Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2207.05587 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909509674336256 |
|---|---|
| author | Eremenko, Alexandre Gui, Changfeng Li, Qinfeng Xu, Lu |
| author_facet | Eremenko, Alexandre Gui, Changfeng Li, Qinfeng Xu, Lu |
| contents | We give a complete classification of solutions bounded from above of the Liouville equation $$-Δu=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty} u(z)/\log|z|:=k(u)<\infty\}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as $z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is symmetric with respect to $x$ and $y$ axes and $u_x<0,\; u_y<0$ in the first quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of ${\mathbf{R}}^2$ with the metric $e^{2u}δ$ is $π$, where $δ$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional.
In addition, we extend the concavity rigidity result on Liouville equation in higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_05587 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rigidity results on Liouville equation Eremenko, Alexandre Gui, Changfeng Li, Qinfeng Xu, Lu Analysis of PDEs 35G20, 35B07 We give a complete classification of solutions bounded from above of the Liouville equation $$-Δu=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty} u(z)/\log|z|:=k(u)<\infty\}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as $z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is symmetric with respect to $x$ and $y$ axes and $u_x<0,\; u_y<0$ in the first quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of ${\mathbf{R}}^2$ with the metric $e^{2u}δ$ is $π$, where $δ$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional. In addition, we extend the concavity rigidity result on Liouville equation in higher dimensions. |
| title | Rigidity results on Liouville equation |
| topic | Analysis of PDEs 35G20, 35B07 |
| url | https://arxiv.org/abs/2207.05587 |