Regularity In L^p,λ Space For Quasi-linear Equations With Discontinuous Coefficients

In my dissertation, I firstly study regularity problems in control-lable condition in Morrey Space for the gradient of weak solutions of degenerate elliptic equations,and in the further condition of improving the integrability of the data, we get the regularity for the weak solution in the Holder Space, Secondly we discuss the quasi-linear parabolic equations of divergence type with discontinu-ous coefficients,and we establish the regularity for its weak solution in the Morrey Space.Compared with the traditional methods,in the process of dealing with the perturbation results,we find that we need the Reverse Holder inequality.In the first chapter,we briefly introduce the development of the regularity of elliptic equations and parabolic equations,and the background of selecting this question. In the second chapter,we will present local regularity in Morrey Space for the gra-dient of weak solutions of degenerate elliptic equations,which the coefficient matrix satisfies uniformly elliptic conditions in independent variables.And then we get the continuity results in Holder Space for the gradient of its weak solutions.In the third chapter,we will present the regularity in Morrey Space for the weak solutions of a class of degenerate elliptic equations when the coefficient matrices satisfy certain VMO conditions in x uniformly with respect to u and the lower order terms satisfy a natural growth conditions.Interior Holder continuity of weak solutions is also derived with the improvement of the given data regularities.