Abstract
Let 
be a bounded C
1,1 domain. In 
we consider strongly elliptic operators A
D,ɛ
and A
N,ɛ
given by the differential expression b(D)*g(x/ɛ)b(D), ɛ > 0, with Dirichlet and Neumann boundary conditions, respectively. Here g(x) is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and b(D) is a first-order differential operator. We find approximations of the operators exp(−A
D,ɛ
t) and exp(−A
N,ɛ
t) for fixed t > 0 and small ɛ in the L
2 → L
2 and L
2 → H
1 operator norms with error estimates depending on ɛ and t. The results are applied to homogenize the solutions of initial boundary value problems for parabolic systems.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 49, No. 1, pp. 88–93, 2015
Original Russian Text Copyright © by Yu. M. Meshkova and T. A. Suslina
Supported by RFBR grant no. 14-01-00760. The first author acknowledges the support of the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026 and JSC “Gazprom Neft.”
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Meshkova, Y.M., Suslina, T.A. Homogenization of solutions of initial boundary value problems for parabolic systems. Funct Anal Its Appl 49, 72–76 (2015). https://doi.org/10.1007/s10688-015-0087-y
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DOI: https://doi.org/10.1007/s10688-015-0087-y