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

數學系

研究成果: 雜誌貢獻 › 期刊論文 › 同行評審

4 引文斯高帕斯（Scopus）

摘要

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).

原文	英語
頁（從 - 到）	1737-1755
頁數	19
期刊	Communications in Analysis and Geometry
卷	27
發行號	8
DOIs	https://doi.org/10.4310/cag.2019.v27.n8.a3
出版狀態	已發佈 - 2019

ASJC Scopus subject areas

分析
統計與概率
幾何和拓撲
統計、概率和不確定性

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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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