Regularity and Singularity Issues in Geometric Variational Problems

几何变分问题中的正则性和奇异性问题

• 批准号: 2055686
• 负责人: Guido De Philippis
• 金额: $ 33.98万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-04-15 至 2025-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055686&HistoricalAwards=false
• 关键词: Regularity; Singularity; Issues; Geometric; Variational

Geometric variational problems are used to describe the behavior of systems driven by surface tension energies, like crystals or soap bubbles. Classical examples are the isoperimetric problem (find the solid of minimal perimeter enclosing a given volume) and the Plateau problem (finding the surface of minimal area spanning a given boundary curve). In the last 50 years, geometric variational problems have found a number of surprising applications in both pure and applied mathematics. Their solutions can describe the equilibrium configurations of physical systems in Mathematical Physics, the behavior of black holes in General Relativity, or they can provide preferred representatives in homology and homotopy classes in Differential Topology. For solutions of geometric variational problems, the presence of singularities is unavoidable and it is linked to the physical behavior of the systems they model or to the concentration of topological obstructions in geometric problems. A fine description of the regular and singular set is of fundamental importance in our understanding of the underlying problem. This project aims to enhance our understanding of the qualitative and quantitative behavior of solutions of geometric variational problems by addressing a series of basic questions concerning their regularity and the description of their singularities. The answer to these questions will require the development of new methods and techniques, which will also be valuable in other areas of mathematics. The project will provide research training opportunities for graduate and undergraduate students. Despite the great amount of study dedicated to geometric variational problems, several basic questions concerning the regular and singular behavior of their solutions are still poorly understood. This project is intended to address these questions and to develop new tools and techniques to tackle them. The project involves work on four deeply interconnected directions of research: boundary regularity for mass minimizing currents, regularity of critical points of anisotropic surface tensions, regularity of solutions to spectral optimization and free boundary problems, and the structure of PDE constrained measures. Their study will require the interaction of techniques coming from different areas of mathematics, such as Partial Differential Equations (PDE), Geometric Analysis, Geometric Measure Theory, Topology, and Harmonic Analysis, as well as the introduction of new ones.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

几何变异问题用于描述由晶体或肥皂泡等表面张力能驱动的系统的行为。经典示例是等等问题（找到给定体积的最小周边的固体）和平稳问题（找到跨越给定边界曲线的最小面积的表面）。在过去的50年中，几何变异问题在纯数学和应用数学中都发现了许多令人惊讶的应用。他们的解决方案可以描述数学物理学中物理系统的均衡配置，一般相对性中黑洞的行为，或者它们可以在差异拓扑中提供同源性和同型类别中的首选代表。对于几何变异问题的解决方案，奇异性的存在是不可避免的，并且与它们建模的系统的物理行为或几何问题中拓扑障碍的浓度有关。对常规和奇异集的精细描述对于我们对潜在问题的理解至关重要。该项目旨在通过解决有关其规律性和对其奇异性的描述的一系列基本问题，来增强我们对几何变异问题解决方案解决方案的定性和定量行为的理解。这些问题的答案将需要开发新的方法和技术，这在其他数学领域也很有价值。该项目将为研究生和本科生提供研究培训机会。尽管有大量研究用于几何变异问题，但有关其解决方案的常规和奇异行为的几个基本问​​题仍然很少了解。该项目旨在解决这些问题，并开发新的工具和技术来解决这些问题。该项目涉及研究的四个互连方向：质量最小的电流的边界规律性，各向异性表面紧张的临界点的规律性，频谱优化的解决方案的规律性和自由边界问题以及PDE约束措施的结构。他们的研究将需要来自不同数学领域的技术的相互作用，例如部分微分方程（PDE），几何分析，几何措施理论，拓扑结构和谐波分析以及新的奖项。这项奖项反映了NSF的法定任务，并通过使用基础的智力效果和宽阔的范围进行评估，并被视为值得通过评估来进行评估。