On the spectrum of the Dirichlet Laplacian in a narrow strip

Leonid Friedlander; Michael Solomyak

doi:10.1007/s11856-009-0032-y

On the spectrum of the Dirichlet Laplacian in a narrow strip

Leonid Friedlander
, Michael Solomyak

Mathematics

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

88 Scopus citations

Abstract

We consider the Dirichlet Laplacian Δ in a family of bounded domains {-a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ.

Original language	English (US)
Pages (from-to)	337-354
Number of pages	18
Journal	Israel Journal of Mathematics
Volume	170
Issue number	1
DOIs	https://doi.org/10.1007/s11856-009-0032-y
State	Published - Mar 2009

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General Mathematics

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title = "On the spectrum of the Dirichlet Laplacian in a narrow strip",

abstract = "We consider the Dirichlet Laplacian Δ in a family of bounded domains \{-a < x < b, 0 < y < εh(x)\}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ.",

author = "Leonid Friedlander and Michael Solomyak",

note = "Funding Information: Acknowledgments. The bulk of the work was done when the first author was the Weston Visiting Professor in the Weizmann Institute of Science in Rehovot, Israel. He thanks the Institute for its hospitality. The work was finished when both authors visited the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK. We acknowledge the hospitality of the Newton Institute. The first author was partially supported by NSF grant DMS 0648786. We are also grateful to V. Maz{\textquoteright}ya and S. Nazarov for their bibliographical advice and to A. Sobolev for discussions.",

year = "2009",

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doi = "10.1007/s11856-009-0032-y",

language = "English (US)",

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pages = "337--354",

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T1 - On the spectrum of the Dirichlet Laplacian in a narrow strip

AU - Friedlander, Leonid

AU - Solomyak, Michael

N1 - Funding Information: Acknowledgments. The bulk of the work was done when the first author was the Weston Visiting Professor in the Weizmann Institute of Science in Rehovot, Israel. He thanks the Institute for its hospitality. The work was finished when both authors visited the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK. We acknowledge the hospitality of the Newton Institute. The first author was partially supported by NSF grant DMS 0648786. We are also grateful to V. Maz’ya and S. Nazarov for their bibliographical advice and to A. Sobolev for discussions.

PY - 2009/3

Y1 - 2009/3

N2 - We consider the Dirichlet Laplacian Δ in a family of bounded domains {-a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ.

AB - We consider the Dirichlet Laplacian Δ in a family of bounded domains {-a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ.

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DO - 10.1007/s11856-009-0032-y

M3 - Article

AN - SCOPUS:65549169108

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SP - 337

EP - 354

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -

On the spectrum of the Dirichlet Laplacian in a narrow strip

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