We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution
u:B_1\subset \mathbb{R}^n \to \mathbb{R}^m
to the elliptic system \begin{equation*} \mbox{div} ((A + (B-A)\chi_D )\nabla u) = 0 \quad \text{in }B_1, \end{equation*} where A and B are Dini continuous, uniformly elliptic matrices, we prove that if \nabla u \in L^{\infty} (D) then u is Lipschitz in B_{1/2}. A similar result is also derived for the parabolic counterpart of this problem.
