On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

TL;DR

This paper establishes the solvability of higher order parabolic and elliptic systems with BMO coefficients in various domains, using new techniques to handle mean oscillation estimates, advancing the theory of PDEs with rough coefficients.

Contribution

It introduces novel methods for mean oscillation estimates in systems with BMO coefficients, extending solvability results to more general settings and boundary conditions.

Findings

Proves solvability in Sobolev spaces for systems with BMO coefficients.

Develops new techniques for mean oscillation estimates on half spaces.

Extends results to bounded domains and different boundary conditions.

Abstract

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.