Regularities Of Generalized Solutions To Several Elliptic And Parabolic Equations With Discontinuous Coefficients

This doctoral dissertation mainly discusses six problems concerning the regulari-ties of generalized solutions to partial differential equations,which contains:First,the regularities in Lorentz space and Orlicz space for the strong solution to nondivergence linear elliptic equations.Second,the regularity in LP(x,t)space for the strong solution to nondivergence linear parabolic equations.Third,the regularities in Lorentz space and Lorentz-Morrey space for the viscosity solution to fully nonlinear elliptic equa-tions.Fourth,the regularity in Lorentz space for the strong solution to fully nonlinear parabolic equations.Fifth,the regularity in Lorentz space for the strong solution to asymptotically regular fully nonlinear parabolic equations.Sixth,the Holder continu-ity of weak solutions to divergence linear parabolic equations.The specific contents include:In Chapters 1 and 2,we respectively introduce the research background,research status,and the basic concepts and properties of several spaces which will be used in this dissertation.In Chapter 3,we prove the interior weighted Lorentz estimates and the interior Or-licz estimates of strong solutions to nondivergence linear elliptic equations aij(x)Diju=f(x)under the assumption that the coefficient aij(x)satisfies the uniformly elliptic con-dition and the small partially BMO condition.The main idea is basecd on the classical perturbation method,the generalized Vitali covering lemma.the Lorentz boundedness and the Orlicz boundedness of Hardy-Littlewood maximal operators,and the equivalent level set measure representations of Lorentz norm and Orlicz norm.In Chapter 4,we employ the large-M-principle to prove the interior L(x,t)regular-ity of strong solutions to nondivergence linear parabolic equations ut-aij(x,t)Diju=f(x,t).Here,we assume that the coefficient aij(x,t)satisfies the uniformly parabol-ic condition and the small partially BMO condition,and the variable exponent p(x,t)satisfies the log-Holder continuity condition.In addition,we also demonstrate that the result is also true for nondivergence linear elliptic equations aij(r)Diju=f(x).In Chapter 5,we study the viscosity solution of the Dirichlet problem for fully nonlinear elliptic equation F(Du,x)= f(x)over a bounded C1.1 domain under the assumption that F(M,x)is convex on M and satisfies the uniformly elliptic condition and the(?,R)-vanishing condition.Based on the interior and boundary W2,p(1<p<?)estimates respectively from Caffarelli and Winter,we establish the global weighted Lorentz regularity via the viscosity method and the finite covering theorem,and further prove the Lorentz-Morrey regularity by taking an appropriate weight function.In Chapter 6,we study the strong solution of the Cauchy-Dirichlet problem for fully nonlinear parabolic equation ut+ F(D2u,x,t)=f(x,t)over a bounded C1,1 domain under the assumption that F(M,x,t)is convex and positive homogeneous of degree one in M and satisfies the uniformly parabolic condition and the(?,R)-vanishing condition.We prove the global Lorentz regularity via the large-M-principle,and this conclusion is also true for the elliptic case.In Chapter 7,we mainly discuss the strong solution of the Cauchy-Dirichlet prob-lem for the asymptotically regular fully nonlinear parabolic equation ut(x,t)+F(D2u,x,t)=f(x,t)over a bounded C1,1 domain under the assumption that the nonlinearity F is asymptotically regular to G which satisfies the assumptions in chapter 6.We prove the global Lorentz regularity by using an appropriate Poisson formula to transform the above asymptotically regular equation to a regular one.Finally,it is demonstrated that this result is also true for the elliptic case.In Chapter 8.we investigate the local Holder regularity of the weak solution to divergence linear parabolic equations under the assumption that the coefficient is inde-pendent of t and satisfies the VMO condition.Here,we first use the natural growth properties of Green's functions and the hole-filling technique to prove the local Morrey regularity of the weak solution,and then obtain our desired result via Morrey lemma.