CAREER: Existence, regularity, uniqueness and stability in anisotropic geometric variational problems

职业：各向异性几何变分问题的存在性、规律性、唯一性和稳定性

• 批准号: 2143124
• 负责人: Antonio De Rosa
• 金额: $ 45万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143124&HistoricalAwards=false
• 关键词: CAREER; Existence; regularity; uniqueness; stability

Minimizing the area functional is one of the most famous examples of geometric variational problems in mathematics and it has had a major impact both in mathematics and in physics. However, for several natural phenomena, the area functional is only a first approximation. In order to capture microstructures, most models in the sciences use directionally dependent functionals, referred to as anisotropic energies. Ever since the introduction of anisotropic energies by Gibbs in the 19th century to model crystals, they have been extensively applied in material sciences and engineering, motivating seminal works in geometric analysis. However, since anisotropic energies are not invariant under translations and rotations, they don’t enjoy the conservation laws of the area functional, which makes them significantly more complicated to study. The PI will advance the anisotropic minimal surfaces theory. Understanding existence, regularity, uniqueness and stability of solutions to anisotropic geometric variational problems plays a major role in analysis, geometry, topology and physics. The PI will also conduct vertically integrated educational activities tied with the research activities. In particular, undergraduate and graduate students will be exposed to the problems and techniques of this project via the organization of seminars, conferences, and a summer school.The PI will study existence, regularity, uniqueness and stability properties of anisotropic minimal surfaces. The existence of anisotropic minimal surfaces in Riemannian manifolds will require extending the min-max theory. In order to determine the regularity of anisotropic minimal surfaces, the PI will study the related geometric nonlinear elliptic PDEs. In addition to the stationary configurations, this research will shed light on the anisotropic Brakke flow and its approximation, through the analysis of the related parabolic PDEs. This project will also address the uniqueness of critical points of the anisotropic isoperimetric problem and investigate the stability properties of the Wulff shapes. Furthermore, part of this research will be devoted to optimal transport, with an emphasis on the regularity and stability properties of branching dynamics.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.

最小化该区域的功能是数学上几何变异问题的最著名示例之一，并且在数学和物理学中都产生了重大影响。但是，对于几种自然现象，该区域功能仅是第一个近似值。为了捕获微观结构，科学中的大多数模型都使用定向依赖的功能，称为各向异性能量。自从19世纪Gibbs引入各向异性能量以模拟晶体以来，它们已被广泛应用于材料科学和工程中，激励了几何分析中的第二幅作品。但是，由于在翻译和旋转下并不是不变的各向异性能量，因此它们不享受该地区功能的保护定律，这使得它们更加复杂。 PI将推进各向异性最小表面理论。了解解决方案对各向异性几何变异问题的存在，规律性，独特性和稳定性在分析，几何，拓扑和物理学中起主要作用。 PI还将进行与研究活动相关的垂直整合的教育活动。特别是，本科生和研究生将通过半手，会议和暑期学校的组织来接触该项目的问题和技术。PI将研究各向异性最小表面的存在，规律性，规律性，独特性和稳定性。黎曼流形中的各向异性最小表面的存在将需要扩展最小值理论。为了确定各向异性最小表面的规律性，PI将研究相关的几何非线性椭圆形PDE。除了固定构型外，这项研究还将通过分析相关的抛物线PDE来阐明各向异性的布拉克流及其近似。该项目还将解决各向异性等值问题的关键点的唯一性，并研究WULFF形状的稳定性。此外，这项研究的一部分将致力于最佳运输，重点是分支动力学的规律性和稳定性。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛的影响来通过评估来获得的支持。

项目成果

Regularity for graphs with bounded anisotropic mean curvature

具有有界各向异性平均曲率图的正则性

• DOI: 10.1007/s00222-022-01129-6
• 发表时间: 2022
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: De Rosa, Antonio;Tione, Riccardo
• 通讯作者: Tione, Riccardo

Efficient joint object matching via linear programming

通过线性规划进行高效的关节对象匹配

• DOI: 10.1007/s10107-023-01932-w
• 发表时间: 2023
• 期刊: Mathematical Programming
• 影响因子: 2.7
• 作者: De Rosa, Antonio;Khajavirad, Aida
• 通讯作者: Khajavirad, Aida