[Paper Review] Mean field equations and premodular forms, I: at critical parameter $16\pi$

Zhijie Chen, Ting-Jung Kuo+1 more|arXiv (Cornell University)|Oct 6, 2016

Nonlinear Partial Differential Equations7 references3 citations

TL;DR

This paper proves the non-existence of solutions to the mean field equation Δu + e^u = 16πδ₀ on a flat torus E_τ when τ ∈ iℝ⁺, resolving a long-standing conjecture for the critical parameter 16π. Using a premodular form Z_{r,s}^{(2)}(τ), the authors establish the zero structure of this form and derive the existence of solutions for τ = 1/2 + ib with b > b* ∈ (√3/2, 6/5).

ABSTRACT

A conjecture about the mean field equation $\Delta u+e^{u}=8n\pi \delta_{0}$ on a flat torus $E_{ au}$ is the non-existence of solutions if $ au\in i\mathbb{R}^{+}$. For any $n\in \mathbb{N}_{\geq 2}$, this conjecture seems very challenging from the viewpoint of PDE theory. In order to solve this conjecture, a premodular form $Z_{r,s}^{(n)}( au)$ was introduced by Wang and the third author \cite{CLW2}, and is used to give necessary and sufficient conditions for the existence of solutions. In this paper, we succeed to prove the conjecture for $n=2$ (i.e. at critical parameter $16\pi$). In turn, we could apply this non-existence result to obtain the structure of zeros of the premodular form $Z_{r,s}^{(2)}% ( au)$. As a consequence, we obtain the existence of solutions for $n=2$ if $ au=\frac{1}{2}+ib$ and $b>b^{\ast}$ for some $b^{\ast}\in(\frac{\sqrt{3}}{2},\frac{6}{5})$.

Motivation & Objective

• To resolve the conjecture on the non-existence of solutions to the mean field equation Δu + e^u = 16πδ₀ on a flat torus E_τ when τ ∈ iℝ⁺.
• To establish a connection between the solvability of the mean field equation and the zero structure of the premodular form Z_{r,s}^{(2)}(τ).
• To use the non-existence result to derive conditions under which solutions exist for n=2, specifically for τ = 1/2 + ib with b > b*.
• To determine the precise range of b* ∈ (√3/2, 6/5) for which solutions exist when τ = 1/2 + ib.

Proposed method

• Introduces the premodular form Z_{r,s}^{(n)}(τ) as a tool to characterize existence conditions for solutions to the mean field equation.
• Applies the premodular form Z_{r,s}^{(2)}(τ) to analyze the zero set of the function, linking it to the solvability of the PDE.
• Employs complex-analytic techniques on the flat torus E_τ with τ ∈ iℝ⁺ to study the behavior of solutions to the mean field equation.
• Uses the non-existence result at critical parameter 16π to infer structural properties of Z_{r,s}^{(2)}(τ), particularly its zeros.
• Analyzes the parameter space τ = 1/2 + ib to determine the threshold b* such that solutions exist for b > b*.
• Combines PDE theory with modular forms to bridge geometric analysis and number theory in the context of mean field equations.

Experimental results

Research questions

• RQ1 Does the mean field equation Δu + e^u = 16πδ₀ on a flat torus E_τ admit solutions when τ ∈ iℝ⁺?
• RQ2 How does the premodular form Z_{r,s}^{(2)}(τ) relate to the existence or non-existence of solutions for n=2?
• RQ3 What is the precise threshold b* such that solutions exist for τ = 1/2 + ib with b > b*?
• RQ4 What is the zero structure of the premodular form Z_{r,s}^{(2)}(τ) derived from the non-existence result?
• RQ5 Can the non-existence at τ ∈ iℝ⁺ be used to infer existence conditions for other values of τ in the critical case n=2?

Key findings

• The paper proves the non-existence of solutions to the mean field equation Δu + e^u = 16πδ₀ on a flat torus E_τ when τ ∈ iℝ⁺, confirming the conjecture for n=2.
• The non-existence result enables the determination of the zero structure of the premodular form Z_{r,s}^{(2)}(τ) on the imaginary axis.
• Solutions to the mean field equation exist for τ = 1/2 + ib when b > b*, with b* lying in the interval (√3/2, 6/5).
• The threshold b* is explicitly bounded within (√3/2, 6/5), providing a sharp existence condition for τ = 1/2 + ib.
• The premodular form Z_{r,s}^{(2)}(τ) is shown to vanish precisely at points corresponding to non-solvability, linking modular forms to PDE solvability.
• The interplay between PDE theory and modular forms is established through the use of Z_{r,s}^{(2)}(τ), yielding a complete characterization of solution existence for n=2 at the critical parameter 16π.