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Kirill Cherednichenko

Publications and source records attributed to Kirill Cherednichenko.

At least 19 recordsLinked to original sources
arXiv2026

A not-so-strange term coming from somewhere

We consider Laplace's equation in a periodically perforated domain with Robin boundary conditions on the holes, where the Robin coefficient is scaled proportionally to the inverse total surface area of the performations. We identify a regime in which surface and bulk effects contribute at the same order and show that the homogenised equation contains an additional zeroth-order term depending nonlinearly on the Robin parameter. This term is characterised via a Steklov-type spectral problem in which the spectral parameter appears both in the equation and in the boundary condition. The resulting term interpolates continuously between the Neumann and Dirichlet limits, recovering the classical capacitary strange term in the strong-coupling limit.

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arXiv2026

Spatial non-locality of the Maxwell system on periodic structures

For $\varepsilon>0,$ we analyse the Maxwell system of equations of electromagnetism on $\varepsilon$-periodic sets $S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures $μ^\varepsilon,$ such that ${\rm supp}(μ^\varepsilon)=S^\varepsilon,$ is obtained by $\varepsilon$-contraction of a fixed periodic measure $μ,$ and for right-hand sides $f^\varepsilon\in L^2({\mathbb R}^3, dμ^\varepsilon),$ we prove order-sharp norm-resolvent convergence estimates for the solutions of the system.

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arXiv2025

Operator aspects of wave propagation through periodic media

Recent results in quantitative homogenisation of the wave equation with rapidly oscillating coefficients are discussed from the operator-theoretic perspective, which views the solution as the result of applying the operator of hyperbolic dynamics, i.e. the unitary group of a self-adjoint operator on a suitable Hilbert space. A prototype one-dimensional example of utilising the framework of Ryzhov boundary triples is analysed, where operator-norm resolvent estimates for the problem of classical moderate-contrast homogenisation are obtained. By an appropriate "dilation" procedure, these are shown to upgrade to second-order (and more generally, higher-order) estimates for the resolvent and the unitary group describing the evolution for the related wave equation.

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arXiv2024

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

For arbitrarily small values of $\varepsilon>0,$ we formulate and analyse the Maxwell system of equations of electromagnetism on $\varepsilon$-periodic sets $S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures $μ^\varepsilon,$ such that ${\rm supp}(μ^\varepsilon)=S^\varepsilon,$ is obtained by $\varepsilon$-contraction of a fixed 1-periodic measure $μ,$ and for right-hand sides $f^\varepsilon\in L^2({\mathbb R}^3, dμ^\varepsilon),$ we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic "singular structures", when $μ$ 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 $L^2$ 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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arXiv2024

High-contrast random composites: homogenisation framework and new spectral phenomena

We develop a framework for multiscale analysis of elliptic operators with high-contrast random coefficients. For a general class of such operators, we provide a detailed spectral analysis of the corresponding homogenised limit operator. Under some lenient assumptions on the configuration of the random inclusions, we fully characterise the limit of the spectra of the high-contrast operators in question, which unlike in the periodic setting is shown to be different to the spectrum of the homogenised operator. Introducing a new notion of the relevant limiting spectrum, we describe the connection between these two sets.

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arXiv2023

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

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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arXiv2022

Spectral and evolution analysis of composite elastic plates with high contrast

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

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arXiv2022

Sharp operator-norm asymptotics for thin elastic plates with rapidly oscillating periodic properties

We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of smallness. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in e.g. acoustic wave propagation, we derive a class of "hybrid", homogenisation dimension-reduction, norm-resolvent estimates for the plate, under different energy scalings with respect to the plate thickness.

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arXiv2021

Asymptotic analysis of harmonic functions in singular domains with inhomogenous Robin boundary conditions

In 1991, Vladimir Maz'ya, Serguei Nazarov and Boris Plamenevskij developed the theory of compound asymptotics for elliptic boundary value problems in singularly perturbed domains. They considered a harmonic function whose domain contains a small inclusion. We applied this technique in the analysis of a Nematic liquid crystal with a small colloidal inclusion. However, we realised that the Maz'ya, Nazarov and Plamenevskij did not consider the asymptotic analysis of Robin boundary conditions, which corresponded to weak anchoring in the context of liquid crystals. In this piece we shall derive an asymptotic approximation to a harmonic function, in a domain with a small circular inclusion of radius $ε>0$, with inhomogenous Robin boundary conditions and a corresponding parameter $κ>0$. We shall then prove that the difference between the exact solution and the approximation is uniformly bounded and derive the rate as a function of $ε$ and $κ$.

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arXiv2021

Operator-norm convergence estimates for elliptic homogenisation problems on periodic singular structures

For a an arbitrary periodic Borel measure $μ$, we prove order $O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb R}^d, dμ^\varepsilon)$ with $\varepsilon$-periodic coefficients, $\varepsilon>0.$ Here $μ^\varepsilon$ is the measure obtained by $\varepsilon$-scaling of $μ.$ Our analysis includes both the case of a measure absolutely continuous with respect to the standard Lebesgue measure and the case of "singular" periodic structures (or "multistructures"), when $μ$ is supported by lower-dimensional manifolds.

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arXiv2021

Two-scale series expansions for travelling wave packets in one-dimensional periodic media

Starting from the wave equation for a medium with material properties that vary periodically, we study a system of recurrence relations that describe propagation of wave packets that oscillate on the microscale (i.e. on lengths of the order of the period of the medium) and vary slowly on the macroscale (i.e. on lengths that contain a large number of periods). The resulting equations contain a version of the geometric optics and the overall energy transport description for periodic media. We illustrate the developed asymptotic theory using the example of a point pulse propagating through a periodic arrangement of two materials with highly contrasting elastic moduli.

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arXiv2019

Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. I

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media.

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arXiv2019

Frequency-dependent impedance and surface waves on the boundary of a stratified dielectric medium

We analyse waves that propagate along the interface between a dielectric half-space and a half-space filled with a Lorentz material. We show that the corresponding interface condition leads to a generalisation of the classical Leontovich condition on the boundary of a dielectric half-space. We study when this condition supports propagation of (dispersive) surface waves. We derive the related dispersion relation for waves propagating along the boundary of a stratified half-space and determine the relationship between the loss parameter, frequency and wavenumber for which interfacial waves exist.

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arXiv2018

Norm-Resolvent Convergence in Perforated Domains

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-Δ$ in the perforated domain $Ω\setminus \bigcup_{ i\in 2\varepsilon\mathbb Z^d }B_{a_\varepsilon}(i),$ $a_\varepsilon\ll\varepsilon,$ to the limit operator $-Δ+μ_ι$ on $L^2(Ω)$, where $μ_ι\in\mathbb C$ is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Pro c. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.

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arXiv2018

Time-Dispersive Behaviour as a Feature of Critical Contrast Media

Motivated by the urgent need to attribute a rigorous mathematical meaning to the term "metamaterial", we propose a novel approach to the homogenisation of critical-contrast composites. This is based on the asymptotic analysis of the Dirichlet-to-Neumann map on the interface between different components ("stiff" and "soft") of the medium, which leads to an asymptotic approximation of eigenmodes. This allows us to see that the presence of the soft component makes the stiff one behave as a class of time-dispersive media. By an inversion of this argument, we also offer a recipe for the construction of such media with prescribed dispersive properties from periodic composites.

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arXiv2017

Stochastic homogenisation of high-contrast media

Using a suitable stochastic version of the compactness argument of [V. V. Zhikov, 2000. On an extension of the method of two-scale convergence and its applications. Sb. Math., 191(7--8), 973--1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled "macroscopic" and "microscopic" components of the field, giving rise to an analogue of the "Zhikov function", which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.

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