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

摘要

This paper studies the Sobolev regularity of weak solution of degenerateelliptic 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 onedimensional $y$-variable as a weight function in the Muckenhoupt class $A_2$ ofweights. Our results give weighted Sobolev regularity estimates ofCalder\'{o}n-Zygmund type for weak solutions of this class of singular,degenerate equations. As an application of these estimates, we establish globalSobolev regularity estimates for solutions of the spectral fractional ellipticequation with measurable coefficients. This result can be considered as theSobolev counterpart of the recently established Schauder regularity theory offractional elliptic equations.