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On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets

Abstract : We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.
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https://hal-u-picardie.archives-ouvertes.fr/hal-03813721
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Soumis le : jeudi 13 octobre 2022 - 15:02:07
Dernière modification le : vendredi 14 octobre 2022 - 03:15:33

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Laura Abatangelo, Veronica Felli, Benedetta Noris. On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets. Communications in Contemporary Mathematics, 2020, 22 (08), ⟨10.1142/S0219199719500718⟩. ⟨hal-03813721⟩

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