Solutions to the $\sigma_k$-Loewner-Nirenberg problem on annuli are locally Lipschitz and not differentiable

TL;DR

This paper investigates the regularity of solutions to the $\sigma_k$-Loewner-Nirenberg problem on annuli, showing they are locally Lipschitz but not differentiable, with specific Hölder continuity properties and a jump in radial derivative.

Contribution

It establishes the optimal regularity and differentiability properties of solutions to the $\sigma_k$-Loewner-Nirenberg problem on annuli, including a precise description of their Hölder continuity.

Findings

Solutions are $C^{1,1/k}_{loc}$ in certain annular regions.

Solutions exhibit a jump in radial derivative at the geometric mean radius.

Solutions are not $C^{1,eta}_{loc}$ for any $eta > 1/k$.

Abstract

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $\sigma_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,\gamma}_{\rm loc}$ for any $\gamma > \frac{1}{k}$. Optimal regularity for solutions to the $\sigma_k$-Yamabe problem on annuli with finite constant boundary values is also established.