A Proof of a Conjecture on a P\'olya Functional

Abstract

Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves the two-dimensional case of Conjecture 4.2 proposed by van den Berg, Buttazzo, and Pratelli. Its planar formulation was later restated as Conjecture 1.1 by Ba\~nuelos and Mariano, who proved it for triangles and rectangles. The lower bound follows by combining Payne's strict estimate for $\lambda_1M$ with the sharp torsion-efficiency inequality $T(\Omega)\geq |\Omega|M/3$. For smooth strictly convex domains, we use an Airy stress potential to construct a convex body whose surface-area measure is the torsional first-variation measure. A sharp one-dimensional inequality for directional maximum profiles then yields a stronger geometric containment. For the upper bound, we factor the P\'olya functional through the second torsion moment. A level-set torsion--perimeter inequality gives the factor $5/6$, while a sharp weighted one-dimensional estimate gives the factor $\pi^2/10$. Collapsing triangles and elongating rectangles show that both endpoint constants, as well as the efficiency constant $1/3$, are sharp.