Asymptotic behavior of u-capacities and singular perturbations for the Dirichlet-Laplacian

¹ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy.
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: paolo.musolino@unive.it

Received: 14 November 2019
Accepted: 11 November 2020

Abstract

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 ${\mathrm{Cap}}_{\Omega }\left(\epsilon \bar{\omega },u\right)$ as ε → 0. As a byproduct, we compute an asymptotic expansion for the Nth eigenvalues of the Dirichlet-Laplacian in the perforated set $\Omega \backslash \left(\epsilon \bar{\omega }\right)$ 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.