A Harnack-type inequality for a perturbed singular Liouville Equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912833433763840 |
|---|---|
| author | Bartolucci, Daniele Cosentino, Paolo Wu, Lina |
| author_facet | Bartolucci, Daniele Cosentino, Paolo Wu, Lina |
| contents | Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we obtain a Harnack-type inequality for sequences of solutions of the following perturbed Liouville equation, \begin{equation}\nonumber
-Δv_n=\left({ε_n^2+|x|^2}\right)^{α_n}V_n(x)e^{\displaystyle v_n} \qquad\text{in} \,\,\, Ω, \end{equation} where $ε_n\to0^+$, $α_n\toα_\infty\in(-1,1)$, $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin and $V_n$ satisfies, \begin{equation}\nonumber
0<a\leq V_n\leq b<+\infty, \,\, V_n\in C^{0}(Ω), \,\,V_n\to V \,\, \text{locally uniformly in}\,\,Ω. \end{equation} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13212 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Harnack-type inequality for a perturbed singular Liouville Equation Bartolucci, Daniele Cosentino, Paolo Wu, Lina Analysis of PDEs 35J61, 35J75, 35R05, 35B45 Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we obtain a Harnack-type inequality for sequences of solutions of the following perturbed Liouville equation, \begin{equation}\nonumber -Δv_n=\left({ε_n^2+|x|^2}\right)^{α_n}V_n(x)e^{\displaystyle v_n} \qquad\text{in} \,\,\, Ω, \end{equation} where $ε_n\to0^+$, $α_n\toα_\infty\in(-1,1)$, $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin and $V_n$ satisfies, \begin{equation}\nonumber 0<a\leq V_n\leq b<+\infty, \,\, V_n\in C^{0}(Ω), \,\,V_n\to V \,\, \text{locally uniformly in}\,\,Ω. \end{equation} |
| title | A Harnack-type inequality for a perturbed singular Liouville Equation |
| topic | Analysis of PDEs 35J61, 35J75, 35R05, 35B45 |
| url | https://arxiv.org/abs/2601.13212 |