Efficiency and localisation for the first Dirichlet eigenfunction

M. van den Berg; Francesco  Della Pietra; Giuseppina di Blasio; Nunzia Gavitone

doi:10.4171/JST/363

Efficiency and localisation for the first Dirichlet eigenfunction

M. van den Berg, Francesco Della Pietra, Giuseppina di Blasio, Nunzia Gavitone

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Abstract

Bounds are obtained for the efficiency or mean to peak ratio E(Ω) for the first Dirichlet eigenfunction (positive) for open, connected sets Ω with finite measure in Euclidean space Rm. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if Ωn is any quadrilateral with perpendicular diagonals of lengths 1 and n respectively, then the sequence of first Dirichlet eigenfunctions localises, and E(Ωn) = O n−2/3 log n. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

Original language	English
Pages (from-to)	981-1003
Number of pages	23
Journal	Journal of Spectral Theory
Volume	11
Issue number	3
DOIs	https://doi.org/10.4171/JST/363
Publication status	Published - 30 Jul 2021

Bibliographical note

Funding Information:
Acknowledgements. The authors acknowledge support by the Leverhulme Trust through Emeritus Fellowship EM-2018-011-9, by GNAMPA of INdAM, and by a MIUR-PRIN 2017 grant “Qualitative and quantitative aspects of nonlinear PDE’s.” Michiel van den Berg wishes to thank Thomas Kappeler for helpful references to the literature.

Publisher Copyright:
© 2021 European Mathematical Society.

Keywords

efficiency
first Dirichlet eigenfunction
localisation

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title = "Efficiency and localisation for the first Dirichlet eigenfunction",

abstract = "Bounds are obtained for the efficiency or mean to peak ratio E(Ω) for the first Dirichlet eigenfunction (positive) for open, connected sets Ω with finite measure in Euclidean space Rm. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if Ωn is any quadrilateral with perpendicular diagonals of lengths 1 and n respectively, then the sequence of first Dirichlet eigenfunctions localises, and E(Ωn) = O n−2/3 log n. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.",

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doi = "10.4171/JST/363",

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AU - Della Pietra, Francesco

AU - di Blasio, Giuseppina

AU - Gavitone, Nunzia

N1 - Funding Information: Acknowledgements. The authors acknowledge support by the Leverhulme Trust through Emeritus Fellowship EM-2018-011-9, by GNAMPA of INdAM, and by a MIUR-PRIN 2017 grant “Qualitative and quantitative aspects of nonlinear PDE’s.” Michiel van den Berg wishes to thank Thomas Kappeler for helpful references to the literature. Publisher Copyright: © 2021 European Mathematical Society.

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N2 - Bounds are obtained for the efficiency or mean to peak ratio E(Ω) for the first Dirichlet eigenfunction (positive) for open, connected sets Ω with finite measure in Euclidean space Rm. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if Ωn is any quadrilateral with perpendicular diagonals of lengths 1 and n respectively, then the sequence of first Dirichlet eigenfunctions localises, and E(Ωn) = O n−2/3 log n. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

AB - Bounds are obtained for the efficiency or mean to peak ratio E(Ω) for the first Dirichlet eigenfunction (positive) for open, connected sets Ω with finite measure in Euclidean space Rm. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if Ωn is any quadrilateral with perpendicular diagonals of lengths 1 and n respectively, then the sequence of first Dirichlet eigenfunctions localises, and E(Ωn) = O n−2/3 log n. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

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KW - localisation

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JO - Journal of Spectral Theory

JF - Journal of Spectral Theory

IS - 3

ER -

Efficiency and localisation for the first Dirichlet eigenfunction

Abstract

Bibliographical note

Keywords

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