$L_p$-estimates of the conormal derivative problem for parabolic equations with time measurable coefficients and $A_p$-weights

TL;DR

This paper establishes weighted mixed-norm estimates for divergence-type parabolic equations with time measurable coefficients on Reifenberg-flat domains, addressing boundary regularity issues using half-time derivative estimates.

Contribution

It provides new weighted mixed-norm boundary estimates for parabolic equations with minimal regularity assumptions on coefficients and domain geometry.

Findings

Weighted mixed-norm estimates are derived for divergence-type parabolic equations.

Boundary regularity is achieved using half-time derivative estimates.

Results apply to equations with coefficients measurable in time and small mean oscillations in space.

Abstract

This paper investigates weighted mixed-norm estimates for divergence-type parabolic equations on Reifenberg-flat domains with the conormal derivative boundary condition. The leading coefficients are assumed to be merely measurable in the time variable and to have small mean oscillations in the spatial variables. In deriving the boundary estimates, we overcome a regularity issue by employing half-time derivative estimates.