Homogenization of the Cauchy problem for parabolic systems with periodic coefficients

Meshkova, Yu.

doi:10.1090/S1061-0022-2014-01326-X

Homogenization of the Cauchy problem for parabolic systems with periodic coefficients
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by Yu. M. Meshkova
Translated by: the author

St. Petersburg Math. J. 25 (2014), 981-1019

DOI: https://doi.org/10.1090/S1061-0022-2014-01326-X

Published electronically: September 8, 2014

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Abstract:

In $L_2(\mathbb {R}^d;\mathbb {C}^n)$, a class of matrix second order differential operators $\mathcal {B}_\varepsilon$ with rapidly oscillating coefficients (depending on $\mathbf {x}/\varepsilon$) is considered. For a fixed $s>0$ and small $\varepsilon >0$, approximation is found for the operator $\exp (-\mathcal {B}_\varepsilon s)$ in the $(L_2\to L_2)$- and $(L_2\to H^1)$-norm with an error term of order of $\varepsilon$. The results are applied to homogenization of solutions of the parabolic Cauchy problem.

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