On localisation of eigenfunctions of the Laplace operator

Michiel van den Berg; Dorin Bucur

doi:10.4171/EMSS/89

On localisation of eigenfunctions of the Laplace operator

Michiel van den Berg, Dorin Bucur

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Abstract

We prove a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet, or Neumann, boundary conditions κ-localise in L2.

Original language	English
Pages (from-to)	1-25
Number of pages	25
Journal	EMS Surveys in Mathematical Sciences
Volume	12
Issue number	1
Early online date	27 Jan 2025
DOIs	https://doi.org/10.4171/EMSS/89
Publication status	Published - 16 May 2025

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© 2025 European Mathematical Society.

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AB - We prove a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet, or Neumann, boundary conditions κ-localise in L2.

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On localisation of eigenfunctions of the Laplace operator

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