Homogenization Of Elliptic And Parabolic Systems With Stratified Or Locally Periodic Structure

In this doctoral dissertation,we consider the quantitative homogenization of two kinds of systems,the second-order elliptic systems with stratified structure on Lipschitz domains (?) and the second-order parabolic systems with locally periodic structure on C1,1 cylinders (?) where e>0 is a parameter.Systems with locally periodic structure are special cases of models with stratified structure.They are both of great significance in many branches of biomechanics and engineering,and are drawing wide attention in recent years.The thesis mainly consists of two parts.The first part focuses on elliptic systems with stratified structure,where we first discuss the fundamental issue of the measurability of coefficients A(x,?(x)/?),as well as the qualitative homogeniza-tion,under much more general assumptions on A than the Catatheodory criteria.Then,to handle with the intrinsic difficulties in stratified structure,we intro-duce the macroscopic smoothing operator and develop a series of sharp essential estimates.With the help of this tool,we establish the optimal scale-invariant convergence rate in L2d/d-1(?)via the duality argument under wide assumption-s.These results greatly extend the corresponding result.s in the normal periodic homogenization,and the conditions imposed here are almost minimum.Then in the next chapter,we study the interior uniform(with respect to?)regularity estimates for the same system.By using the Meyers estimate,we obtain another suboptimal convergence rate without the symmetry assumption on A,which leads to the large-scale uniform interior Lipschitz estimate by the large-scale scheme.Furthermore,the uniform interior W1,p and Holder estimates are also obtained by the real variable method.The second part is concerned with parabolic systems with locally periodic structure.As like the former part,armed with the macroscopic smoothing op-erator for multi-variable functions,we establish the sharp convergence rate in L2(0,T;L2d/d-1(?))under wide assumptions where the smoothness condition on A in t is 1/2-Holder continuity.During this process,we build a new approach to deal with integrals on temporal boundary layers from the point of view of functional analysis,which earns us better results compared to the pioneering work.Our job here solves a general kind of problem in the quantitative homoge-nization of linear systems and is the first achievement on the optimal convergence rates for systems with stratified structure.The ideas and skills shall be instructive to the homogenization of some other complicated problems,such as the reiterated homogenization,the homogenization of nonlinear systems.