Quantitative Estimates in Homogenization of Parabolic Systems of Elasticity in Lipschitz Cylinders

Abstract

In a Lipschitz cylinder, this paper is devoted to establish an almost sharp error estimate $O(\varepsilon\log_2(1/\varepsilon))$ in $L^2$-norm for parabolic systems of elasticity with initial-Dirichlet conditions, arising in the homogenization theory. To achieve the goal, with the parabolic distance function being a weight, we first developed some new weighted-type inequalities for the smoothing operator at scale $\varepsilon$ in terms of t-anisotropic Sobolev spaces, and then all the problems may be reduced to three kinds of estimate for the homogenized system, in which a weighted-type Caccioppoli's inequality on time-layer has also been found. Throughout the paper, we do not introduce any smoothness on coefficients compared to the arguments investigated by C.Kenig, F. Lin and Z. Shen in \cite{SZW2}, while this study can be considered to be a further development of \cite{GZS} and \cite{QX2}.