Some Regularity Problems Of Quasi-linear Elliptic Equations (or Systems) With Discontinuous Coefficients

In my dissertation, I mainly study some regularity problems in Morrey Space for the gradient of weak solutions of nonlinear elliptic equations with discontinuous coefficients. We will present regularity or partial regularity in Morrey spaces for the gradient of weak solutions for the system of quasi-linear elliptic equations and degenerate elliptic equations, which the coefficient matrix satisfies uniformly VMO conditions in independent variables. The paper is made up of four parts as follows :In the first chapter, we briefly introduce the development of the regularity of elliptic equation(systems) and the background of selecting this question .The second chapter, we shall extend the partial regularity in Morrey Space for minimizers of quasilinear functionals to the following quasilinear elliptic systems with uniformly VMO coefficients in independent variables.-D_α(A_ij^αβ(x, u)D_βu^j) = f_i(x). (α,β=1,2,...,n; i,j = 1,2,..., N.) (1)we get the partial regularity in Morrey space for the gradient of its weak solutions. The third chapter, we shall consider the regularity in Morrey Space for quasi-linear elliptic equations ( or systems) as follows.-D_α(A_ij^αβ(x,u)D_βu^j) = b_i(x,u,Du), (α,β= 1,2,...,n; i,j = 1,2,..., N.) (2)which the coefficient satisfies uniformly VMO condition and the lower order item satisfies natural growth. We obtain the partial regularity result in Morrey spaces of the gradient for the weak solutions of systems (2).The forth chapter, we shall study the following quasi-linear degenerate elliptic equation.Making use of elliptic equation (3) compares with A-harmonic functions, and the methods of freezing coefficients, we present the partial regularity result of the weak solutions for quasi-linear degenerate elliptic equations with the coefficient matrix A(x,u) satisfying uniformly VMO conditions in independent variables.Remark:In the above three classes of elliptic equation(or systems) , the coefficient operator A_ij^αβ(x,u) and A(x,u) are different from usual operator,it need not continuous with respect to x, but satisfies uniformly VMO conditions in x for fixed...