Spectral Bounds for the Torsion Function

Let Ω be an open set in Euclidean space R^m; m = 2, 3, ..., and let v_Ω denote the torsion function for Ω. It is known that v_Ω is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in L²(Ω), denoted by λ(Ω), is bounded away from 0. It is shown that the previously obtained bound ||v_Ω||_L∞(Ω)λ(Ω) ≥ 1 is sharp: for m ε {2, 3, ...}, and any ε > 0 we construct an open, bounded and connected set Ω_ε ⊂ R^m such that ||v_Ω||_L∞(Ωε)λ(Ω_ε) < 1 + ε. An upper bound for v_Ω is obtained for planar, convex sets in Euclidean space M = R², 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_Ω is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in L²(Ω) is bounded away from 0.