两组分玻色-爱因斯坦凝聚问题的基态解研究

In 1997, the research group of E.A. Cornell, a Nobel laureate in Physics, realized successfully the two component Bose-Einstein condensates, in which some novel physical phenomena were observed, such as phase separations, phase coherences and so on. With the further development of BEC theory, a lot of challenging and difficult mathematics problems arise in two component BEC. This project is focused on the following two types of variational problems arising in the two component BEC: (1) When the intraspecies interaction of the atoms inside each component is attractive, and the interspecies interaction between two components is however repulsive, by employing some new ideas of the blow-up analysis and energy estimates, we analyze the mass concentration, symmetry breaking and phase separations of minimizers for the corresponding constraint variational problem; (2) When the interatomic interactions inside each component and between two component are both inhomogeneous, the properties of the associated minimizers are studied, including the existence, uniqueness and the limit behavior. Investigating deeply these variational problems can not only contribute to deepening our understanding of the physical phenomena of two component BEC, but also promote to the applications of nonlinear functional analysis to the nonlinear elliptic PDEs.

1997年，诺贝尔物理学奖获得者E.A. Cornell的研究小组成功地实现了两组分玻色-爱因斯坦凝聚（BEC）实验，并观测到了一些新奇的物理现象，比如两组分之间的相位分离、相位相干等。随着BEC理论的深入研究，两组分BEC问题中出现了许多十分具有挑战性的数学难题。本项目主要探讨两组分BEC中的如下两类变分问题：（1）当各组分内部冷原子间相互吸引而两组分之间相互排斥时，通过探讨一些新的爆破分析和能量估计等方法，分析相应约束变分问题极小元的质量集中、对称破缺及相位分离等现象；（2）当各组分内部及两组分之间冷原子间的相互作用力具有某种非齐次性时，探讨相关泛函极小的分析性质，包括存在性，唯一性以及极限行为等。上述变分问题的深入研究，不仅有助于对两组分BEC物理现象的深刻理解，而且也能够促进非线性泛函分析理论在非线性椭圆方程中的应用。

本项目旨在分析探讨源于BEC中的一些约束变分问题的极小元的存在性、质量集中性等性质，具体来说，主要探讨了如下两个问题：（a）、探讨了一类带有非齐次扰动的质量次临界约束变分问题，详细分析了非齐次扰动对于极小元的存在性与集中性的影响；（b）、通过探讨相应的约束变分问题极小元的性质，详细分析了非齐次相互作用力下两组分BEC问题的基态解的性质。给出了极小元存在性与非存在性的分类条件。尤其在某些特殊条件下，我们还给出了极小元存在性与非存在性的完整分类，并进一步分析了某些参数靠近临界值时极小元的质量集中现象。整体上完成了项目拟定的目标，取得了一些有意义的成果。