Abstract
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 L^{2}(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 = R^{2}, which is sharp in the limit of elongation. For a complete, noncompact, mdimensional Riemannian manifold M with nonnegative Ricci curvature, and without boundary it is shown that v_{Ω} is bounded if and only if the bottom of the spectrum of the DirichletLaplaceBeltrami operator acting in L^{2}(Ω) is bounded away from 0.
Original language  English 

Pages (fromto)  387400 
Number of pages  14 
Journal  Integral Equations and Operator Theory 
Volume  88 
Issue number  3 
Early online date  26 Apr 2017 
DOIs  
Publication status  Published  Jul 2017 
Keywords
 Torsion function
 Dirichlet Laplacian
 Riemannian manifold
 nonnegative Ricci curvature
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Profiles

Professor Michiel Van Den Berg
 School of Mathematics  Emeritus Professor
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member, Honorary and Visiting Academic