Global Lorentz Estimates To Asymptotical Regular Nonlinear Equations Of Elliptic And Parabolic Type

In this thesis,we are mainly devoted to studying Calderon-Zygmund type estimates for two classes of problems of asymptotically regular partial differential equations.The first one is to establish a global regularity in Lorentz spaces for Hessian estimates of strong solution to asymptotically regular fully nonlinear elliptic equations.The second one is to get a global estimate in weighted Lorentz spaces for the zero Dirichlet problems of p(z)-Laplacian type parabolic equations with asymptotical regular coefficients defined in Reifenberg flat domains.More precisely,we organize them as follows.In Chapter 1,we summarize the background of the thesis,and recall the newest de-velopment of the literature.Additionally,we introduce the definitions of Lorentz spaces and weighted Lorentz spaces,and some related primary facts.We also recall the bounded-ness in Lorentz space for the Hardy-Littlewood maximal operators and the modified Vitali covering,etc.In Chapter 2,we are to consider the zero Dirichlet problem for the following fully nonlinear elliptic equations:F(x,D2u)= f(x),x∈Ω,(3)where the nonlinearity F(x,D2u)is asymptotically regular with respective to the term G(x,D2u)that satisfies uniformly ellipticity and small BMO regular in the independent variable x ∈ Ω with(?)Ω∈C1,1.If the nonhomogeneous term f(x)belongs to the Lorentz spaces,then we prove that the Hessian of strong solutions u also belongs to the same Lorentz spaces,which implies that where the positive constant C = C(n,λ,Λ,γ,q,Ω);while q = ∞ the constant C depends only on n,λ,Λ,γ,Ω.Here,our ingredient is mainly to make use of the Poisson formula to transform an asymptotical regular problem into the regular one with a small perturbation.Then,by flat-tening boundary due to a local diffeomorphism on the boundary neighborhood,Lemma 2.2 and an extension argument with even or odd reflection,we obtain a local Lorentz esti-mate on flat boundary.Finally,by combining the interior Lorentz and boundary estimates,it follows from a finite covering argument that we derive global Lorentz estimate for the original asymptotically regular problem.In Chapter 3,we consider the following zero initial-boundary problem of divergence parabolic equations with nonstandard growth defined in nonsmooth parabolic domains:where the nonlinearity a(x,t,Du)is assumed to be asymptotically δ-regular,p(x,t)satis-fies log-Holder continuity,and the underlying domain Ω is Reifenberg flatness.By ex-tending the Poisson formula from elliptic problem to parabolic setting,we obtain global gradient estimate of weak solution to parabolic problem in weighted Lorentz spaces:where the weight function ω Am*,and the positive constant C = C(n,δ,γ1,γ2,γ,q,θ,|ΩT|).In the case q = ∞,the constant C depends only on n,δ,γ1,γ2,γ,θ,|ΩT|.