On localisation of eigenfunctions of the Laplace operator

Michiel van den Berg; Dorin Bucur

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arXiv · 2206.00479

On localisation of eigenfunctions of the Laplace operator

Michiel van den Berg·Dorin Bucur

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Abstract

We prove (i) 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 (ii) 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 $\kappa$-localise in $L^2$.

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Michiel van den Berg, Dorin Bucur. 2022-06-01. On localisation of eigenfunctions of the Laplace operator. https://arxiv.org/abs/2206.00479

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