Lower Bounds on Eigenfunctions and the First Eigenvalue Gap

R. M. Brown; P. D. Hislopt; A. Martinez

doi:10.1016/S0076-5392(08)62370-X

Lower Bounds on Eigenfunctions and the First Eigenvalue Gap

R. M. Brown, P. D. Hislopt, A. Martinez

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We give upper and lower bounds on the difference ΔE ≡E₂ −E₁, of the first two Dirichlet eigenvalues for a dumbbell region in Rⁿ. These bounds are exponentially small in the diameter ɛ of the straight tube connecting two identical bounded cavities as ɛ → 0. The proof relies on a lower bound for the first Dirichlet eigenfunction for one cavity with a thin tube attached.

Original language	English
Pages (from-to)	33-49
Number of pages	17
Journal	Mathematics in Science and Engineering
Volume	192
Issue number	C
DOIs	https://doi.org/10.1016/S0076-5392(08)62370-X
State	Published - Jan 1 1993

Bibliographical note

Funding Information:
*Supported in part by NSF DMS 91-03046. 'Supported in part by NSF DMS 91-06479 and INT 90-15895.

Funding

*Supported in part by NSF DMS 91-03046. 'Supported in part by NSF DMS 91-06479 and INT 90-15895.

Funders	Funder number
NSF CAREER DMS-1149054	91-03046, 91-06479, INT 90-15895

ASJC Scopus subject areas

General Mathematics
General Engineering

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10.1016/S0076-5392(08)62370-X

Cite this

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abstract = "We give upper and lower bounds on the difference ΔE ≡E2 −E1, of the first two Dirichlet eigenvalues for a dumbbell region in Rn. These bounds are exponentially small in the diameter ɛ of the straight tube connecting two identical bounded cavities as ɛ → 0. The proof relies on a lower bound for the first Dirichlet eigenfunction for one cavity with a thin tube attached.",

author = "Brown, \{R. M.\} and Hislopt, \{P. D.\} and A. Martinez",

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AU - Hislopt, P. D.

AU - Martinez, A.

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Y1 - 1993/1/1

N2 - We give upper and lower bounds on the difference ΔE ≡E2 −E1, of the first two Dirichlet eigenvalues for a dumbbell region in Rn. These bounds are exponentially small in the diameter ɛ of the straight tube connecting two identical bounded cavities as ɛ → 0. The proof relies on a lower bound for the first Dirichlet eigenfunction for one cavity with a thin tube attached.

AB - We give upper and lower bounds on the difference ΔE ≡E2 −E1, of the first two Dirichlet eigenvalues for a dumbbell region in Rn. These bounds are exponentially small in the diameter ɛ of the straight tube connecting two identical bounded cavities as ɛ → 0. The proof relies on a lower bound for the first Dirichlet eigenfunction for one cavity with a thin tube attached.

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ER -

Lower Bounds on Eigenfunctions and the First Eigenvalue Gap

Abstract

Bibliographical note

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