Parabolic Systems with measurable coefficients in weighted Sobolev spaces

TL;DR

This paper develops a weighted $L_p$-theory for parabolic systems in half spaces, accommodating measurable coefficients in time and BMO conditions in space, even with boundary blow-up of lower order terms.

Contribution

It introduces a novel weighted Sobolev space framework for parabolic systems with minimal regularity assumptions on coefficients.

Findings

Established weighted $L_p$ estimates for solutions.

Extended theory to coefficients with small BMO in space.

Handled boundary blow-up of lower order coefficients.

Abstract

In this paper we present a weighted $L_p$-theory of parabolic systems on a half space. The leading coefficients are assumed to be only measurable in $t$ and have small bounded mean oscillations (BMO) with respect to $x$, and the lower order coefficients are allowed to blow up near the boundary.