# Spectral Bounds for the Torsion Function

Spectral Bounds for the Torsion Function Let $$\Omega$$ Ω be an open set in Euclidean space $$\mathbb {R}^m,\, m=2,3,...$$ R m , m = 2 , 3 , . . . , and let $$v_{\Omega }$$ v Ω denote the torsion function for $$\Omega$$ Ω . It is known that $$v_{\Omega }$$ v Ω is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $${\mathcal {L}}^2(\Omega )$$ L 2 ( Ω ) , denoted by $$\lambda (\Omega )$$ λ ( Ω ) , is bounded away from 0. It is shown that the previously obtained bound $$\Vert v_{\Omega }\Vert _{{\mathcal {L}}^{\infty }(\Omega )}\lambda (\Omega )\ge 1$$ ‖ v Ω ‖ L ∞ ( Ω ) λ ( Ω ) ≥ 1 is sharp: for $$m\in \{2,3,...\}$$ m ∈ { 2 , 3 , . . . } , and any $$\epsilon >0$$ ϵ > 0 we construct an open, bounded and connected set $$\Omega _{\epsilon }\subset \mathbb {R}^m$$ Ω ϵ ⊂ R m such that $$\Vert v_{\Omega _{\epsilon }}\Vert _{{\mathcal {L}}^{\infty }(\Omega _{\epsilon })} \lambda (\Omega _{\epsilon })<1+\epsilon$$ ‖ v Ω ϵ ‖ L ∞ ( Ω ϵ ) λ ( Ω ϵ ) < 1 + ϵ . An upper bound for $$v_{\Omega }$$ v Ω is obtained for planar, convex sets in Euclidean space $$\mathbb {R}^2$$ 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 $$v_{\Omega }$$ v Ω is bounded if and only if the bottom of the spectrum of the Dirichlet–Laplace–Beltrami operator acting in $${\mathcal {L}}^2(\Omega )$$ L 2 ( Ω ) is bounded away from 0. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

# Spectral Bounds for the Torsion Function

, Volume 88 (3) – Apr 26, 2017
14 pages

/lp/springer_journal/spectral-bounds-for-the-torsion-function-RZ00MoTMAO
Publisher
Springer International Publishing
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-017-2371-0
Publisher site
See Article on Publisher Site

### Abstract

Let $$\Omega$$ Ω be an open set in Euclidean space $$\mathbb {R}^m,\, m=2,3,...$$ R m , m = 2 , 3 , . . . , and let $$v_{\Omega }$$ v Ω denote the torsion function for $$\Omega$$ Ω . It is known that $$v_{\Omega }$$ v Ω is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $${\mathcal {L}}^2(\Omega )$$ L 2 ( Ω ) , denoted by $$\lambda (\Omega )$$ λ ( Ω ) , is bounded away from 0. It is shown that the previously obtained bound $$\Vert v_{\Omega }\Vert _{{\mathcal {L}}^{\infty }(\Omega )}\lambda (\Omega )\ge 1$$ ‖ v Ω ‖ L ∞ ( Ω ) λ ( Ω ) ≥ 1 is sharp: for $$m\in \{2,3,...\}$$ m ∈ { 2 , 3 , . . . } , and any $$\epsilon >0$$ ϵ > 0 we construct an open, bounded and connected set $$\Omega _{\epsilon }\subset \mathbb {R}^m$$ Ω ϵ ⊂ R m such that $$\Vert v_{\Omega _{\epsilon }}\Vert _{{\mathcal {L}}^{\infty }(\Omega _{\epsilon })} \lambda (\Omega _{\epsilon })<1+\epsilon$$ ‖ v Ω ϵ ‖ L ∞ ( Ω ϵ ) λ ( Ω ϵ ) < 1 + ϵ . An upper bound for $$v_{\Omega }$$ v Ω is obtained for planar, convex sets in Euclidean space $$\mathbb {R}^2$$ 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 $$v_{\Omega }$$ v Ω is bounded if and only if the bottom of the spectrum of the Dirichlet–Laplace–Beltrami operator acting in $${\mathcal {L}}^2(\Omega )$$ L 2 ( Ω ) is bounded away from 0.

### Journal

Integral Equations and Operator TheorySpringer Journals

Published: Apr 26, 2017

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