中国科学院数学与系统科学研究院
数学研究所
数学科学全国重点实验室
非线性泛函分析研讨班
Speaker: Prof. Bernhard Ruf
（Istituto Lombardo - Accademia di Scienze e Lettere, Milano, Italy）
Inviter: 张志涛
Language: English
Title：On some inequalities of Bliss-Moser type
Time&Venue：2026年5月25日（星期一）10:00-11:00 & 南楼N818
Abstract: From the Bliss inequalities, which are a generalization of the Hardy inequality, one can derive a limiting Bliss inequality, which is of Trudinger-Moser type. We then consider a critical version of this inequality, and discuss related non-compactness properties. Furthermore, we show that this inequality is related to critical boundary growth for radial functions on a disk in two dimensions. Finally, we prove the existence of solutions for associated critical boundary value problems.
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Speaker: 郭琪 (中国人民大学)
Inviter: 张志涛
Language: Chinese
Title：Positivity Criterion and Extensions for the Ekeland–Nirenberg Problem
Time&Venue：2026年5月25日（星期一）11:00-12:00 & 南楼N818
Abstract:In this talk, I wil address the positivity conjecture raised by Ekeland and Nirenberg and later recorded in Long's collection of problems from ICVAM. The problem originates from the work of Bouchard, Ekeland, and Touzi, where Malliavin integration by parts is used to rewrite conditional expectations as unconditional expectations with localizing weights. This leads to the minimization of an unusual quadratic functional. Ekeland and Nirenberg proved the existence and uniqueness of the minimizer, and conjectured that it is always positive. We show that this conjecture is false in general. More precisely, for the two-dimensional diagonal family, we prove a sharp positivity criterion: the minimizer is positive if and only if d≤ac. When d>ac, the minimizer necessarily changes sign. The proof combines a Bessel–Laplace representation in the subcritical regime with an analytic-continuation and boundary-layer analysis in the supercritical regime. We will also discuss two further problems from Long's collection: a domain capacity criterion for the existence of constrained minimizers, and a decaying-branch evolution analogue exhibiting well-posedness without Sobolev smoothing.
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