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

Zhijie Chen; Ting Jung Kuo; Chang Shou Lin

doi:10.4310/cag.2019.v27.n8.a3

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

Zhijie Chen, Ting Jung Kuo, Chang Shou Lin

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

It is known from [17] that the solvability of the mean field equation ∆u + 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 [17]. 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).

Original language	English
Pages (from-to)	1737-1755
Number of pages	19
Journal	Communications in Analysis and Geometry
Volume	27
Issue number	8
DOIs	https://doi.org/10.4310/cag.2019.v27.n8.a3
Publication status	Published - 2019

ASJC Scopus subject areas

Analysis
Statistics and Probability
Geometry and Topology
Statistics, Probability and Uncertainty

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10.4310/cag.2019.v27.n8.a3

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abstract = "It is known from [17] that the solvability of the mean field equation ∆u + eu = 8nπδ0 with n ∈ N≥ 1 on a flat torus Eτ essentially depends on the geometry of Eτ. A conjecture is the non-existence of solutions for this equation if Eτ is a rectangular torus, which was proved for n = 1 in [17]. 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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AB - It is known from [17] that the solvability of the mean field equation ∆u + eu = 8nπδ0 with n ∈ N≥ 1 on a flat torus Eτ essentially depends on the geometry of Eτ. A conjecture is the non-existence of solutions for this equation if Eτ is a rectangular torus, which was proved for n = 1 in [17]. 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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Non-existence of solutions for a mean field equation on flat tori at critical parameter 16π

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