• Corpus ID: 119116580

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

@article{Meshkova2017HomogenizationOT,
  title={Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates},
  author={Yu. M. Meshkova and Tatiana Suslina},
  journal={arXiv: Analysis of PDEs},
  year={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 $B_{D,\varepsilon}$, $0<\varepsilon\leqslant 1$, 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… 
Homogenization of the first initial boundary-value problem for parabolic systems: operator 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 consider a selfadjoint matrix second order elliptic differential operator
Variations on the theme of the Trotter-Kato theorem for homogenization of periodic hyperbolic systems
TLDR
The proof of the Trotter-Kato theorem is adopted by introduction of some correction term and hyperbolic results from elliptic ones are derived from elliptIC ones.
On homogenization of periodic hyperbolic systems in $L_2(\mathbb{R}^d;\mathbb{C}^n)$. Variations on the theme of the Trotter-Kato theorem
In L2(R d;Cn), we consider a matrix elliptic second order differential operator Bε > 0. Coefficients of the operator Bε are periodic with respect to some lattice in Rd and depend on x/ε. We study the
Homogenization of the Neumann problem for higher order elliptic equations with periodic coefficients
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
On homogenization of the first initial-boundary value problem for periodic hyperbolic systems
ABSTRACT Let be a bounded domain of class . In , we consider a self-adjoint matrix strongly elliptic second-order differential operator , , with the Dirichlet boundary condition. The coefficients of

References

SHOWING 1-10 OF 36 REFERENCES
OPERATOR ERROR ESTIMATES FOR HOMOGENIZATION OF THE ELLIPTIC DIRICHLET PROBLEM IN A BOUNDED DOMAIN
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
Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients
TLDR
A sharp order estimate for the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$ is obtained.
Homogenization of the first initial boundary-value problem for parabolic systems: operator 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 consider a selfadjoint matrix second order elliptic differential operator
Homogenization for Non-self-adjoint Periodic Elliptic Operators on an Infinite Cylinder
  • N. N. Senik
  • Mathematics, Computer Science
    SIAM J. Math. Anal.
  • 2017
TLDR
The problem of homogenization for non-self-adjoint second-order elliptic differential operators of divergence form on L_{2}(\mathbb{R}^{d_{1}}\times\mathbb {T}^{D_{2}})$, where $d_1}$ is positive and~$d_2$ is non-negative, is considered.
Convergence Rates in L2 for Elliptic Homogenization Problems
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
Convergence Rates for General Elliptic Homogenization Problems in Lipschitz Domains
  • Qiang Xu
  • Mathematics
    SIAM J. Math. Anal.
  • 2016
TLDR
The paper finds the new weighted-type estimates for the smoothing operator at scale $\varepsilon$ and obtains sharp convergence rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Shen for elasticity systems in [preprint, arXiv:1505.00694v1, 2015].
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
...
...