Geodesic
On localisation of eigenfunctions of the Laplace operator
Michiel van den Berg1 orcid ; Dorin Bucur2 orcid
1University of Bristol, Bristol, UK
2Université Savoie Mont Blanc, Le-Bourget-du-Lac, France
EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 1-25

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DOI

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 κ-localise in L2.
DOI : 10.4171/emss/89
Classification : 35J25, 35P99
Mots-clés : first Dirichlet eigenfunction, localisation, κ-localisation, Hardy inequality

Michiel van den Berg  1   ; Dorin Bucur  2

1 University of Bristol, Bristol, UK
2 Université Savoie Mont Blanc, Le-Bourget-du-Lac, France 
Michiel van den Berg; Dorin Bucur. On localisation of eigenfunctions of the Laplace operator. EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 1-25. doi: 10.4171/emss/89
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