Some impact of topology on variational problems

拓扑对变分问题的一些影响

• 批准号: 0209504
• 负责人: Alice Chang
• 金额: $ 9.05万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0209504&HistoricalAwards=false
• 关键词: Some; impact; topology; variational; problems

PI: S.-Y. Alice Chang and Fengbo Hang, Princeton UniversityDMS-0209504Abstract:Some impact of topology on variational problemsIn this project, we will focus on variational problems involving vector- valued maps. In particular, we will study upper and lower bounds ofthe energy functional for maps with suitable topological information,the weak sequential approximation of Sobolev maps by smooth maps andthe structure of singular sets for solutions of some variationalproblems. Various tools from geometric measure theory and differentialand algebraic topology are used in handling these issues. On the otherhand, our approach to the problems sheds light on new relations amongthese topics.Variational problems involving vector-valued maps appear naturally in analysis, geometry and physics. Besides their mathematical interests and importance, they are crucial to various models in mathematical physics. Typical examples include models for superconductivity, magnetic material and liquid crystals. The research in this area will lead to better understanding of certain type of physics phenomenon as well as new impetus in mathematics in this direction.

PI：S.- Y. 11张丽，杭丰波，普林斯顿大学DMS-0209504摘要：拓扑学对变分问题的一些影响在这个项目中，我们将集中研究涉及向量值映射的变分问题。特别地，我们将研究具有适当拓扑信息的映射的能量泛函的上、下界，Sobolev映射的光滑弱序列逼近以及某些变分问题解的奇异集结构。在处理这些问题时，使用了几何测度论和微分代数拓扑学的各种工具.另一方面，我们的研究方法揭示了与这些问题相关的新的关系。涉及向量值映射的变分问题自然地出现在分析、几何和物理中。除了它们的数学兴趣和重要性之外，它们对数学物理中的各种模型至关重要。典型的例子包括超导模型，磁性材料和液晶。这一领域的研究将有助于更好地理解某些类型的物理现象，并为数学在这一方向的发展提供新的动力。