Regularity In Generalized Morrey Spaces For Degenerate Elliptical Equations With Discontinuous Coefficients

The paper is constituted by three questions. I firstly study the local regularity prob-lems in controllable condition in the generalized Morrey space Lφp,λfor the gradient of weak solutions of the A-harmonic equation with VMO coefficients. In the process of dealing with the perturbation results, we use the property of the VMO function. In or-der to use the Campanato's turbulent argument, we need a series of iterate inequalities that are complicated. These inequalities are different from those which are used before. In order to construct the inequalities we use the concept called almost-increasing. Sec-ondly we discuss the quasi-linear degenerate elliptic equation of divergence type and we establish the regularity for its weak solution in the generalized Morry space when we allow VMO dependence on the variable x and continuous dependence on the variable u. The method we use is very different from that used in the first part. Otherwise we also need to use the continuity assumption with respect to u and some related conclusions that come from Ladyzhenskaya and Ural'tseva. Thirdly we study the regularity prob-lems in the LX2,λspace for the generalized gradient of weak solutions of the subelliptic equations with discontinuous coefficients in the Carnot-Caratheodary space.We divided my dissertation into four parts.In the first chapter, we briefly introduce the development of the regularity of elliptic equations and the background of selecting this question.In the second chapter, we will present regularity in the generalized Morrey space for the gradient of weak solutions of the A-harmonic equation with VMO coefficients.In the third chapter, we will present regularity in the generalized Morrey space for the weak solutions of quasi-linear degenerate equations of divergence type. We allow VMO dependence on the variable x and continuous dependence on the variable u.In the forth chapter, we will present regularity in the LX2,λspaces for the generalized gradient of weak solutions of subelliptic equation with VMO coefficients.