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

Loading next page...
 
/lp/springer_journal/spectral-bounds-for-the-torsion-function-RZ00MoTMAO
Publisher
Springer International Publishing
Copyright
Copyright © 2017 by The Author(s)
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off