Weighted Calder\'{o}n-Zygmund estimates for weak solutions of quasi-linear degenerate elliptic equations

TL;DR

This paper establishes weighted Sobolev regularity estimates for weak solutions of degenerate and singular quasi-linear elliptic equations, extending known results to more general coefficient behaviors and solution-dependent cases.

Contribution

It provides new weighted Sobolev regularity estimates for degenerate and singular quasi-linear elliptic equations, including cases where coefficients depend on the solution.

Findings

Established global and interior weighted W^{1,p} estimates

Extended regularity results to solution-dependent coefficients

Provided new insights even for the case when weight μ=1

Abstract

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form $\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients $\mathbf{A}$ could be be singular, and degenerate or both in $x$ in the sense that they behave like some weight function $\mu$, which is in the $A_2$ class of Muckenhoupt weights. Global and interior weighted $W^{1,p}(\Omega, \omega)$-regularity estimates are established for weak solutions of these equations with some other weight function $\omega$. The results obtained are even new for the case $\mu =1$ because of the dependence on the solution $u$ of $\mathbf{A}$. In case of linear equations, our $W^{1,p}$-regularity estimates can be viewed as the Sobolev's counterpart of the H\"{o}lder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.