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 $O(\epsilon {\log }_2(1/\epsilon ))$ 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 $\epsilon$ 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}.

37 pages, 3 figures

References in corpus (2)

• Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity
• Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

Cited by in corpus (1)

• Homogenization of locally periodic parabolic operators with non-self-similar scales