Regularity near the fixed boundary for transmission systems

Given Ω⊂ Rⁿ with n≥ 2, D⊂ Ω open, and u:Ω → Rᵐ, we study elliptic systems of the type div ( ( A + (B- A)χD)∇ u) = 0 inΩ∩ 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 B₁ ∩ D is transmitted to B1/2∩ Ω up to the boundary of Ω. This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022.