Non-existence of solutions for a mean field equation on flat tori at critical parameter 16π

2016/10/06 by Zhijie Chen, Chen, Zhijie, Ting-Jung Kuo +3 · 2 citations
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1610.01787

openalex publication_date 2016/10/06 · openalex created_date 2017/09/15 · openalex updated_date 2026/07/28

Abstract

It is known from \citeLW that the solvability of the mean field equation Δu+e^u=8nπδ₀ with n∈ℕ_{≥ 1} on a flat torus E_τ essentially depends on the geometry of E_τ. A conjecture is the non-existence of solutions for this equation if E_τ is a rectangular torus, which was proved for n=1 in \citeLW. For any n∈ ℕ_≥2, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for n=2 (i.e. at critical parameter 16π).

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