Let $\Omega \subseteq \mathbb{R}^d$ be an open, bounded, connected domain, and suppose that $\partial \Omega $ is smooth. Let $g$ be a smooth Riemannian metric on $\overline{\Omega}$ (which is smooth up to the boundary.)
Let $h$ be a smooth Riemannian metric on $\mathbb{R}^d$, and let $f:(\overline \Omega,g) \to (\mathbb{R}^d,h)$ be a Lipschitz map, which is smooth on the interior $\Omega$, and satisfies $df \in \text{SO}(g,h)$ on $\Omega$. (i.e. $df$ is an orientation-preserving isometry at interior points.)
Is $f$ smooth up to the boundary? (i.e. as a map $\overline \Omega \to \mathbb{R}^d$)?
Comment:
It is in fact suffice to assume $f$ is Lipschitz and satisfies $df \in \text{SO}(g,h)$ a.e. - since known interior regularity results give smoothness of $f$ on the interior. The question is whether this smoothness extends to the boundary.