Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

• Giuseppe Di Fazio ^[1] ; Truyen Nguyen ^[2]
  1. [1] University of Catania

     University of Catania

     Catania, Italia

     
  2. [2] University of Akron

     University of Akron

     City of Akron, Estados Unidos

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 6, 2020, págs. 1627-1658
• Idioma: inglés
• DOI: 10.4171/rmi/1178
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• Resumen

  • 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.