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

@article{Meshkova2018HomogenizationOT,
  title={Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates},
  author={Yu. M. Meshkova and Tatiana Suslina},
  journal={St. Petersburg Mathematical Journal},
  year={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 $B_{D,\varepsilon}$, $0 0$, as $\varepsilon\rightarrow 0$. We obtain approximations for the exponential $e^{-B_{D,\varepsilon}t}$ in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ 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… 
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
Homogenization of periodic parabolic systems in the $L_2(\mathbb {R}^d)$-norm with the corrector taken into account
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a self-adjoint matrix second order elliptic differential operator $\mathcal{B}_\varepsilon$, $0<\varepsilon \leqslant 1$. The principal part of the
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
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 51 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 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
Two-parametric error estimates in homogenization of second order elliptic systems in $\mathbb{R}^d$ including lower order terms
TLDR
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.
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
Quantitative Estimates in Homogenization of Parabolic Systems of Elasticity in Lipschitz Cylinders
In a Lipschitz cylinder, this paper is devoted to establish an almost sharp error estimate $O(\varepsilon\log_2(1/\varepsilon))$ in $L^2$-norm for parabolic systems of elasticity with
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].
...
...