# Spectral bounds for the torsion function

## TL;DR

This paper investigates bounds on the torsion function in Euclidean and Riemannian settings, establishing sharp bounds and conditions for boundedness related to the spectrum of the Laplacian.

## Contribution

It proves the sharpness of a known spectral bound for the torsion function and extends the analysis to convex planar sets and Riemannian manifolds.

## Key findings

- The bound  v_{\u03a9} _{\u211d^{}} \u22a5  is sharp.
- Constructs sets where the product of the torsion function's supremum and the spectrum approaches 1.
- Provides a characterization of bounded torsion functions on Riemannian manifolds with non-negative Ricci curvature.

## Abstract

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(\Omega)$, denoted by $\lambda(\Omega)$, is bounded away from $0$. It is shown that the previously obtained bound $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\ge 1$ is sharp: for $m\in\{2,3,...\}$, and any $\epsilon>0$ we construct an open, bounded and connected set $\Omega_{\epsilon}\subset \R^m$ such that $\|v_{\Omega_{\epsilon}}\|_{\Leb^{\infty}(\Omega_{\epsilon})} \lambda(\Omega_{\epsilon})<1+\epsilon$. An upper bound for $v_{\Omega}$ is obtained for planar, convex sets in Euclidean space $M=\R^2$, which is sharp in the limit of elongation. For a complete, non-compact, $m$-dimensional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary it is shown that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in $\Leb^2(\Omega)$ is bounded away from $0$.