Given \Omega\subset \mathbb{R}^n with n\geq 2, D\subset \Omega open, and u:\Omega \to \mathbb{R}^m, we study elliptic systems of the type {\rm div} \big( ( A + (B- A)\chi_D)\nabla u\big) = 0 \quad \text{in $\Omega\cap B_1$,} for some uniformly elliptic tensors A and B with Hölder continuous entries. We show that, given appropriate boundary data, the Lipschitz regularity of u inside B_1 \cap D is transmitted to B_{1/2}\cap \Omega up to the boundary of \Omega. This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022.
