Regularity theory for parabolic equations with singular degenerate coefficients

Hongjie Dong, Tuoc Phan

Published 2018-02-26Version 1

In this paper, we study regularity and solvability in weighted Sobolev spaces for a class of parabolic equations in divergence form with coefficients singular or degenerate in one space variable. Under certain conditions, reverse H\"{o}lder's inequalities are established. Lipschitz estimates for weak solutions are proved for a class of homogeneous equations whose coefficients depend only on one space variable, but they can be singular and degenerate. These estimates are then used to establish interior, boundary, and global estimates of the Calder\'{o}n-Zygmund type for weak solutions assuming that the coefficients are partially VMO (vanishing mean oscillations) with respect to the considered weights. The solvability in weighted Sobolev spaces for this class of equations is also achieved. Our results are new even for elliptic equations, and extend some recent results for uniformly elliptic and parabolic equations.