Efficiency and localisation for the first Dirichlet eigenfunction

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

Research output: Contribution to journalArticle (Academic Journal)peer-review

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
Number of pages16
JournalJournal of Spectral Theory
Publication statusAccepted/In press - 28 Jan 2021

Keywords

  • efficiency
  • first Dirichlet eigenfunction
  • localisation

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