Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I

Cherednichenko, Kirill; Ershova, Yulia; Kiselev, Alexander V.

doi:10.1007/s00220-020-03696-2

“…An overview of the existing approaches to obtaining operator-norm estimates would not be complete without mentioning also the works [41,42,44] that use, respectively, the method of periodic unfolding and the analysis of boundary integral representations, as well as the paper [70] cited above, based on the analysis of the homogenisation corrector via "Steklov smoothing", and the recent papers [26,27], which employ an analysis of appropriate Dirichletto-Neumann (or Poincaré-Steklov) operators. The methods of these works could also be considered in the context of thin and singular structures, however we refrain from pursuing the related discussion here.…”

Section: Introductionmentioning

confidence: 99%

Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures

Cherednichenko

¹

,

D'Onofrio

²

2022

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For arbitrarily small values of $$\varepsilon >0,$$ ε > 0 , we formulate and analyse the Maxwell system of equations of electromagnetism on $$\varepsilon $$ ε -periodic sets $$S^\varepsilon \subset {{\mathbb {R}}}^3.$$ S ε ⊂ R 3 . Assuming that a family of Borel measures $$\mu ^\varepsilon ,$$ μ ε , such that $$\mathrm{supp}(\mu ^\varepsilon )=S^\varepsilon ,$$ supp ( μ ε ) = S ε , is obtained by $$\varepsilon $$ ε -contraction of a fixed 1-periodic measure $$\mu ,$$ μ , and for right-hand sides $$f^\varepsilon \in L^2({\mathbb {R}}^3, d\mu ^\varepsilon ),$$ f ε ∈ L 2 ( R 3 , d μ ε ) , we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic “singular structures”, when $$\mu $$ μ is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for $$L^2$$ L 2 functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method.

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“…In our analysis of BVP, we adopt the approach to the operator‐theoretic treatment of BVP suggested by [53], which appears to be particularly convenient for obtaining sharp quantitative information about scattering properties of the medium, cf. , for example, [17], where this same approach is used as a framework for the asymptotic analysis of homogenisation problems in resonant composites.…”

Section: Introductionmentioning

confidence: 99%

“…The present paper is a development of the recent activity [16][17][18][19] aimed at implementing the above strategy in the context of problems of materials science and wave propagation in inhomogeneous media. Our recent papers [20,21] have shown that the language of boundary triples is particularly fitting for direct and inverse scattering problems on quantum graphs, as one of the key challenges to their analysis stems from the presence of interfaces through which energy exchange between different components of the medium takes place.…”

mentioning

confidence: 99%

“…As in [20,21], the ideas of [40,47] concerning the functional model allow one to efficiently incorporate into the analysis information about the mentioned energy exchange, by employing a suitable Dirichlet-to-Neumann map. In our analysis of BVP, we adopt the approach to the operatortheoretic treatment of BVP suggested by [53], which appears to be particularly convenient for obtaining sharp quantitative information about scattering properties of the medium, cf., for example, [17], where this same approach is used as a framework for the asymptotic analysis of homogenisation problems in resonant composites.…”

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confidence: 99%

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Functional Model for Boundary‐value Problems

Cherednichenko

¹

,

Kiselev

²

,

Silva

³

2021

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We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We then provide explicit formulae for the resolvents of the associated extensions of symmetric operators in terms of appropriate Dirichletto-Neumann maps, which can be utilised in the analysis of the properties of parameter-dependent problems, including the study of their spectra. §1. Introduction. The need to understand and quantify the behaviour of solutions to problems of mathematical physics has been central in driving the development of theoretical tools for the analysis of boundary-value problems (BVP). On the other hand, the second part of the last century witnessed several substantial advances in the abstract methods of spectral theory in Hilbert spaces, stemming from the groundbreaking achievement of John von Neumann in laying the mathematical foundations of quantum mechanics. Some of these advances have made their way into the broader context of mathematical physics [22,36,44]. In spite of these obvious successes of spectral theory applied to concrete problems, the operator-theoretic understanding of BVP has been lacking. However, in models of short-range interactions, the idea of replacing the original complex system by an explicitly solvable one, with a zero-radius potential (possibly with an internal structure), has proved to be highly valuable [5, 7, 14, 33, 34, 48, 57]. This facilitated an influx of methods of the theory of extensions (both self-adjoint and non-selfadjoint) of symmetric operators to problems of mathematical physics, culminating in the theory of boundary triples.The theory of boundary triples introduced in [23, 25, 31, 32] has been successfully applied to the spectral analysis of BVP for ordinary differential operators and related setups, for example, that of finite "quantum graphs", where the Dirichlet-to-Neumann maps act on finitedimensional "boundary" spaces, see [20] and references therein. However, in its original form this theory is not suited for dealing with BVP for partial differential equations (PDE), see [12, Section 7] for a relevant discussion. The key obstacle to such analysis is the lack of boundary traces 0 u and 1 u for functions u : → R (where is a bounded open set with a smooth boundary) in the domain of the maximal operator A corresponding to the differential expression considered (e.g., the operator − on the domain of L 2 ( )-functions u such that u is in L 2 ( )) entering the Green identity

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“…Interesting results on the norm-resolvent asymptotics in high-contrast homogenisation are obtained in [7], [8], [9].…”

Section: Introductionmentioning

confidence: 99%

Operator-norm resolvent estimates for thin elastic periodically heterogeneous rods in moderate contrast

Cherednichenko¹,

Velčić²,

Žubrinić³

2021

Preprint

0

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We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing the thin infinite elastic rod with material coefficients which periodically highly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the thickness of the rod and the period of material oscillations. These two parameters are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under some additional assumptions, as well as in the general case where these two sorts of displacements intertwine inseparably.

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Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I

Cited by 30 publications

References 90 publications

Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures

Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures

Functional Model for Boundary‐value Problems

Operator-norm resolvent estimates for thin elastic periodically heterogeneous rods in moderate contrast

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