Weighted $W^{1,p}$-estimates for Parabolic Equations of Fabes-Kenig-Seraponi singular-degenerate type

Abstract

We investigate Dirichlet boundary value problems for a class of second-order parabolic equations in divergence-form with coefficient matrices that exhibit singular and degenerate behaviors characterized by a Muckenhoupt weight class. This framework serves as the parabolic analogue to the singular-degenerate elliptic equations pioneered by Fabes, Kenig, and Seraponi. Under a smallness assumption on the partially weighted mean oscillation of the coefficients, we establish the existence, uniqueness, and local interior and boundary regularity estimates for weak solutions within appropriately defined weighted Sobolev spaces. The proofs rely on the freezing coefficient technique alongside the level-set method introduced by Caffarelli and Peral. Additionally, we develop the necessary weighted Sobolev space framework and related weighted inequalities. Finally, a compactness argument is utilized to demonstrate that solutions to these equations remain locally close, in the weighted Sobolev norm, to their frozen-coefficient counterparts.