Regularity near the fixed boundary for transmission systems

Abstract

Given $\Omega \subset {\mathbb{R}}^n$ with $n\ge 2$, $D\subset \Omega$ open, and $u:\Omega \to {\mathbb{R}}^m$, we study elliptic systems of the type $$\mathrm{div}((A+(B-A){\chi }_D)\nabla u)=0\text{in Ω∩B1,}$$ 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 $B_{1/2}\cap \Omega$ up to the boundary of $\Omega$. This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022.

In honor of Nina Nikolaevna Uraltseva for her 90th birthday