Regularity near the fixed boundary for transmission systems

Abstract

Given $\Omega \subset {\mathbb{R}}^n$ with $n\ge 2$, $D\subset \Omega$ Borel, and $u:\Omega \to {\mathbb{R}}^m$, we study elliptic systems of the type

for some uniformly elliptic tensors $\mathcal{A}$ and $\mathcal{B}$ 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 by A. Figalli, S. Kim, and H. Shahgholian [Nonlinear Anal. 221 (2022), article no. 112911].