玻色-爱因斯坦凝聚中的若干变分问题

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。

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.