A weighted $L_p$-theory for parabolic PDEs with BMO coefficients on $C^1$-domains

TL;DR

This paper develops a weighted $L_p$-theory for second-order parabolic PDEs on $C^1$ domains, allowing for BMO coefficients with relaxed conditions and unbounded lower order coefficients near the boundary.

Contribution

It introduces a slightly relaxed BMO condition for coefficients and handles unbounded lower order coefficients near the boundary in the $L_p$-theory.

Findings

Established weighted $L_p$-estimates for parabolic PDEs with BMO coefficients.

Extended the theory to include unbounded lower order coefficients near the boundary.

Relaxed BMO conditions compared to existing literature.

Abstract

In this paper we present a weighted $L_p$-theory of second-order parabolic partial differential equations defined on $C^1$ domains. The leading coefficients are assumed to be measurable in time variable and have VMO (vanishing mean oscillation) or small BMO (bounded mean oscillation) with respect to space variables, and lower order coefficients are allowed to be unbounded and to blow up near the boundary. Our BMO condition is slightly relaxed than the others in the literature.