Second-order elliptic and parabolic equations with $B(\mathbb R^{2}, VMO)$ coefficients

TL;DR

This paper proves solvability of certain second-order elliptic and parabolic equations with coefficients that are measurable in some variables and VMO in others, extending known results to near $p=2$ in Sobolev spaces.

Contribution

It establishes $W^{1,2}_p$ and $W^{2}_p$ solvability for equations with mixed regularity coefficients, including near $p=2$, which was previously unaddressed.

Findings

Solvability in Sobolev spaces for $p>2$ close to 2.

Extension of solvability results to elliptic equations with coefficients measurable in two variables.

Results also hold at $p=2$ under slightly different assumptions.

Abstract

The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergence form second order parabolic equations for $p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $W^{2}_p$-solvability for the same $p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $p=2$.