Solutions to the ${\sigma }_k$-Loewner-Nirenberg problem on annuli are locally Lipschitz and not differentiable

Abstract

We show for $k\ge 2$ that the locally Lipschitz viscosity solution to the ${\sigma }_k$-Loewner-Nirenberg problem on a given annulus $\{a<|x|<b\}$ is $C_{\mathrm{loc}}^{1,\frac{1}{k}}$ in each of $\{a<|x|\le \sqrt{ab}\}$ and $\{\sqrt{ab}\le |x|<b\}$ and has a jump in radial derivative across $|x|=\sqrt{ab}$. Furthermore, the solution is not $C_{\mathrm{loc}}^{1,\gamma }$ 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.

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