玻色-爱因斯坦凝聚中的若干变分问题

Bose-Einstein condensation (BEC) is a state of the matter in which a large number of particles in a Boson system occupy macroscopically the quantum ground state, when the temperature of the system is below a certain critical value. Since USA Physicists in 1995 first realized BEC in cold atomic gases, there appear a lot of challenging and difficult mathematics problems in the studies of BEC, including many interesting and new variational problems. This project is focussed on the following three types of variational problems arising in BEC: (1) We consider constraint minimizers of attractive BEC in bounded domains, and study their existence and local uniqueness, collapse, and dynamical stability as well. (2) Variational problems of two-dimensional nonhomogeneous BEC with attractive interactions are investigated, including the existence, symmetry-breaking, bifurcation phenomena, precised blow-up behavior and energy expansions of constraint minimizers. (3) We analyze constraint minimizers of two component BEC with attractive interactions, and discuss their analytic properties including the existence, collapse, and phase separations or phase coherences. Investigating deeply these variational problems cannot only contribute to deepening our understanding of BEC physical phenomena, but also contribute to expanding the application ranges of modern nonlinear functional analysis.

玻色-爱因斯坦凝聚(BEC)是在温度小于某一临界值时, 玻色子体系中大量粒子宏观地占据量子基态的一种物质状态。自1995年美国物理学家首次实现冷原子气体的BEC以来，BEC现象的实验与理论研究中出现了许多具有挑战性的数学难题，包括一些新的且引人注目的变分问题。本项目将重点考虑BEC中的如下三类变分问题：(1) 研究有界区域中吸引力作用下的BEC泛函极小化问题，包括泛函极小的存在性与局部唯一性、坍塌行为以及动态稳定性等；(2) 探索平面中吸引力作用下的非齐次BEC变分问题，包括泛函极小的存在性、对称破缺性、分支行为、精细爆破行为与精确能量展式等；(3) 分析吸引力作用下的两组分BEC泛函问题，包括泛函约束极小的存在性、坍塌行为、相位分离或相位相干及其相关解析性质等。这些变分问题的深入研究，不仅可以帮助我们加深对BEC物理现象的理解，而且有助于拓展现代非线性泛函分析理论的应用范围。

应用非线性泛函分析方法与偏微分方程理论，本项目主要围绕玻色-爱因斯坦凝聚（BEC）及其相关的几类变分问题，努力开展原创性研究，取得了一系列科研成果，具体包括：1).发展了归纳性对称方法，首次证明了吸引力作用下旋转BEC基态解的涡旋非存在性; 2).通过引入合适的辅助函数、建立椭圆方程组的局部Pohozaev恒等式，证明了两组分BEC基态解的唯一性与半平凡性；3).通过采用Green函数给出分数阶非局部线性化算子特征函数的精细一致估计，在一定条件下解决了著名数学家E.Lieb于1987年提出的“Lieb-Yau猜想”。上述相关成果发表在ARMA、Trans-AMS、AIHP-AN等国际权威数学期刊上。本项目的研究成果，不仅帮助我们加深了BEC物理现象的进一步理解，而且丰富了非线性泛函分析的理论体系，拓展了非线性泛函分析的应用范围。