Regularity near the fixed boundary for transmission systems

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 H\"{o}lder 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.