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}
}
• T. Suslina
• Published 5 December 2012
• Mathematics
• SIAM J. Math. Anal.
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…
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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
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
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
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