The Calderon-Zygmund Type Estimates For Divergence Form Elliptic And Parabolic Equations With Weak Regular Datum

This dissertation is devoted to studying global Calderon-Zygmund type estimates of weak solutions for the Dirichlet problems of elliptic equations,the Cauchy-Dirichlet problems of parabolic equations and various relevant obstacle problems in divergence form with standard growth and non-standard growth,respectively,under weaker regular assumptions on the coefficients and the boundary of domains.More precisely,we are to part this dissertation in eight Chapters as follows:In Chapter 1 we are to introduce the background of our study,related concepts and notations.Further,we also describe various recent progressive methods concerning the Calderon-Zygmund theory of PDEs.Finally,we summarize our main research contents in this dissertation.Chapter 2 is to study global regularity in the weighted Lorentz-Sobolev spaces for the Dirichlet problems of elliptic equations in general form.Here,the leading coeffi-cients are assumed to be merely measurable in one variable and have small BMO semi-norms in the remaining variables(or call partially bounded mean oscillations,shortly partially BMO),while a geometric structure on the boundary of the underlying domain is locally bounded Reifenberg flat.As its direct consequence,we also present global Lorentz-Morrey estimate for the gradient of weak solution to such elliptic problems un-der the same main assumptions.Finally,we obtain local Holder continuity of its weak solutions with optimal exponent if there is a higher regularity for nonhomogeneous.In Chapter 3,by making use of an elementary estimate instead of the weighted Lp estimate with a special weight,we prove global Morrey estimate for the weak deriva-tives to the Dirichlet problems of linear elliptic equations with small partially BMO co-efficients in a half space.Here,the leading coefficients aij(x)are assumed to be merely measurable in one variable,and have small BMO in the remaining spatial variables.In Chapter 4,we derive global Lorentz estimate for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coeffi-cients over a bounded Reifenberg flat domain.Here,we mainly assume that the leading coefficients are measurable in one variable and have small BMO semi-norms in the other variables,while the variable exponents p(x)satisfy log-Holder continuity.We are in Chapter 5 to prove global Morrey estimate for the gradient of weak so-lutions to divergence nonlinear elliptic equations under controlled growth with minimal regular nonlinearities in Reifenberg flat domains.We mainly suppose that the non-linearity is merely measurable in one variable and has a small BMO seminorm in the remaining variables;while the lower order terms are subject to be controlled growth.This aim of this study is to weaken the key assumption on nonlinearity from small BMO to minimal partially BMO in order to attain the same global estimate as recent papers concerning quasilinear elliptic equations under controlled growth.Chapter 6 is to establish global regularity in weighted Lorentz spaces for the Cauchy-Dirichlet problems of nonlinear parabolic equations of p-Laplacian type in a bounded Reifenberg flat domain.Here,we mainly assume that the nonlinearity is merely mea-surable in the time variable t,and has a small BMO semi-norm in the spatial variables x.Our result extends from the regularity in Lebesgue spaces to that in refined weighted Lorentz spaces for p-Laplacian parabolic problems.In Chapter 7,we prove global Lorentz estimate for the variable power of the gradi-ent for weak solution of parabolic obstacle problems with p(t,x)-growth over a bounded quasiconvex domain.Here,we mainly suppose that the variable exponents p(t,x)satis-fy a strong type log-Holder continuity,the associated nonlinearities are merely measur-able in the time variable and have small BMO semi-norms in the spatial variables.As a consequence,we not only develop Lp theory for parabolic problems with non-standard growths in Lorentz spaces,but also extend the underlying domain from a Reifenberg flat domain to rough quasiconvex one.In the last Chapter,we summary up our main achievements in our dissertation,and present some further problems so as to investigate in the coming days.