A Harnack-Type Inequality for a Perturbed Singular Liouville Equation

IF 1 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2026-06-05 Epub Date: 2026-04-14 DOI:10.1002/mana.70149

Daniele Bartolucci, Paolo Cosentino, Lina Wu

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引用次数: 0

Abstract

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,

- Δ v_{n} = {(ε_{n}^{2} + {| x |}^{2})}^{α_{n}} V_{n} (x) e^{v_{n}} in Ω,

where

ε_{n} \to 0^{+}

α_{n} \to α_{\infty} \in (- 1, 1)

Ω

is a bounded domain in

R^{2}

containing the origin and

V_{n}

satisfies

0 < a \leq V_{n} \leq b < + \infty, V_{n} \in C^{0} (Ω), V_{n} \to V locally uniformly in Ω .

查看原文本刊更多论文

一类微扰奇异Liouville方程的harnack型不等式

根据具有点奇点的湍流欧拉流的Onsager统计力学描述，我们得到了以下扰动Liouville方程解序列的一个harnack型不等式：−Δ v n = ε n+ | x | 2α n V n (x) e在Ω中的vn，$$\begin{equation*} \nonumber -\Delta v_n={\left({\epsilon _n^2+|x|^2}\right)}^{\alpha _n}V_n(x)e^{\displaystyle v_n} \qquad \text{in } \,\,\, \Omega, \end{equation*}$$其中ε n→0 + $\epsilon _n\rightarrow 0^+$，α n→α∞∈(−1,1)$\alpha _n\rightarrow \alpha _\infty \in (-1,1)$，Ω $\Omega$是r2中包含原点和V n的有界域$\mathbb {R}^2$$V_n$满足0 ＆lt；a≤V n≤b ＆lt; +∞；V n∈c0 （Ω），vn→V在Ω中局部一致。 $$\begin{equation*} \nonumber 0<a\le V_n\le b<+\infty, \,\, V_n\in C^{0}(\Omega), \,\,V_n\rightarrow V \;\; \text{locally uniformly in }\,\,{\Omega }. \end{equation*}$$

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来源期刊

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index