Regularity near the fixed boundary for transmission systems

Given $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 2$, $D\subset \mathrm{\Omega }$ open, and $u:\mathrm{\Omega }\to {\mathbb{R}}^m$, we study elliptic systems of the type $$\mathrm{d}\mathrm{i}\mathrm{v}((A+(B-A){\chi }_D)\mathrm{\nabla }u)=0\text{in }\mathrm{\Omega }\cap B_1\text{,}$$ 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 $B_{1\mathrm{/}2}\cap \mathrm{\Omega }$ up to the boundary of $\mathrm{\Omega }$. This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022.