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

@article{Suslina2017HomogenizationOT, title={Homogenization of the Neumann problem for higher order elliptic equations with periodic coefficients}, author={T. A. Suslina}, journal={Complex Variables and Elliptic Equations}, year={2017}, volume={63}, pages={1185 - 1215} }

Let be a bounded domain of class . In , we study a self-adjoint strongly elliptic operator of order 2p given by the expression , , with Neumann boundary conditions. Here, is a bounded and positive definite matrix-valued function in , periodic with respect to some lattice; is a differential operator of order p. The symbol is subject to some condition ensuring strong ellipticity of the operator . We find approximations for the resolvent in different operator norms with error estimates depending…

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## References

SHOWING 1-10 OF 47 REFERENCES

Homogenization of initial boundary value problems for parabolic systems with periodic coefficients

- Mathematics
- 2015

Let be a bounded domain of class . In the Hilbert space , we consider matrix elliptic second-order differential operators and with the Dirichlet or Neumann boundary condition on , respectively. Here…

Homogenization of the elliptic Dirichlet problem: operator error estimates in $L_2$

- Mathematics
- 2012

Let O ⊂ R be a bounded domain of class C. In the Hilbert space L2(O;C ), we consider a matrix elliptic second order differential operator AD,ε with the Dirichlet boundary condition. Here ε > 0 is the…

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…

Homogenization of periodic differential operators of high order

- Mathematics
- 2011

A periodic differential operator of the form Aε = (Dp)∗g(x/ε)Dp is considered on L2(R); here g(x) is a positive definite symmetric tensor of order 2p periodic with respect to a lattice Γ. The…

Homogenization with corrector term for periodic elliptic differential operators

- Mathematics
- 2006

We continue to study the class of matrix periodic elliptic differential operators Aε in Rd with coefficients oscillating rapidly (i.e., depending on x/ε). This class was introduced in the authors’…

Operator estimates in homogenization theory

- Mathematics
- 2005

This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method. Though substantially different in mathematical technique and…

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…

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…

Homogenization of Differential Operators and Integral Functionals

- Mathematics
- 1994

1 Homogenization of Second Order Elliptic Operators with Periodic Coefficients.- 1.1 Preliminaries.- 1.2 Setting of the Homogenization Problem.- 1.3 Problems of Justification Further Examples.- 1.4…