Regularity of viscosity solutions of the σk$\sigma _k$ ‐Loewner–Nirenberg problem

We study the regularity of the viscosity solution u$u$ of the σk$\sigma _k$ ‐Loewner–Nirenberg problem on a bounded smooth domain Ω⊂Rn$\Omega \subset \mathbb {R}^n$ for k⩾2$k \geqslant 2$ . It was known that u$u$ is locally Lipschitz in Ω$\Omega$ . We prove that, with d$d$ being the distance function to ∂Ω$\partial \Omega$ and δ>0$\delta > 0$ sufficiently small, u$u$ is smooth in {0