# Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients

@article{Suslina2013HomogenizationOT, title={Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients}, author={Tatiana Suslina}, journal={SIAM J. Math. Anal.}, year={2013}, volume={45}, pages={3453-3493} }

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…

## 70 Citations

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

- Mathematics
- 2017

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…

Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates

- MathematicsSt. Petersburg Mathematical Journal
- 2018

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 a selfadjoint matrix second order elliptic differential operator…

Homogenization for locally periodic elliptic problems on a domain

- Mathematics
- 2020

Let $\Omega$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $\Omega$. We suppose that…

Homogenization of the Stationary Maxwell System with Periodic Coefficients in a Bounded Domain

- Mathematics, Computer ScienceArchive for Rational Mechanics and Analysis
- 2019

The classical results are improved and approximations for the homogenized Maxwell system with perfect conductivity boundary conditions are found.

Two-parametric error estimates in homogenization of second order elliptic systems in $\mathbb{R}^d$ including lower order terms

- Mathematics, Computer Science
- 2015

Approximations for the generalized resolvent in the L_2-norms with two-parametric error estimates (with respect to the parameters $\varepsilon$ and $\zeta$) are obtained.

Uniform boundary estimates in homogenization of higher-order elliptic systems

- Mathematics
- 2017

AbstractThis paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators $$\mathcal {L}_\varepsilon $$Lε, with rapidly oscillating periodic…

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

- Mathematics
- 2014

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…

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

- Mathematics, Computer ScienceSt. Petersburg Mathematical Journal
- 2019

It is shown that the solutions of the Maxwell system, namely, the electric field intensity, electric displacement vector, and the magnetic field intensity converge to the corresponding homogenized fields and the error terms do not exceed $C \varepsilon \| {\mathbf r}\|_{L_2}$.

On homogenization estimates in Neuman boundary value problem for an elliptic equation with multiscale coefficients

- Mathematics
- 2015

Homogenization of a scalar elliptic equation in a bounded domain with Neuman boundary condition is studied. Coefficients of the operator are oscillating over two different groups of variables with…

Periodic homogenization of elliptic systems with stratified structure

- MathematicsDiscrete & Continuous Dynamical Systems - A
- 2019

This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix,…

## References

SHOWING 1-10 OF 28 REFERENCES

OPERATOR ERROR ESTIMATES FOR HOMOGENIZATION OF THE ELLIPTIC DIRICHLET PROBLEM IN A BOUNDED DOMAIN

- Mathematics
- 2012

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator…

Operator error estimates in L2 for homogenization of an elliptic dirichlet problem

- Mathematics
- 2012

AbstractIn a bounded domain O ⊂ ℝd with C1,1 boundary a matrix elliptic second-order operator AD,ɛ with Dirichlet boundary condition is studied. The coefficients of this operator are periodic and…

Convergence Rates in L2 for Elliptic Homogenization Problems

- Mathematics
- 2011

We study rates of convergence of solutions in L2 and H1/2 for a family of elliptic systems $${\{\mathcal{L}_\varepsilon\}}$$ with rapidly oscillating coefficients in Lipschitz domains with Dirichlet…

Error estimate and unfolding for periodic homogenization

- Mathematics
- 2004

This paper deals with the error estimate in problems of periodic homogenization. The methods used are those of the periodic unfolding. We give the upper bound of the distance between the unfolded…

Threshold Effects near the Lower Edge of the Spectrum for Periodic Differential Operators of Mathematical Physics

- Mathematics
- 2001

In L 2 \(\left( {{\mathbb{R}^d}} \right), \) we consider vector periodic DO A admitting a factorization A = X*X, where X is a homogeneous DO of first order. Many operators of mathematical physics…

HOMOGENIZATION OF THE DIRICHLET PROBLEM FOR ELLIPTIC SYSTEMS: ‐OPERATOR ERROR ESTIMATES

- Mathematics
- 2013

Let OR d be a bounded domain of class C 1,1 . In L2(O;C n ), we consider a matrix elliptic differential operator A" = b(D) � g(x/")b(D) with the Dirichlet boundary condition. We assume that an…

Second order periodic differential operators. Threshold properties and homogenization

- Mathematics
- 2004

The vector periodic differential operators (DO’s) A admitting a factorization A = X ∗X , where X is a first order homogeneous DO, are considered in L2(R). Many operators of mathematical physics have…

Homogenization of the elliptic Dirichlet problem: Error estimates in the (L2 → H1)-norm

- Mathematics
- 2012

Let ⊂ ℝd be a bounded domain with boundary of class C1,1. In L2(;ℂn), consider a matrix elliptic second-order differential operator AD,ɛ with Dirichlet boundary condition. Here ɛ > 0 is a small…

Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials

- Mathematics
- 1989

1. Formulation of Elementary Boundary Value Problems.- 1. The Concept of the Classical Formulation of a Boundary Value Problem for Equations with Discontinuous Coefficients.- 2. The Concept of…

Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class ¹(ℝ^{})

- Mathematics
- 2007

Investigation of a class of matrix periodic elliptic second-order differential operators Aε in Rd with rapidly oscillating coefficients (depending on x/ε) is continued. The homogenization problem in…