Regularity Estimates For Elliptic And Parabolic Systems In Homogenization And Elliptic Systems With Drift Terms

This doctoral thesis studies the regularity estimates for parabolic systems and the systems of linear elasticity in homogenization and elliptic systems with drift terms in non-smooth domains,and consists of five chapters.Chapter 1 is of an introductory chapter,here we give an introduction to the problem and show the main results and key ideas of this dissertation.In chapter 2,we introduce some notations,and briefly recall some needed estimates and preliminaries.In chapter 3,we consider a family of second-order parabolic systems in divergence form with rapidly oscillating periodic and time-dependent coefficients.We establish the pointwise estimates of the Green's matrices.We investigate uniform boundary Lipschitz estimates and boundary Holder estimates as well as non-tangential maximum function estimates in C1,? or C1 cylinders.Consequently,we also obtain the AgmonMiranda maximum principle.In chapter 4,for a fixed bounded Lipschitz domain and a family of systems of linear elasticity with rapidly oscillating periodic coefficients,we investigate a necessary and sufficient condition that an A1 weight ? must satisfy in order for the weighted W1,2(w)estimates for weak solutions of Neumann problems to be true.Moreover,in any Lipschitz domain,under the assumption that A?C?,we prove that the uniform W1,p estimates for solutions to the Neumann problem hold for 2d/(d+1)-?