Department of Mathematics - Seminar on PDE - Non-smoothness of solutions to the $\sigma_k$-Loewner– Nirenberg problem in domains with nontrivial topology

I will talk about a joint work with Yanyan Li and Luc Nguyen about the regularity of viscosity solutions to the $\sigma_k$-Loewner–Nirenberg problem in bounded domains, $k\ge 2$. When the domains' boundaries have more than one connected component, we show that viscosity solutions are not $C^1$. However, the Lipschitz regularity was known. On the other hand, we establish smoothness of the viscosity solutions near the boundaries.