On Pólya's inequality for torsional rigidity and first Dirichlet eigenvalue

Michiel van den Berg; Vincenzo Ferone; Carlo Nitsch; Cristina Trombetti

doi:10.1007/s00020-016-2334-x

On Pólya's inequality for torsional rigidity and first Dirichlet eigenvalue

Michiel van den Berg
, Vincenzo Ferone
, Carlo Nitsch
, Cristina Trombetti

Research output: Contribution to journal › Article (Academic Journal) › peer-review

36 Citations (Scopus)

350 Downloads (Pure)

Abstract

Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω|. We obtain some properties of the set function F:Ω→R⁺ defined by
F(Ω)=T(Ω)λ1(Ω)/|Ω|, where T(Ω) and λ₁(Ω) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F(Ω)≤1, and show that F(Ω)≤1−ν_mT(Ω)|Ω|^−1−2/m, where ν_m depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ω_ϵ⊂ R^m such that F(Ω_ϵ)≥1−ϵ.

Original language	English
Pages (from-to)	579-600
Number of pages	23
Journal	Integral Equations and Operator Theory
Volume	86
Issue number	4
Early online date	9 Nov 2016
DOIs	https://doi.org/10.1007/s00020-016-2334-x
Publication status	Published - Dec 2016

Keywords

Torsional rigidity
first Dirichlet eigenvalue

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Emeritus Professor Michiel van den Berg
- M.vandenBergbristol.acuk
- School of Mathematics - Emeritus Professor
- Probability, Analysis and Dynamics
- Pure Mathematics
- Analysis
Person: Member, Honorary and Visiting Academic

Cite this

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title = "On P{\'o}lya's inequality for torsional rigidity and first Dirichlet eigenvalue",

abstract = "Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω|. We obtain some properties of the set function F:Ω→R+ defined by F(Ω)=T(Ω)λ1(Ω)/|Ω|, where T(Ω) and λ1(Ω) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P{\'o}lya bound F(Ω)≤1, and show that F(Ω)≤1−νmT(Ω)|Ω|−1−2/m, where νm depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ωϵ ⊂ Rm such that F(Ωϵ)≥1−ϵ. ",

keywords = "Torsional rigidity, first Dirichlet eigenvalue",

author = "\{van den Berg\}, Michiel and Vincenzo Ferone and Carlo Nitsch and Cristina Trombetti",

year = "2016",

month = dec,

doi = "10.1007/s00020-016-2334-x",

language = "English",

volume = "86",

pages = "579--600",

journal = "Integral Equations and Operator Theory",

issn = "0378-620X",

publisher = "Birkhauser Verlag Basel",

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TY - JOUR

T1 - On Pólya's inequality for torsional rigidity and first Dirichlet eigenvalue

AU - van den Berg, Michiel

AU - Ferone, Vincenzo

AU - Nitsch, Carlo

AU - Trombetti, Cristina

PY - 2016/12

Y1 - 2016/12

N2 - Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω|. We obtain some properties of the set function F:Ω→R+ defined by F(Ω)=T(Ω)λ1(Ω)/|Ω|, where T(Ω) and λ1(Ω) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F(Ω)≤1, and show that F(Ω)≤1−νmT(Ω)|Ω|−1−2/m, where νm depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ωϵ ⊂ Rm such that F(Ωϵ)≥1−ϵ.

AB - Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω|. We obtain some properties of the set function F:Ω→R+ defined by F(Ω)=T(Ω)λ1(Ω)/|Ω|, where T(Ω) and λ1(Ω) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F(Ω)≤1, and show that F(Ω)≤1−νmT(Ω)|Ω|−1−2/m, where νm depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ωϵ ⊂ Rm such that F(Ωϵ)≥1−ϵ.

KW - Torsional rigidity

KW - first Dirichlet eigenvalue

U2 - 10.1007/s00020-016-2334-x

DO - 10.1007/s00020-016-2334-x

M3 - Article (Academic Journal)

SN - 0378-620X

VL - 86

SP - 579

EP - 600

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

IS - 4

ER -

On Pólya's inequality for torsional rigidity and first Dirichlet eigenvalue

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Emeritus Professor Michiel van den Berg

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