Global gradient estimates in weighted Lebesgue spaces for parabolic operators

TL;DR

This paper establishes gradient regularity estimates for linear parabolic equations with measurable coefficients in weighted Lebesgue spaces, under minimal boundary smoothness assumptions, extending classical Calderón–Zygmund theory.

Contribution

It develops a Calderón–Zygmund type theory for parabolic operators in weighted Lebesgue spaces with non-smooth domains and coefficients, providing new regularity results.

Findings

Gradient estimates in weighted Lebesgue spaces

Regularity results in parabolic Morrey scales

Extension of Calderón–Zygmund theory to non-smooth settings

Abstract

We deal with the regularity problem for linear, second order parabolic equations and systems in divergence form with measurable data over non-smooth domains, related to variational problems arising in the modeling of composite materials and in the mechanics of membranes and films of simple non-homogeneous materials which form a linear laminated medium. Assuming partial BMO smallness of the coefficients and Reifenberg flatness of the boundary of the underlying domain, we develop a Calder\'{o}n--Zygmund type theory for such parabolic operators in the settings of the weighted Lebesgue spaces. As consequence of the main result, we get regularity in parabolic Morrey scales for the spatial gradient of the weak solutions to the problems considered.