[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