On $W^{2,\varepsilon}$-estimates for a class of singular-degenerate parabolic equations

TL;DR

This paper establishes weighted second-order derivative estimates for a class of singular-degenerate parabolic equations with measurable coefficients, using a stochastic-geometric approach and intrinsic weighted cylinders.

Contribution

It introduces a novel approach combining stochastic interpretation, intrinsic weighted cylinders, and perturbation techniques to obtain regularity results for singular-degenerate parabolic equations.

Findings

Weighted $W^{2, ext{ε}}$-estimates established

Applicable to equations with polynomial blow-up or vanishing coefficients

Results extend classical regularity theory to broader singular-degenerate cases

Abstract

We study a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in the $A_{1+\frac{1}{n}}$-Muckenhoupt class. Under a smallness assumption on a weighted mean oscillation of the weights, we establish weighted $W^{2,\varepsilon}$-estimates in the spirit of F.-H. Lin. Our results particularly holds for equations whose leading coefficients are of logistic-type singularities, as well as to those with polynomial blow-up or vanishing with sufficiently small exponents. A central component of our approach is the development of local quantitative lower estimates for solutions, which are interpreted as the mean sojourn time of sample paths, a stochastic-geometric perspective that generalizes the seminal work of L. C. Evans. We address the singular-degenerate nature of the operators by employing a class of intrinsic weighted parabolic cylinders, combined with a perturbation argument and parabolic Aleksandrov-Bakelman-Pucci (ABP) estimates. Furthermore, we conduct a rigorous analysis of weight regularization and truncation to ensure that the estimates are independent of the regularization and truncation parameters. The results extend classical regularity theory to a broad class of second-order parabolic equations and provide a functional analytic foundation for further study of fully nonlinear parabolic equations with singular-degenerate structure