arXiv subjects

Tatiana Suslina

Publications and source records attributed to Tatiana Suslina.

17 recordsLinked to original sources

arXiv2026

Homogenization of Lévy-type operators: operator estimates with correctors

The goal of the paper is to study in $L_2(\R^d)$ a self-adjoint operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} μ(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+α}}\,d\y $$ with $1< α< 2$; here the function $μ(\x,\y)$ is $\Z^d$-periodic in the both variables, satisfies the symmetry relation $μ(\x,\y) = μ(\y,\x)$ and the estimates $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. The rigorous definition of the operator ${\mathbb A}_\eps$ is given in terms of the corresponding quadratic form. In the previous work of the authors it was shown that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\mathbb R^d)$ to the resolvent of the effective operator $A^0$, and the estimate $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} \| = O(\eps^{2-α})$ holds. In the present work we achieve a more accurate approximation of the resolvent of ${\mathbb A}_\eps$ which takes into account the correctors. Namely, for $N\in\mathbb N$ such that $2-1/N < α\le 2-1/(N+1)$, we obtain $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-α)} \mathbb{K}_m \bigr\| = O(\eps). $$

math.AP↗

arXiv2025

Homogenization of non-symmetric convolution type operators

The paper studies homogenization problem for a bounded in $L_2(\mathbb R^d)$ convolution type operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \eps^{-d-2} \int_{\R^d} a((\x-\y)/\eps) μ(\x/\eps, \y/\eps) \left( u(\x) - u(\y) \right)\,d\y. $$ It is assumed that $a(\x)$ is a non-negative function from $L_1(\R^d)$, and $μ(\x,\y)$ is a periodic in $\x$ and $\y$ function such that $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. No symmetry assumption on $a(\cdot)$ and $μ(\cdot)$ is imposed, so the operator ${\mathbb A}_\eps$ need not be self-adjoint. Under the assumption that the moments $M_k = \int_{\R^d} |\x|^k a(\x)\,d\x$, $k=1,2,3$, are finite we obtain, for small $\eps>0$, sharp in order approximation of the resolvent $({\mathbb A}_\eps + I)^{-1}$ in the operator norm in $L_2(\mathbb R^d)$, the discrepancy being of order $O(\eps)$. The approximation is given by an operator of the form $({\mathbb A}^0 + \eps^{-1} \langle \boldsymbolα,\nabla \rangle + I)^{-1}$ multiplied on the right by a periodic function $q_0(\x/\eps)$; here ${\mathbb A}^0 = - \operatorname{div}g^0 \nabla$ is the effective operator, and $\boldsymbolα$ is a constant vector.

math.FA↗

arXiv2024

Operator estimates in homogenization of Lévy-type operators with periodic coefficients

The paper deals with homogenization of self-adjoint operators in $L_2(\mathbb R^d)$ of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} μ(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+α}}\,d\y, $$ where $0< α< 2$, and $\eps>0$ is a small parameter. It is assumed that the function $μ(\x,\y)$ is $\Z^d$-periodic in each variable, $μ(\x,\y)=μ(\y,\x)$ for all $\x$ and $\y$, and $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. Under these assumptions we show that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\R^d)$ to the resolvent $({\mathbb A}^0 + I)^{-1}$ of the limit operator ${\mathbb A}^0$ given by $$ ({\mathbb A}^0 u) (\x) = \int_{\R^d} μ^0 \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+α}}\,d\y, $$ where $μ^0$ is the mean value of $μ(\x,\y)$. We also show that the operator norm of the discrepancy $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1}\|_{L_2(\mathbb R^d)\to L_2(\mathbb R^d)}$ can be estimated by $O(\eps^α)$, if $0< α< 1$, by $O(\eps (1 + | \operatorname{ln} \eps|)^2)$, if $ α=1$, and by $O(\eps^{2- α})$, if $1< α< 2$.

math.AP↗

arXiv2023

Homogenization of nonlocal convolution type operators: Approximation for the resolvent with corrector

In $L_2(\mathbb{R}^d)$, we consider a selfadjoint bounded operator ${\mathbb A}_\varepsilon$, $\varepsilon >0$, of the form $$ ({\mathbb A}_\varepsilon u) (\mathbf{x}) = \varepsilon^{-d-2} \int_{\mathbb{R}^d} a((\mathbf{x} - \mathbf{y} )/ \varepsilon ) μ(\mathbf{x} /\varepsilon, \mathbf{y} /\varepsilon) \left( u(\mathbf{x}) - u(\mathbf{y}) \right)\, d\mathbf{y}. $$ It is assumed that $a(\mathbf{x})$ is a nonnegative function of class $L_1(\mathbb{R}^d)$ such that \hbox{$a(-\mathbf{x}) = a(\mathbf{x})$} and $μ(\mathbf{x},\mathbf{y})$ is $\mathbb{Z}^d$-periodic in each variable and such that $μ(\mathbf{x},\mathbf{y}) = μ(\mathbf{y},\mathbf{x})$ and $0< μ_- \leqslant μ(\mathbf{x},\mathbf{y}) \leqslant μ_+< \infty$. Moreover, it is assumed that the moments $M_k (a)= \int_{\mathbb{R}^d} | \mathbf{x} |^k a(\mathbf{x})\,d\mathbf{x}$, $k=1,2,3,4,$ are finite. We obtain approximation of the resolvent $({\mathbb A}_\varepsilon + I)^{-1}$ for small $\varepsilon$ in the operator norm on $L_2(\mathbb{R}^d)$ with error of order $O(\varepsilon^2)$.

math.AP↗

arXiv2020

Homogenization of the higher-order Schrödinger-type equations with periodic coefficients

In $L_2({\mathbb R}^d; {\mathbb C}^n)$, we consider a matrix strongly elliptic differential operator ${A}_\varepsilon$ of order $2p$, $p \geqslant 2$. The operator ${A}_\varepsilon$ is given by ${A}_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$, $\varepsilon >0$, where $g(\mathbf{x})$ is a periodic, bounded, and positive definite matrix-valued function, and $b(\mathbf{D})$ is a homogeneous differential operator of order $p$. We prove that, for fixed $τ\in {\mathbb R}$ and $\varepsilon \to 0$, the operator exponential $e^{-i τ{A}_\varepsilon}$ converges to $e^{-i τ{A}^0}$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d; {\mathbb C}^n)$ (with a suitable $s$) into $L_2({\mathbb R}^d; {\mathbb C}^n)$. Here $A^0$ is the effective operator. Sharp-order error estimate is obtained. The results are applied to homogenization of the Cauchy problem for the Schrödinger-type equation $i \partial_τ{\mathbf u}_\varepsilon = {A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$, ${\mathbf u}_\varepsilon\vert_{τ=0} = \boldsymbolϕ$.

math.AP↗

arXiv2020

Homogenization of nonstationary periodic Maxwell system in the case of constant permeability

In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator ${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential expression $μ_0^{-1/2}\operatorname{curl} η(\mathbf{x}/\varepsilon)^{-1} \operatorname{curl} μ_0^{-1/2} - μ_0^{1/2}\nabla ν(\mathbf{x}/\varepsilon) \operatorname{div} μ_0^{1/2}$, where $μ_0$ is a constant positive matrix, a matrix-valued function $η(\mathbf{x})$ and a real-valued function $ν(\mathbf{x})$ are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions $\cos (τ{\mathcal L}_\varepsilon^{1/2})$ and ${\mathcal L}_\varepsilon^{-1/2} \sin (τ{\mathcal L}_\varepsilon^{1/2})$ for $τ\in {\mathbb R}$ and small $\varepsilon$. It is shown that these operators converge to the corresponding operator-valued functions of the operator ${\mathcal L}^0$ in the norm of operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here ${\mathcal L}^0$ is the effective operator with constant coefficients. Also, an approximation with corrector in the $(H^s \to H^1)$-norm for the operator ${\mathcal L}_\varepsilon^{-1/2} \sin (τ{\mathcal L}_\varepsilon^{1/2})$ is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on $τ$. The results are applied to homogenization of the Cauchy problem for the nonstationary Maxwell system in the case where the magnetic permeability is equal to $μ_0$, and the dielectric permittivity is given by the matrix $η(\mathbf{x}/\varepsilon)$.

math.AP↗

arXiv2020

Homogenization of hyperbolic equations with periodic coefficients in ${\mathbb R}^d$: sharpness of the results

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal A}_\varepsilon$ is considered. It is assumed that the coefficients of the operator ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. We find approximations for the operators $\cos ( {\mathcal A}_\varepsilon^{1/2}τ)$ and ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}τ)$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d)$ to $L_2({\mathbb R}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}τ)$ in the $(H^s \to H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $τ$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation $\partial_τ^2 {\mathbf u}_\varepsilon = - {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$.

math.AP↗

arXiv2018

Homogenization of the stationary Maxwell system with periodic coefficients in a bounded domain

In a bounded domain $\mathcal{O}\subset\mathbb{R}^3$ of class $C^{1,1}$, we consider a stationary Maxwell system with the perfect conductivity boundary conditions. It is assumed that the dielectric permittivity and the magnetic permeability are given by $η({\mathbf x}/ \varepsilon )$ and $μ({\mathbf x}/ \varepsilon )$, where $η( {\mathbf x})$ and $μ({\mathbf x})$ are symmetric $(3 \times 3)$-matrix-valued functions; they are periodic with respect to some lattice, bounded and positive definite. Here $\varepsilon >0$ is the small parameter. We use the following notation for the solutions of the Maxwell system: ${\mathbf u}_\varepsilon$ is the electric field intensity, ${\mathbf v}_\varepsilon$ is the magnetic field intensity, ${\mathbf w}_\varepsilon$ is the electric displacement vector, and ${\mathbf z}_\varepsilon$ is the magnetic displacement vector. It is known that ${\mathbf u}_\varepsilon$, ${\mathbf v}_\varepsilon$, ${\mathbf w}_\varepsilon$, and ${\mathbf z}_\varepsilon$ weakly converge in $L_2({\mathcal O})$ to the corresponding homogenized fields ${\mathbf u}_0$, ${\mathbf v}_0$, ${\mathbf w}_0$, and ${\mathbf z}_0$ (the solutions of the homogenized Maxwell system with the effective coefficients), as $\varepsilon \to 0$. We improve the classical results and find approximations for ${\mathbf u}_\varepsilon$, ${\mathbf v}_\varepsilon$, ${\mathbf w}_\varepsilon$, and ${\mathbf z}_\varepsilon$ in the $L_2({\mathcal O})$-norm. The error terms do not exceed $C \sqrt{\varepsilon} (\| {\mathbf q}\|_{L_2}+\|{\mathbf r}\|_{L_2})$, where the divergence free vector-valued functions ${\mathbf q}$ and ${\mathbf r}$ are the right-hand sides of the Maxwell equations.

math.AP↗

arXiv2018

Homogenization of a stationary periodic Maxwell system in a bounded domain in the case of constant magnetic permeability

In a bounded domain $\mathcal{O}\subset\mathbb{R}^3$ of class $C^{1,1}$, we consider a stationary Maxwell system with the boundary conditions of perfect conductivity. It is assumed that the magnetic permeability is given by a constant positive $(3\times 3)$-matrix $μ_0$ and the dielectric permittivity is of the form $η({\mathbf x}/ \varepsilon)$, where $η({\mathbf x})$ is a $(3 \times 3)$-matrix-valued function with real entries, periodic with respect to some lattice, bounded and positive definite. Here $\varepsilon >0$ is the small parameter. Suppose that the equation involving the curl of the magnetic field intensity is homogeneous, and the right-hand side $\mathbf r$ of the second equation is a divergence-free vector-valued function of class $L_2$. It is known that, as $\varepsilon \to 0$, the solutions of the Maxwell system, namely, the electric field intensity ${\mathbf u}_\varepsilon$, the electric displacement vector ${\mathbf w}_\varepsilon$, the magnetic field intensity ${\mathbf v}_\varepsilon$, and the magnetic displacement vector ${\mathbf z}_\varepsilon$ weakly converge in $L_2$ to the corresponding homogenized fields ${\mathbf u}_0$, ${\mathbf w}_0$, ${\mathbf v}_0$, ${\mathbf z}_0$ (the solutions of the homogenized Maxwell system with effective coefficients). We improve the classical results. It is shown that ${\mathbf v}_\varepsilon$ and ${\mathbf z}_\varepsilon$ converge to ${\mathbf v}_0$ and ${\mathbf z}_0$, respectively, in the $L_2$-norm, the error terms do not exceed $C \varepsilon \| {\mathbf r}\|_{L_2}$. We also find approximations for ${\mathbf v}_\varepsilon$ and ${\mathbf z}_\varepsilon$ in the energy norm with error $C\sqrt{\varepsilon} \|{\mathbf r}\|_{L_2}$. For ${\mathbf u}_\varepsilon$ and ${\mathbf w}_\varepsilon$ we obtain approximations in the $L_2$-norm with error $C\sqrt{\varepsilon} \| {\mathbf r}\|_{L_2}$.

math.AP↗

arXiv2017

Spectral approach to homogenization of hyperbolic equations with periodic coefficients

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/ \varepsilon$, $\varepsilon>0$. We study the behavior of the operators $\cos( {\mathcal A}^{1/2}_\varepsilon τ)$ and ${\mathcal A}^{-1/2}_\varepsilon \sin( {\mathcal A}^{1/2}_\varepsilon τ)$, $τ\in \mathbb{R}$, for small $\varepsilon$. Approximations for these operators in the $(H^s\to L_2)$-operator norm with a suitable $s$ are obtained. The results are used to study the behavior of the solution ${\mathbf v}_\varepsilon$ of the Cauchy problem for the hyperbolic equation $\partial^2_τ{\mathbf v}_\varepsilon = - \mathcal{A}_\varepsilon {\mathbf v}_\varepsilon +\mathbf{F}$. General results are applied to the acoustics equation and the system of elasticity theory.

math.AP↗

arXiv2017

Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we study a selfadjoint matrix elliptic second order differential operator $B_{D,\varepsilon}$, $0<\varepsilon\leqslant 1$, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves lower order terms with unbounded coefficients. The coefficients of $B_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $\left(B_{D,\varepsilon}-ζQ_0(\cdot/\varepsilon)\right)^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $ζ$ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the $L_2(\mathcal{O};\mathbb{C}^n)$-operator norm and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$, with two-parametric error estimates (depending on $\varepsilon$ and $ζ$).

math.AP↗

arXiv2017

Homogenization of the Neumann problem for higher-order elliptic equations with periodic coefficients

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{2p}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we study a selfadjoint strongly elliptic operator $A_{N,\varepsilon}$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$, with the Neumann boundary conditions. Here $g({\mathbf x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function in ${\mathbb R}^d$, periodic with respect to some lattice; $b({\mathbf D})=\sum_{|α|=p} b_α{\mathbf D}^α$ is a differential operator of order $p$ with constant coefficients; $b_α$ are constant $(m\times n)$-matrices. It is assumed that $m\geqslant n$ and that the symbol $b({\boldsymbol ξ})$ has maximal rank for any $0 \ne {\boldsymbol ξ}\in {\mathbb C}^d$. We find approximations for the resolvent $\left(A_{N,\varepsilon}-ζI \right)^{-1}$ in the $L_2(\mathcal{O};\mathbb{C}^n)$-operator norm and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^p(\mathcal{O};\mathbb{C}^n)$, with error estimates depending on $\varepsilon$ and $ζ$.

math.AP↗

arXiv2016

Homogenization of hyperbolic equations with periodic coefficients

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/ \varepsilon$, $\varepsilon>0$. We study the behavior of the operator cosine $\cos( {\mathcal A}^{1/2}_\varepsilon τ)$, $τ\in \mathbb{R}$, for small $\varepsilon$. Approximations for this operator in the $(H^s\to L_2)$-operator norm with a suitable $s$ are obtained. The results are used to study the behavior of the solution ${\mathbf v}_\varepsilon$ of the Cauchy problem for the hyperbolic equation $\partial^2_τ{\mathbf v}_\varepsilon = - \mathcal{A}_\varepsilon {\mathbf v}_\varepsilon +\mathbf{F}$. General results are applied to the acoustics equation and the system of elasticity theory.

math.AP↗

arXiv2015

Homogenization of high order elliptic operators with periodic coefficients

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function in ${\mathbb R}^d$; it is assumed that $g({\mathbf x})$ is periodic with respect to some lattice. Next, $b({\mathbf D})=\sum_{|α|=p}^d b_α{\mathbf D}^α$ is a differential operator of order $p$ with constant coefficients; $b_α$ are constant $(m\times n)$-matrices. It is assumed that $m\ge n$ and that the symbol $b({\boldsymbol ξ})$ has maximal rank. For the resolvent $(A_\varepsilon - ζI)^{-1}$ with $ζ\in {\mathbb C} \setminus [0,\infty)$, we obtain approximations in the norm of operators in $L_2({\mathbb R}^d;{\mathbb C}^n)$ and in the norm of operators acting from $L_2({\mathbb R}^d;{\mathbb C}^n)$ to the Sobolev space $H^p({\mathbb R}^d;{\mathbb C}^n)$, with error estimates depending on $\varepsilon$ and $ζ$.

math.AP↗

arXiv2015

Homogenization of nonstationary Schrödinger type equations with periodic coefficients

In $L_2(\mathbb{R}^d;{\mathbb C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/\varepsilon$. We study the behavior of the operator exponential $\exp(-i {\mathcal A}_\varepsilon τ)$, $τ\in {\mathbb R}$, for small $\varepsilon$. Approximations for this exponential in the $(H^s\to L_2)$-operator norm with a suitable $s$ are obtained. The results are applied to study the behavior of the solution ${\mathbf u}_\varepsilon$ of the Cauchy problem for the Schrödinger type equation $i \partial_τ{\mathbf u}_\varepsilon = {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon$.

math.AP↗

arXiv2014

Homogenization of elliptic problems: error estimates in dependence of the spectral parameter

We consider a strongly elliptic differential expression of the form $b(D)^* g(x/\varepsilon) b(D)$, $\varepsilon >0$, where $g(x)$ is a matrix-valued function in ${\mathbb R}^d$ assumed to be bounded, positive definite and periodic with respect to some lattice; $b(D)=\sum_{l=1}^d b_l D_l$ is the first order differential operator with constant coefficients. The symbol $b(ξ)$ is subject to some condition ensuring strong ellipticity. The operator given by $b(D)^* g(x/\varepsilon) b(D)$ in $L_2({\mathbb R}^d;{\mathbb C}^n)$ is denoted by $A_\varepsilon$. Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, we consider the operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ given by $b(D)^* g(x/\varepsilon) b(D)$ with the Dirichlet or Neumann boundary conditions, respectively. For the resolvents of the operators $A_\varepsilon$, $A_{D,\varepsilon}$, and $A_{N,\varepsilon}$ in a regular point $ζ$ we find approximations in different operator norms with error estimates depending on $\varepsilon$ and the spectral parameter $ζ$.

math.AP↗

arXiv2012

Homogenization of the Neumann problem for elliptic systems with periodic coefficients

Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, a matrix elliptic second order differential operator ${\mathcal A}_{N,\varepsilon}$ with the Neumann boundary condition is considered. Here $\varepsilon>0$ is a small parameter, the coefficients of ${\mathcal A}_{N,\varepsilon}$ are periodic and depend on ${\mathbf x} /\varepsilon$. There are no regularity assumptions on the coefficients. It is shown that the resolvent $({\mathcal A}_{N,\varepsilon}+λI)^{-1}$ converges in the $L_2({\mathcal O};{\mathbb C}^n)$-operator norm to the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$. A sharp order error estimate $|({\mathcal A}_{N,\varepsilon}+λI)^{-1} - ({\mathcal A}_{N}^0 +λI)^{-1}|_{L_2\to L_2} \le C\varepsilon$ is obtained. Approximation for the resolvent $({\mathcal A}_{N,\varepsilon}+λI)^{-1}$ in the norm of operators acting from $L_2({\mathcal O};{\mathbb C}^n)$ to the Sobolev space $H^1({\mathcal O};{\mathbb C}^n)$ with an error $O(\sqrt{\varepsilon})$ is found. Approximation is given by the sum of the operator $({\mathcal A}^0_N +λI)^{-1}$ and the first order corrector. In a strictly interior subdomain ${\mathcal O}'$ a similar approximation with an error $O(\varepsilon)$ is obtained.

math.AP↗