Weighted [formula omitted] estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable

Abstract

This paper studies Sobolev regularity of weak solution of degenerate elliptic equations in divergence form div[A(X)∇u]=div[F(X)], where X=(x,y)∈Rn×R. The coefficient matrix A(X) is a symmetric, measurable (n+1)×(n+1) matrix, and it could be degenerate or singular in the one dimensional y-variable as a weight function in the A2 Muckenhoupt class. Our results give weighted Sobolev regularity estimates of Calderón–Zygmund type for weak solutions of this class of degenerate/singular equations. As an application of these estimates, we establish global fractional 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.