Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||43|
|Published online||01 March 2021|
Asymptotic behavior of u-capacities and singular perturbations for the Dirichlet-Laplacian
Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca,
Via Cozzi 55,
2 Département de mathématiques et applications, École normale supérieure, CNRS, PSL University, 45 rue d’Ulm, 75005 Paris, France.
3 Institut de Mathématiques, Bâtiment UniMail Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland.
4 Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca’ Foscari Venezia, via Torino 155, 30172 Venezia Mestre, Italy.
* Corresponding author: firstname.lastname@example.org
Accepted: 11 November 2020
In this paper we study the asymptotic behavior of u-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets Ω and ω of ℝ2, containing the origin. First, if ε is close to 0 and if u is a function defined on Ω, we compute an asymptotic expansion of the u-capacity as ε → 0. As a byproduct, we compute an asymptotic expansion for the Nth eigenvalues of the Dirichlet-Laplacian in the perforated set for ε close to 0. Such formula shows explicitly the dependence of the asymptotic expansion on the behavior of the corresponding eigenfunction near 0 and on the shape ω of the hole.
Mathematics Subject Classification: 35P20 / 31C15 / 31B10 / 35B25 / 35C20
Key words: Dirichlet-Laplacian / eigenvalues / small capacity sets / asymptotic expansion / perforated domain
© EDP Sciences, SMAI 2021
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