On a P\'olya functional for rhombi, isosceles triangles, and thinning convex sets

TL;DR

This paper establishes sharp bounds for the product of the maximum torsion function value and the spectral bottom in convex sets, with specific results for rhombi and isosceles triangles, highlighting asymptotic sharpness in thinning sequences.

Contribution

It provides a new upper bound for the product of torsion function maximum and spectral bottom, and demonstrates sharpness for certain convex shapes like rhombi and triangles.

Findings

Upper bound for $ orm{v_ ext{Omega}}_{L^ty}\,mbda( ext{Omega})$ in convex sets.

Sharpness of the bound in the limit of thinning convex sets.

Specific lower bound for rhombi and isosceles triangles with area 1.

Abstract

Let $\Omega$ be an open convex set in ${\mathbb R}^m$ with finite width, and let $v_{\Omega}$ be the torsion function for $\Omega$, i.e. the solution of $-\Delta v=1, v\in H_0^1(\Omega)$. An upper bound is obtained for the product of $\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega)$, where $\lambda(\Omega)$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Omega)$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_{\Omega}\Vert_{L^{1}(\Omega)}\lambda(\Omega)\ge \frac{\pi^2}{24}$, and that this bound is sharp.