Spectral Bounds for the Torsion Function

Let Omega be an open set in Euclidean space R-m, r = 2, 3,..., and let nu Omega denote the torsion function for Omega. It is known that nu Omega is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in L-2(Omega), denoted by lambda(Omega), is bounded away from 0. It is shown that the previously obtained bound parallel to nu Omega parallel to L infinity(Omega) >= 1 is sharp: for in m is an element of{2, 3, and any epsilon > 0 we construct an open, bounded and connected set Omega(is an element of) subset of R-m such that parallel to nu Omega(is an element of)parallel to L infinity(Omega(is an element of))lambda(Omega(is an element of)) < 1 + is an element of. An upper bound for nu Omega is obtained for planar, convex sets in Euclidean space 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 nu Omega is bounded if and only if the bottom of the spectrum of the Dirichlet Laplace Beltrami operator acting in L-2 (Omega) is bounded away from 0.