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On the first eigenvalue of Liouville-type problems

Daniele Bartolucci, Paolo Cosentino, Aleks Jevnikar, Chang-Shou Lin

TL;DR

The paper characterizes when the first eigenvalue ν̂1 of the linearized Liouville problem vanishes, revealing a sharp mass threshold ∫_Ω e^{w} dx = 4π and a geometric portrait in terms of conformal maps to geodesic disks on the sphere S_{√2}. The authors develop a refined Alexandrov-Bol inequality that, together with rearrangement against the Liouville bubble U, yields a precise equality framework and pointwise information on the first eigenfunction. They extend the AB inequality from simply connected to multiply connected domains, proving that ν̂1 > 0 in the latter under standard boundary conditions, and show that ν̂1 = 0 corresponds to the surface (Ω, e^{w} dx^2) being conformally a hemisphere, with explicit eigenfunction forms. The results bridge Liouville-type PDE analysis, conformal geometry, and sharp isoperimetric inequalities, with potential implications for nondegeneracy and uniqueness in Liouville-type problems.

Abstract

The aim of this note is to study the spectrum of a linearized Liouville-type problem, characterizing the case in which the first eigenvalue is zero. Interestingly enough, we obtain also point-wise information on the associated first eigenfunction. To this end, we refine the Alexandrov-Bol inequality suitable for our problem and characterize its equality case.

On the first eigenvalue of Liouville-type problems

TL;DR

The paper characterizes when the first eigenvalue ν̂1 of the linearized Liouville problem vanishes, revealing a sharp mass threshold ∫_Ω e^{w} dx = 4π and a geometric portrait in terms of conformal maps to geodesic disks on the sphere S_{√2}. The authors develop a refined Alexandrov-Bol inequality that, together with rearrangement against the Liouville bubble U, yields a precise equality framework and pointwise information on the first eigenfunction. They extend the AB inequality from simply connected to multiply connected domains, proving that ν̂1 > 0 in the latter under standard boundary conditions, and show that ν̂1 = 0 corresponds to the surface (Ω, e^{w} dx^2) being conformally a hemisphere, with explicit eigenfunction forms. The results bridge Liouville-type PDE analysis, conformal geometry, and sharp isoperimetric inequalities, with potential implications for nondegeneracy and uniqueness in Liouville-type problems.

Abstract

The aim of this note is to study the spectrum of a linearized Liouville-type problem, characterizing the case in which the first eigenvalue is zero. Interestingly enough, we obtain also point-wise information on the associated first eigenfunction. To this end, we refine the Alexandrov-Bol inequality suitable for our problem and characterize its equality case.
Paper Structure (5 sections, 5 theorems, 99 equations)

This paper contains 5 sections, 5 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. The Alexandrov-Bol inequality on simply connected domains
  3. The Alexandrov-Bol inequality on multiply connected domains
  4. On the first eigenvalue
  5. Appendix

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^2$ be an open, bounded domain whose boundary is the union of finitely many rectifiable Jordan curves and $w\in C^2(\Omega )\cap C^{0}(\overline{\Omega })$ be a solution of (eq0). Let $\hat{\nu}_1$ be the first eigenvalue of lin1 and assume that $\int_\Omega e^w\,dx\leq $(j)$ Assume that $w$ satisfies, for $c\in\mathbb{R}$, then $\hat{\nu}_1=0$ happens if and only if

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • Remark 3.1
  • Lemma 3.2
  • ...and 1 more