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

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 $x_d^2{a}_{ij}$, 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.

arXiv comment42 pages, minor revision, to appear in Trans. Amer. Math. Soc