Calderón-zygmund Type Estimates For Non-uniformly Elliptic Equations Of Divergence Form

This doctoral dissertation is devoted to the Calderon-Zygmund type estimates for various elliptic equations of divergence form.We mainly discuss four problems.First,we prove the W1,?(·)-regularity for weak solutions to nonlinear non-uniformly elliptic e-quations and the L?-estimates to corresponding double obstacle problems and asymptot-ically regular problems.Second,we study the estimates in Lorentz spaces to nonlinear elliptic equation with Lp(·)log L-growth and the estimates in Lebesgue spaces of cor-responding double obstacle problems.Third,we prove a Lorentz estimate for variable power of the gradients to weak solutions of nonlinear elliptic equations with partially regular nonlinearities and a Orlicz estimate for the corresponding obstacle problems.Fourth,we prove a Lorentz estimate for variable power of the gradients to weak solu-tion pair(u,P)of stationary steady Stokes systems.This paper construct by seven parts,the specific contents are summarized as follows:In Chapter 1,we briefly recall the research background,the recent development involving our research topic and research content.In Chapter 2,we consider nonlinear non-uniformly elliptic problems div A(Du,x)=div G(F,x)with the model case of A(Du,x)?|Du|p-2 Du+a(x)|Du|q-2 Du and G(F,x)?|F|p-2F+a(x)|F|q-2F.Based on large-M-inequality principle,the geometric approach,various approximate estimates and flattening argument,we prove a global W1,?(·)-estimate,under the sharp assumptions that ?(x)?1 satisfy the log-Holder continuity,a(·)is C0,?-Holder continuous with ??(0,1],1<p<q<p+?p/n and the boundary of domain is of class C1,? with ??[?,1].In Chapter 3,we still study non-uniformly elliptic problems,which is an extension and further study of the problems in Chapter 2.Under the same assumptions,we first obtain interior estimates in the setting of Lebesgue spaces and weighted Lebesgue s-paces for the corresponding double obstacle problem.And we further prove a global L?estimate for non-uniformly elliptic equations with an asymptotically regular nonlinear-ity,the nonlinearity is asymptotically regular to the operator considered in Chapter 2 Here,a key ingredient of our proof is to make use of approximating the solutions of asymptotically regular problems by the solutions of regular problems while the gradi-ent of the solution close to infinityIn Chapter 4,we consider the nonlinear elliptic equation with Lp(·)log L-growth div A(x,Du)=div H(x,F)with A(x,?)=D?(a(x)|?|p(x)log(e+|?|))for any ??0,A(x,0)=0,and |H(x,?)|(?)|?|p(x)-1 log(e+|?|).Under the assumptions that the co-efficient A(x,?)satisfies small BMO(bounded mean oscillation)condition,p(x)sat-isfies strong log-Holder continuity and the underlying domain is Reifenberg flat,we use large-M-inequality principle,the geometric approach and iteration argument to get global Calderon-Zygmund type estimates in the framework of Lorentz spaces.More-over,we obtain a global estimate in the setting of Lebesgue spaces to the corresponding double obstacle problemsIn Chapter 5,we employ large-M-inequality principle and the geometric approach to obtain a global Lorentz estimate for variable power of the gradients to weak solutions of nonlinear elliptic equations with partially regular nonlinearities div a(Du,x)=div F We mainly assume that the nonlinearities a(?,x)are partially BMO,which means that the coefficients are merely measurable in one spatial variable and have sufficiently s-mall BMO seminorm in the other variables,and the boundary of domain belongs to Reifenberg flatness.Also,we attain a global Orlicz estimate for the corresponding ob-stacle problem based on making use of the Hardy-Littlewood maximal functions and an equivalent representation of Orlicz normIn Chapter 6,we study the stationary steady Stokes systems D?(A??D?u)+?P=D?f? with ?,?=1,2,…,d.It is also assumed that the leading coefficients A?? are par-tially BMO,the boundary of domain belongs to Reifenberg flatness.Here,we consider two kinds of boundary problems,Dirichlet problem and conormal derivative problem.We prove global variable Lorentz estimates for weak solution pair(u,P)for both of the two boundary problems via large-M-inequality principle and the geometric approach,where the variable exponents satisfy the log-Holder continuity.In Chapter 7,we summary up our main achievements in the dissertation and present some further problems.