TY - JOUR
T1 - Solutions to the $\sigma_k$-Loewner-Nirenberg Problem on Annuli are Locally Lipschitz and Not Differentiable
AU - Li , Yanyan
AU - Nguyen , Luc
JO - Journal of Mathematical Study
VL - 2
SP - 123
EP - 141
PY - 2021
DA - 2021/02
SN - 54
DO - http://doi.org/10.4208/jms.v54n2.21.01
UR - https://global-sci.org/intro/article_detail/jms/18612.html
KW - $\sigma_k$-Loewner-Nirenberg problem, $\sigma_k$-Yamabe problem, viscosity solution, regularity, conformal invariance.
AB - <p style="text-align: justify;">We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $\sigma_k$-Loewner-Nirenberg problem on a given annulus $\{a &lt; |x| &lt; b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a &lt; |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| &lt; b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,\gamma}_{\rm loc}$ for any $\gamma &gt; \frac{1}{k}$. Optimal regularity for solutions to the $\sigma_k$-Yamabe problem on annuli with finite constant boundary values is also established.</p>