Homogenization of initial boundary value problems for parabolic systems with periodic coefficients

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.

Keywords:

• periodic differential operators
• parabolic systems
• homogenization
• operator error estimates

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by RFBR [project number 14-01-00760] and SPbSU [project number 11.38.263.2014]. The first author is supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026 and by JSC “Gazprom Neft”.