by $\pi^{2}/24$ and above by $\pi^{2}/12$ and that these bounds are sharp. Remarkably, the conjecture remains open even within the class of triangles. The purpose of this paper is to prove the conjecture in this case. The conjecture is also proved for rectangles where a stronger monotonicity property is verified. Finally, the upper bound also holds for tangential quadrilateral.