Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve

Борисов, Д. И.; Cardone, Giuseppe; Durante, Tiziana

doi:10.1017/s0308210516000019

Cited by 54 publications

(32 citation statements)

References 47 publications

(112 reference statements)

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“…There are many works in which related geometries are studied: [3] analyzes Robin boundary conditions in cases where the scale of the inclusion is smaller than ε; nontrivial effective equations are obtained for appropriate (exponential) scalings. For the case that there is no perforation inside the domain, but rather the boundary ∂Ω is perturbed in a periodic fashion, results are available in [14]: As in our case, the lowest order approximation of the solution is given by a trivial limit problem, the first order corrector solves a modified macroscopic problem.…”

Section: Literaturementioning

confidence: 99%

Effective Helmholtz problem in a domain with a Neumann sieve perforation

Schweizer

2020

Journal de Mathématiques Pures et Appliquées

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A first order model for the transmission of waves through a sound-hard perforation along an interface is derived. Mathematically, we study the Neumann problem for the Helmholtz equation in a complex geometry, the domain contains a periodic array of inclusions of size ε > 0 along a co-dimension 1 manifold. We derive effective equations that describe the limit as ε → 0. At leading order, the Neumann sieve perforation has no effect; the corrector is given by a Helmholtz equation on the unperturbed domain with jump conditions across the interface. The corrector equations are derived with unfolding methods in L 1-based spaces.

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Section: Literaturementioning

confidence: 99%

Effective Helmholtz problem in a domain with a Neumann sieve perforation

Schweizer

2020

Journal de Mathématiques Pures et Appliquées

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“…III,Sec. 4], where the norm-resolvent convergence for problems with a fast periodically oscillating boundary was proved, and in [13,14], where elliptic operators in perforated domains were studied.…”

Section: Introductionmentioning

confidence: 99%

Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model

Cherednichenko

Kiselev

2016

Commun. Math. Phys.

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Abstract:We prove operator-norm resolvent convergence estimates for one-dimensional periodic differential operators with rapidly oscillating coefficients in the non-uniformly elliptic high-contrast setting, which has been out of reach of the existing homogenisation techniques. Our asymptotic analysis is based on a special representation of the resolvent of the operator in terms of the M-matrix of an associated boundary triple ("Krein resolvent formula"). The resulting asymptotic behaviour is shown to be described, up to a unitary transformation, by a non-standard version of the Kronig-Penney model on R.

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“…For the case n = 2, the norm resolvent convergence with estimates on the rate of convergence were obtained in [BCD16], where even more general elliptic operators were treated. The proofs in [BCD16] rely on variational formulations for the pre-limit and the homogenized resolvent equations (the key object of their analysis is a certain integral identity associated with the difference of the resolvents). Note that the method we use in the current works allows to treat surface distributions of holes as well.…”

Section: Introductionmentioning

confidence: 99%

Operator estimates for the crushed ice problem

Khrabustovskyi

Post

2018

Asymptotic Analysis

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Let ∆ Ω ε be the Dirichlet Laplacian in the domain Ω ε := Ω \ (∪ i D iε ).Here Ω ⊂ R n and {D iε } i is a family of tiny identical holes ("ice pieces") distributed periodically in R n with period ε. We denote by cap(D iε ) the capacity of a single hole. It was known for a long time that −∆ Ω ε converges to the operator −∆ Ω + q in strong resolvent sense provided the limit q := lim ε→0 cap(D iε )ε −n exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded Ω) an estimate for the difference of the k-th eigenvalue of −∆ Ω ε and −∆ Ω ε + q. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.

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Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve

Cited by 54 publications

References 47 publications

Effective Helmholtz problem in a domain with a Neumann sieve perforation

Effective Helmholtz problem in a domain with a Neumann sieve perforation

Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model

Operator estimates for the crushed ice problem

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