*Submitted Paper*

**Inserted:** 16 mar 2021

**Last Updated:** 16 mar 2021

**Year:** 2020

**Abstract:**

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
$$
{\rm div} ((A + (B-A)\chi_{D} )\nabla u) = 0 \quad \text{in }B_{1,}
$$
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.

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