Resumen de Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

Giuseppe Di Fazio, Truyen Nguyen

• We study regularity for solutions of quasilinear elliptic equations of the form divA(x,u,∇u)=divF in bounded domains in Rn. The vector field A is assumed to be continuous in u, and its growth in ∇u is like that of the p-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions u to the equation under a small BMO condition in x for A. As a consequence, we obtain that ∇u is in the classical Morrey space Mq,λ or weighted space Lqw whenever |F|1/(p−1) is respectively in Mq,λ or Lqw, where q is any number greater than p and w is any weight in the Muckenhoupt class Aq/p. In addition, our two-weight estimate allows the possibility to acquire the regularity for ∇u in a weighted Morrey space that is different from the functional space that the data |F|1/(p−1) belongs to.