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arXiv · 1911.06686

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

Laura Abatangelo·Virginie Bonnaillie-Noël·Corentin Léna·Paolo Musolino
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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 $\mathbb{R}^2$, containing the origin. First, if $\varepsilon$ is positive and small enough and if $u$ is a function defined on $Ω$, we compute an asymptotic expansion of the $u$-capacity $\mathrm{Cap}_Ω(\varepsilon ω, u)$ as $\varepsilon \to 0$. As a byproduct, we compute an asymptotic expansion for the $N$-th eigenvalues of the Dirichlet-Laplacian in the perforated set $Ω\setminus (\varepsilon \overlineω)$ for $\varepsilon$ 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.

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Laura Abatangelo, Virginie Bonnaillie-Noël, Corentin Léna, Paolo Musolino. 2019-11-15. Asymptotic behavior of $u$-capacities and singular perturbations for the Dirichlet-Laplacian. https://arxiv.org/abs/1911.06686

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