Efficiency and localisation for the first Dirichlet eigenfunction

  • Michiel van den Berg

    University of Bristol, UK
  • Francesco Della Pietra

    Università degli studi di Napoli Federico II, Italy
  • Giuseppina di Blasio

    Università degli Studi della Campania "L. Vanvitelli", Caserta, Italy
  • Nunzia Gavitone

    Università degli studi di Napoli Federico II, Italy
Efficiency and localisation for the first Dirichlet eigenfunction cover
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Abstract

Bounds are obtained for the efficiency or mean to max ratio for the first Dirichlet eigenfunction (positive) for open, connected sets with finite measure in Euclidean space . 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 is any quadrilateral with perpendicular diagonals of lengths and respectively, then the sequence of first Dirichlet eigenfunctions localises and . This disproves some claims in the literature. A key technical tool is the Feynman–Kac formula.

Cite this article

Michiel van den Berg, Francesco Della Pietra, Giuseppina di Blasio, Nunzia Gavitone, Efficiency and localisation for the first Dirichlet eigenfunction. J. Spectr. Theory 11 (2021), no. 3, pp. 981–1003

DOI 10.4171/JST/363