Volume 54, Issue 2
Solutions to the $\sigma_k$-Loewner-Nirenberg Problem on Annuli are Locally Lipschitz and Not Differentiable

J. Math. Study, 54 (2021), pp. 123-141.

Published online: 2021-02

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• 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.

• Keywords

$\sigma_k$-Loewner-Nirenberg problem, $\sigma_k$-Yamabe problem, viscosity solution, regularity, conformal invariance.

35J60, 35J75, 35B65, 35D40, 53C21

yyli@math.rutgers.edu (Yanyan Li)

luc.nguyen@maths.ox.ac.uk (Luc Nguyen)

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@Article{JMS-54-123, author = {Yanyan and Li and yyli@math.rutgers.edu and 22848 and and Yanyan Li and Luc and Nguyen and luc.nguyen@maths.ox.ac.uk and 10322 and Mathematical Institute and St Edmund Hall, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK and Luc Nguyen}, title = {Solutions to the $\sigma_k$-Loewner-Nirenberg Problem on Annuli are Locally Lipschitz and Not Differentiable}, journal = {Journal of Mathematical Study}, year = {2021}, volume = {54}, number = {2}, pages = {123--141}, 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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v54n2.21.01}, url = {http://global-sci.org/intro/article_detail/jms/18612.html} }
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 -

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.

Yanyan Li & Luc Nguyen. (2021). Solutions to the $\sigma_k$-Loewner-Nirenberg Problem on Annuli are Locally Lipschitz and Not Differentiable. Journal of Mathematical Study. 54 (2). 123-141. doi:10.4208/jms.v54n2.21.01
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