Gradient estimates for divergence form parabolic systems

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 Hölder 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.

41 pages. Submitted