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.
Categories: math.AP