Regularity near the fixed boundary for transmission systems

Given Ω⊂ℝ^n with n≥ 2, D⊂Ω open, and u:Ω→ℝ^m, we study elliptic systems of the type div( ( A + (B- A)χ_D)∇ u) = 0 in Ω∩ 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 ∩ D is transmitted to B_1/2∩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.