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 languageEnglish
Pages (from-to)981-1003
Number of pages23
JournalJournal of Spectral Theory
Volume11
Issue number3
DOIs
Publication statusPublished - 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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