Global Weighted Lorentz Estimates For Homogenization Of Elliptic And Parabolic Obstacle Problems

In this thesis,we study global estimates for gradients of weak solutions in Lorentz and Orlicz spaces to the homogenization of linear elliptic obstacle problem,linear parabolic obstacle problem and the Dirichlet problem of a class of nonlinear elliptic e-quations with discontinuous coefficients in non-smooth domains.The specific contents and conclusions are as follows.In Chapter 1,we introduce the research background and the present situation at home and abroad.Besides,we also give the related concepts and summarize the main results of this article.In Chapter 2,under the assumptions that coefficients satisfy small BMO seminorm and the underlying domains are Reifenberg flat,we obtain global Lorentz estimate for the gradient of the weak solution to the homogenization of elliptic obstacle problem.This is an extension of the classic Calderon-Zygmund theory for the homogenization of linear elliptic equation.In Chapter 3,under the conditions of discontinuous coefficients and rough domain-s,we establish a global Lorentz estimate for the gradient of the weak solution for the homogenization of parabolic obstacle problem.This work is to extend the Calderon-Zygmund estimate for the homogenization of elliptic equation to the setting of parabolic problem.In addition,it also extends the Lebesgue space to the refined Lorentz space.In Chapter 4,we establish global weighted Lorentz estimates for the gradients of weak solutions to the Dirichlet problem of nonlinear elliptic equations in Reifenberg flat domains when the nonlinearities are satisfy small BMO.This is an extension from the Lebesgue space to the refined Lorentz space.In the Chapter 5,a summary of main work of the whole thesis is given and some problems to be further studied are discussed.