Abstract
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
is the small parameter. The coefficients of the operators are periodic and depend on
. The behaviour of the operator
,
, for small
is studied. It is shown that, for fixed
, the operator
converges in the
-operator norm to
, as
. Here
is the effective operator with constant coefficients. For the norm of the difference of the operators
and
, a sharp order estimate (of order
) is obtained. Also, we find approximation for the exponential
in the
-norm with error estimate of order
; in this approximation, a corrector is taken into account. The results are applied to homogenization of solutions of initial boundary value problems for parabolic systems.
Notes
No potential conflict of interest was reported by the authors.