Spectral Bounds for the Torsion Function

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Abstract

Let Ω be an open set in Euclidean space Rm; m = 2, 3, ..., and let vΩ denote the torsion function for Ω. It is known that vΩ 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 ||vΩ||L∞(Ω)λ(Ω) ≥ 1 is sharp: for m ε {2, 3, ...}, and any ε > 0 we construct an open, bounded and connected set Ωε Rm such that ||vΩ||L∞(Ωε)λ(Ωε) < 1 + ε. An upper bound for vΩ 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 vΩ 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.
Original languageEnglish
Pages (from-to)387-400
Number of pages14
JournalIntegral Equations and Operator Theory
Volume88
Issue number3
Early online date26 Apr 2017
DOIs
Publication statusPublished - Jul 2017

Keywords

  • Torsion function
  • Dirichlet Laplacian
  • Riemannian manifold
  • non-negative Ricci curvature

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