两类质量约束变分问题的研究

The L^2-constraint variational theory has wide applications in the fields of quantum many-body systems, nonlinear optics, mechanics and so on, which has attracted lots of experts to do the related researches over the past few years. Following the further investigations of Bose-Einstein condensation (BEC) and Kirchhoff models, there appear many challenging and difficult mathematics problems in studying the related L^2-constraint variational problems. By making full use of the nonlinear functional analysis, the present project is focussed on the following two classes of problems: (1) We study constraint minimizers of the L^2-constraint variational problems modeling two-component BEC in bounded domains, including the existence and non-existence, blow-up behavior, and the local uniqueness as well. (2) We analyze the L^2-constraint variational problems of Kirchhoff type involving two L^2-critical exponents, for which we investigate the existence, mass concentration and uniqueness of minimizers. Studying deeply the above L^2-constraint variational problems cannot only provide theoretical interpretations of the relevant physical experiments, but also contribute to enriching the theory and applications of nonlinear functional analysis.

L^2-约束变分理论在量子多体系统、非线性光学、力学等领域有着非常广泛的应用背景，近年来吸引了大量的国内外专家开展了相关的理论研究。特别是随着玻色-爱因斯坦凝聚(BEC)现象、Kirchhoff模型的深入探讨，相关的L^2-约束变分问题的研究中涌现了许多具有挑战性的数学难题。本项目将充分应用非线性泛函分析理论与方法，重点考虑如下两类问题: (1)研究有界区域上两组分BEC中的L^2-约束变分问题，包括泛函极小的存在性与非存在性、爆破行为及局部唯一性等；(2)分析Kirchhoff型L^2-约束泛函问题，包括具有两个L^2-临界指标情形下极小元的存在性、质量集中行为、唯一性等。上述两类变分问题的深入研究，不仅可以为相关物理现象的解释提供理论上的支撑，而且有助于丰富非线性泛函分析的理论与应用。

质量约束变分理论在生态学系统中基尔霍夫问题以及物理实验中的BEC问题等领域有着非常广泛的应用背景，近年来吸引了大量的国内外专家开展了相关的理论研究。本项目研究了玻色-爱因斯坦凝聚(BEC)问题、Kirchhoff模型相关的L^2-约束变分问题的泛函极小的存在性与非存在性、爆破行为及局部唯一性等。这两类变分问题的深入研究，不仅可以为相关物理现象的解释提供理论上的支撑，而且有助于丰富非线性泛函分析的理论与应用。