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A Harnack-type inequality for a perturbed singular Liouville Equation

Daniele Bartolucci, Paolo Cosentino, Lina Wu

TL;DR

The paper studies a perturbed singular Liouville equation $-\Delta v_n = (\epsilon_n^2+|x|^2)^{\alpha_n} V_n(x) e^{v_n}$ in a bounded domain, motivated by Onsager vortex statistics. It proves a Harnack-type inequality of the form $\sup_K v_n + C_1\inf_\Omega v_n \le C_2$ for $\alpha_\infty\in(-1,1)$ and $\epsilon_n\to0^+$, with $C_1>\max\{1,(1+\alpha_\infty)/(1-\alpha_\infty)\}$, based on a Minimal Mass Lemma and a detailed blow-up analysis across three regimes. The work shows that blow-up implies a minimal mass at the origin and rules out concentration beyond certain scales, yielding a robust control on the solutions. It also demonstrates the inequality is almost sharp in certain regimes, illustrating nuanced blow-up/concentration behaviors and outlining open problems about the exact sharp constant. An appendix collects known Liouville-type mass results in the perturbed setting, which support the technical arguments.

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, \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}

A Harnack-type inequality for a perturbed singular Liouville Equation

TL;DR

The paper studies a perturbed singular Liouville equation in a bounded domain, motivated by Onsager vortex statistics. It proves a Harnack-type inequality of the form for and , with , based on a Minimal Mass Lemma and a detailed blow-up analysis across three regimes. The work shows that blow-up implies a minimal mass at the origin and rules out concentration beyond certain scales, yielding a robust control on the solutions. It also demonstrates the inequality is almost sharp in certain regimes, illustrating nuanced blow-up/concentration behaviors and outlining open problems about the exact sharp constant. An appendix collects known Liouville-type mass results in the perturbed setting, which support the technical arguments.

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, \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 , , is a bounded domain in containing the origin and 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}
Paper Structure (5 sections, 3 theorems, 144 equations)

This paper contains 5 sections, 3 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. A Minimal Mass Lemma
  3. A "Sup+ C Inf" estimate
  4. The inequality \ref{['sup+inf.intro']} is almost sharp.
  5. Appendix: Known results

Key Result

Theorem 1.1

Let $\Omega$ be an open bounded domain in $\mathbb{R}^2$ which contains the origin, $\{0\}\subset \Omega$. Assume that $v_n$ is a sequence of solutions of intro:eq satisfying intro:alphan and intro: Vn with Then, for any and for any compact set $K\subset\Omega$, there exists a constant $C_2>0$, which depends only by $a,b,dist(K,\partial\Omega ), \alpha_\infty$ and by the uniform modulus of conti

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1: Minimal Mass Lemma
  • proof
  • proof
  • Lemma 5.1