A Harnack type inequality for singular Liouville type equations
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912832336953344 |
|---|---|
| author | Cosentino, Paolo |
| author_facet | Cosentino, Paolo |
| contents | We obtain a Harnack type inequality for solutions of the Liouville type equation, \begin{equation}\nonumber
-Δu=|x|^{2α}K(x)e^{\displaystyle u} \qquad\text{in} \,\,\, Ω, \end{equation} where $α\in(-1,0)$, $Ω$ is a bounded domain in $\mathbb{R}^2$ and $K$ satisfies, \begin{equation}\nonumber
0<a\leq K(x)\leq b<+\infty. \end{equation} This is a generalization to the singular case of a result by C.C. Chen and C.S. Lin [Comm. An. Geom. 1998], which considered the regular case $α=0$. Part of the argument of Chen-Lin can be adapted to the singular case by means of an isoperimetric inequality for surfaces with conical singularities. However, the case $α\in(-1,0)$ turns out to be more delicate, due to the lack of traslation invariance of the singular problem, which requires a different approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21499 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Harnack type inequality for singular Liouville type equations Cosentino, Paolo Analysis of PDEs We obtain a Harnack type inequality for solutions of the Liouville type equation, \begin{equation}\nonumber -Δu=|x|^{2α}K(x)e^{\displaystyle u} \qquad\text{in} \,\,\, Ω, \end{equation} where $α\in(-1,0)$, $Ω$ is a bounded domain in $\mathbb{R}^2$ and $K$ satisfies, \begin{equation}\nonumber 0<a\leq K(x)\leq b<+\infty. \end{equation} This is a generalization to the singular case of a result by C.C. Chen and C.S. Lin [Comm. An. Geom. 1998], which considered the regular case $α=0$. Part of the argument of Chen-Lin can be adapted to the singular case by means of an isoperimetric inequality for surfaces with conical singularities. However, the case $α\in(-1,0)$ turns out to be more delicate, due to the lack of traslation invariance of the singular problem, which requires a different approach. |
| title | A Harnack type inequality for singular Liouville type equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.21499 |