法贝斯 - 凯尼格 - 塞拉波尼奇异 - 退化型抛物方程的加权$W^{1,p}$估计
Weighted $W^{1,p}$-estimates for Parabolic Equations of Fabes-Kenig-Seraponi singular-degenerate type
AI总结：
研究一类具奇异和退化行为的二阶抛物方程狄利克雷边值问题，在系数部分加权平均振荡小性假设下，用冻结系数技术等建立弱解相关性质，发展加权索伯列夫空间框架，证明解在加权范数下局部接近冻结系数对应物。
AI中文摘要：
我们研究一类散度形式的二阶抛物方程的狄利克雷边值问题，其系数矩阵呈现由穆肯霍普特权重类表征的奇异和退化行为。该框架是法贝斯、凯尼格和塞拉波尼开创的奇异 - 退化椭圆方程的抛物类似物。在系数的部分加权平均振荡的小性假设下，我们在适当定义的加权索伯列夫空间中建立了弱解的存在性、唯一性以及局部内部和边界正则性估计。证明依赖于冻结系数技术以及卡法雷利和佩拉尔引入的水平集方法。此外，我们发展了必要的加权索伯列夫空间框架和相关加权不等式。最后，利用紧致性论证表明这些方程的解在加权索伯列夫范数下局部接近其冻结系数对应物。
英文摘要：
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.
