Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients

TL;DR

This paper establishes Sobolev estimates and regularity results for degenerate parabolic and elliptic equations with specific coefficient structures in divergence form, focusing on the upper half space.

Contribution

It introduces new Sobolev estimates for equations with coefficients degenerate in the normal direction and analyzes their well-posedness in weighted mixed-norm Sobolev spaces.

Findings

Proved well-posedness of solutions in weighted Sobolev spaces.

Derived regularity estimates for degenerate divergence form equations.

Extended classical results to degenerate coefficient scenarios.

Abstract

We study a class of degenerate parabolic and elliptic equations in divergence form in the upper half space $\{x_d>0\}$. The leading coefficients are of the form $x_d^2a_{ij}$, where $a_{ij}$ are bounded, uniformly elliptic, and measurable in $(t,x_d)$ except $a_{dd}$, which is measurable in $t$ or $x_d$. Additionally, they have small bounded mean oscillations in the other spatial variables. We obtain the well-posedness and regularity of solutions in weighted mixed-norm Sobolev spaces.