Weighted $W^{1,p}$- estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable

TL;DR

This paper establishes weighted Sobolev regularity estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable, extending regularity theory to a class of singular, degenerate PDEs with applications to fractional elliptic equations.

Contribution

It provides new weighted Sobolev regularity estimates for degenerate elliptic equations with coefficients degenerate in one variable, including applications to spectral fractional elliptic equations.

Findings

Weighted Sobolev estimates of Calderón-Zygmund type for degenerate equations

Global Sobolev regularity for solutions of fractional elliptic equations

Extension of regularity theory to equations with coefficients in Muckenhoupt class

Abstract

This paper studies the Sobolev regularity of weak solution of degenerate elliptic equations in divergence form $\text{div}[\mathbf{A}(X) \nabla u] = \text{div}[\mathbf{F}(X)]$, where $X = (x,y) \in \mathbb{R}^{n} \times \mathbb{R}$ . The coefficient matrix $\mathbf{A}(X)$ is a symmetric, measurable $(n+1) \times (n+1)$ matrix, and it could be degenerate or singular in the one dimensional $y$-variable as a weight function in the Muckenhoupt class $A_2$ of weights. Our results give weighted Sobolev regularity estimates of Calder\'{o}n-Zygmund type for weak solutions of this class of singular, degenerate equations. As an application of these estimates, we establish global Sobolev regularity estimates for solutions of the spectral fractional elliptic equation with measurable coefficients. This result can be considered as the Sobolev counterpart of the recently established Schauder regularity theory of fractional elliptic equations.