Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

Abstract

This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form div[A(x,u,∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, degenerate or both in x in the sense that they behave like some weight function μ, which is in the A2 class of Muckenhoupt weights. Global and interior weighted W1,p(Ω,ω)-regularity estimates are established for weak solutions of these equations with some other weight function ω. The results obtained are even new for the case μ = 1 because of the dependence on the solution u of A. In case of linear equations, our W1,p-regularity estimates can be viewed as the Sobolev’s counterpart of the Holder’s regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.