Solutions to the $σ_k$-Loewner-Nirenberg Problem on Annuli are Locally Lipschitz and Not Differentiable (Journal of Mathematical Study, 2021)
We show for $k ≥ 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| ≤ \sqrt{ab}\}$ and $\{\sqrt{ab} ≤ |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,γ}_{\rm loc}$ for any $γ > \frac{1}{k}$. Optimal regularity for solutions to the $σ_k$-Yamabe problem on annuli with finite constant boundary values is also established.
Keywords: $σ_k$-Loewner-Nirenberg problem; $σ_k$-Yamabe problem; viscosity solution; regularity; conformal invariance.