Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains

TL;DR

This paper proves solvability of higher order elliptic and parabolic systems with coefficients that are only partially regular, in both regular and irregular domains, extending the theory to more general coefficient behaviors.

Contribution

It introduces new solvability results for complex-valued systems with variably partially BMO coefficients in diverse domain types, including irregular ones.

Findings

Solvability established in Sobolev spaces for complex-valued systems.

Results apply to domains like the whole space, half space, and Reifenberg flat domains.

Coefficients are measurable in one direction and have small mean oscillations in others.

Abstract

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.