Spectral Bounds for the Torsion Function

Abstract

Let Omega be an open set in Euclidean space mathbb {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 {mathcal {L}}^2(Omega ), denoted by lambda (Omega ), is bounded away from 0. It is shown that the previously obtained bound Vert v_{Omega }Vert _{{mathcal {L}}^{infty }(Omega )}lambda (Omega )ge 1 is sharp: for min {2,3,...}, and any epsilon >0 we construct an open, bounded and connected set Omega _{epsilon }subset mathbb {R}^m such that Vert v_{Omega _{epsilon }}Vert _{{mathcal {L}}^{infty }(Omega _{epsilon })} lambda (Omega _{epsilon })<1+epsilon . An upper bound for v_{Omega } is obtained for planar, convex sets in Euclidean space mathbb {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 {mathcal {L}}^2(Omega ) is bounded away from 0.