The Dirichlet problem in a planar domain with two moderately close holes

Riva, Matteo Dalla; Musolino, Paolo

doi:10.1016/j.jde.2017.04.006

Cited by 4 publications

(3 citation statements)

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“…As we emphasize in a comment at the end of Subsection 4.1, the reason for this simpler behavior is that in the toy problem we do not have an exterior boundary ∂ + Ω. It is worth noting that a quotient similar to (1.9) plays a fundamental role also in the two-dimensional Dirichlet problem with moderately close small holes, which was investigated in [14] and where it was shown that an analog of the limiting value λ (cf. (1.11)) appears explicitly in the second term of the asymptotic expansion of the solution.…”

Section: 32mentioning

confidence: 98%

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A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Bonnaillie‐Noël

Riva

Dambrine³

et al. 2018

Journal de Mathématiques Pures et Appliquées

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We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair ε = (ε 1 , ε 2 ) of positive parameters, we consider a perforated domain Ω ε obtained by making a small hole of size ε 1 ε 2 in an open regular subset Ω of R n at distance ε 1 from the boundary ∂Ω. As ε 1 → 0, the perforation shrinks to a point and, at the same time, approaches the boundary. When ε → (0, 0), the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by u ε the solution of a Dirichlet problem for the Laplace equation in Ω ε . For a space dimension n ≥ 3, we show that the function mapping ε to u ε has a real analytic continuation in a neighborhood of (0, 0). By contrast, for n = 2 we consider two different regimes: ε tends to (0, 0), and ε 1 tends to 0 with ε 2 fixed. When ε → (0, 0), the solution u ε has a logarithmic behavior; when only ε 1 → 0 and ε 2 is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of ε 1 . We also show that for n = 2, the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials.

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Section: 32mentioning

confidence: 98%

“…The function η → δ(η) is defined as in (1.12). The pair ε ∈ ]0, ε ad [ is small enough to yield 14) and η can be any number in ]0, 1[ such that…”

Section: 32mentioning

confidence: 99%

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Bonnaillie‐Noël

Riva

Dambrine³

et al. 2018

Journal de Mathématiques Pures et Appliquées

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“…The approximations are accurate up to and including the boundaries of the medium and are efficient in the low-frequency regime, where the frequency of vibration does not compete with the defect size. Configurations with several moderately close holes have also been analysed with the so-called functional analytic approach [ 42 ] in [ 43 , 44 ].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions

Movchan

2022

Phil. Trans. R. Soc. A.

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We present formal asymptotic approximations of fields representing the in-plane dynamic response of elastic solids containing clusters of closely interacting small rigid inclusions. For finite densely perforated bodies, the asymptotic scheme is developed to approximate the eigenfrequencies and the associated eigenmodes of the elastic medium with clamped boundaries. The asymptotic algorithm is also adapted to address the scattering of in-plane waves in infinite elastic media containing dense clusters. The results are accompanied by numerical simulations that illustrate the accuracy of the asymptotic approach. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

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