W^{1, P} Estimates For Degenerate Elliptic Equations With BMO Coefficients In Non-smooth Domain

This paper mainly considers the W^l,p global estimates for the weak solutions of a class of degenerate elliptic equations. The principal coefficients are supposed to be in the Jhon-Nirenberg space with BMO semi-norms and the domain is a Reifenberg domain. Under this weaker restriction to the coefficients and the domain, the regularity result of this class of equations is obtained, which modified the Lipsticz continuity requirement to the coefficients and the boundary of the domain. The preliminary tools to investigate this question are the modified Vitali covering lemma and Hardy-Littlewood maximal function. By using the inclusion of level sets and the standard arguments of measure theory about the class of L^p functions established by Caffarelli, we shall obtain the regularity estimates of the equation on Reifenberg flat domain.The paper is made up of four chapters. As follows:The first chapter will introduce two important historical backgrounds about this question. Firstly, we shall investigate the classic Schauder regularity of the basic linear elliptic equations when the elliptic coefficients are Holder continuous. Secondly, by studying the Holder continuity to the solutions of the divergent elliptic equations with bounded measurable coefficients, we established the main outline of De Giorgi iteration and Moser iteration. Moreover, the skills to study inhomogeneous equations by this conclusion are obtained.In the second chapter, we shall consider all kinds of cutting-edge results about the weak solution regularity of elliptic equations when assumptions on the coefficients are between the two extreme conditions above. At the same time, the corresponding research methods are briefly summarized. After that the topic of the paper is put forward and the preliminary tools are given.In the third chapter, we shall consider the interior W^l,p estimates for the elliptic equations as follows:where the matrix of coefficients A(x) is (δ, R)-vanishing andΩis a bounded open domain.In the forth chapter, we shall make observations on the minimal regularity requirementsto the coefficients and the domain of the Dirichlet problem for the above mentioned equation and prove the global regularity under the assumptions that A(x) is (δ,R)-vanishing and the domain is (δ, R)-Reifenberg flat. We will use the modified Vitali covering lemma and the equivalence theory for L^p space established by Caffarelli.At last, the main conclusions of this paper are summarized, and the further research problems are presented.