Regularities Of The Gradient In Lorentz Spaces To Weak Solutions For Two Classes Of Elliptic Obstacle Problems

In this dissertation,we are devoted to the regularities of the gradient in the Lorentz spaces to weak solutions for two classes of elliptic obstacle problems under the weak as-sumptions imposed on the leading coefficients and boundary of domain.The first one is to get a global weighted Lorentz estimate of gradients to weak solution of linear elliptic obstacle problems with partially BMO coefficients over a bounded Reifenberg domain,The other one is to obtain a local regularity of the gradients in the Lorentz spaces to weak solutions to nonlinear obstacle elliptic problems with(p1,p2)type nonstandard growth.This paper will be organized as follows.In the first chapter,we briefly introduce the background and the recent development concerning the topic of this paper,recall the definitions of Lorentz spaces and weighted Lorentz spaces,and Vitali covering lemma.In the second chapter,we focus on the global weighted Lorentz estimate of the gradients in the weighted Lorentz spaces to weak solution of the following linear ob-stacle elliptic system satisfying the given obstacle ?=(?1,…?m)? W1,2(?,Rm)over Reifenberg flat domain ?:where the leading coefficients Aij??(x)satisfying uniformly ellipticity and measurable in one variable and have small BMO-norms in the other variables.We have the following global weighted Lorentz estimate:Our ingredient is mainly to use the modified Vitali covering lemma,the boundedness of Hardy-Littlewood maximal function in the weighted Lorentz spaces,and the equivalent norm of the weighted Lorentz space in the sense of measure theory.In the third chapter,we consider the following nonlinear obstacle elliptic problems with nonstandard growth the vector field A(x,z):?×Rn?Rn is assumed to be continuous in x ? Q and is C1(Rn\{0})regular with respect to z ? Rn,with the following conditions:andWe have the following Lorentz estimate:Here,we use the large-M-inequality to deal with the upper level set E(?,BS)={x ? Bs:H(x,Du)>?} such that its energy decay estimate being realized.In fact,the energy decay estimate can be achieved while the A increases by way of using Vitali covering lemma and large-M-inequality.This argument was originated from Colombo-Mingione's recent work.