Gradient estimates for divergence form parabolic systems

TL;DR

This paper establishes gradient estimates and regularity results for divergence form parabolic systems with piecewise smooth coefficients and interfaces, extending previous work and providing new weak type-$(1,1)$ estimates under weighted norms.

Contribution

It improves existing gradient regularity results for parabolic systems with complex interfaces and introduces new weak type-$(1,1)$ estimates for systems with certain coefficient conditions.

Findings

Gradient estimates and piecewise regularity are established.

A global weak type-$(1,1)$ estimate with Muckenhoupt weights is proved.

Optimal regularity of solutions to transmission problems with smooth interfaces is demonstrated.

Abstract

We consider divergence form, second-order strongly parabolic systems in a cylindrical domain with a finite number of subdomains under the assumption that the interfacial boundaries are $C^{1,\text{Dini}}$ and $C^{\gamma_{0}}$ in the spatial variables and the time variable, respectively. Gradient estimates and piecewise $C^{1/2,1}$-regularity are established when the leading coefficients and data are assumed to be of piecewise Dini mean oscillation or piecewise H\"{o}lder continuous. Our results improve the previous results in \cite{ll,fknn} to a large extent. We also prove a global weak type-$(1,1)$ estimate with respect to $A_{1}$ Muckenhoupt weights for the parabolic systems with leading coefficients which satisfy a stronger assumption. As a byproduct, we give a proof of optimal regularity of weak solutions to parabolic transmission problems with $C^{1,\mu}$ or $C^{1,\text{Dini}}$ interfaces. This gives an extension of a recent result in \cite{css} to parabolic systems.