[Author]
 Birman, M.Sh.; Suslina, T.A.

[Title]
 Operator error estimates in homogenization of nonstationary periodic equations

[AMS Subj-class]
 35B27 Homogenization; partial differential equations in media with periodic structure

[Abstract]
 We study matrix periodic differential operators $A=A(x,D)$ acting in
 $L_2(R^d;C^n)$ and admitting a factorization of the form $A=X^*X$, where $X$
 is a homogeneous first order differential operator. We put $A_\epsilon =
 A(\epsilon^{-1}x,D)$, $\epsilon >0$, and consider the Cauchy problem for the
 Schrodinger equation $i\partial_\tau u_\epsilon = A_\epsilon u_\epsilon$ and
 the Cauchy problem for the hyperbolic equation $\partial^2_\tau u_\epsilon =
 -A_\epsilon u_\epsilon$. We study the behavior of solutions $u_\epsilon$ as
 $\epsilon \to 0$. Let $u_0$ be the solution of the corresponding homogenized
 problem. We obtain the estimates of order $\epsilon$ for the norm of the
 difference $u_\epsilon - u_0$ in $L_2(\R^d;\C^n)$ for a fixed $\tau \in R$.
 The estimates are uniform with respect to the norm of initial data in the
 Sobolev space $H^s(\R^d;\C^n)$, where $s=3$ in the case of the Schrodinger
 equation and $s=2$ in the case of the hyperbolic equation. We trace the
 dependence of constants in estimates on $\tau$, which allows us to obtain
 qualified error estimates for small $\epsilon$ and large $|\tau| =
 O(\epsilon^{-\alpha})$ with appropriate $\alpha <1$.

[Keywords]
 periodic operators, nonstationary equations, Cauchy problem, 
 threshold effect, homogenization, effective operator 

[Comments]
 Russian, 70 pp.

[Contact e-mail]
 suslina@list.ru