Anisotropic Energy Functionals in Geometric Analysis

几何分析中的各向异性能量泛函

• 批准号: 2112311
• 负责人: Antonio De Rosa
• 金额: $ 12.42万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-12-15 至 2022-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2112311&HistoricalAwards=false
• 关键词: Anisotropic; Energy; Functionals; Geometric; Analysis

Anisotropic energies were introduced by Gibbs in the 19th century to model the equilibrium shape of crystals and, more generally, the surface tension at the interfaces of any two different materials. An increasing interest has been devoted to the corresponding geometric variational problems in mathematics. Minimizing the area functional, the simplest anisotropic energy, is one of the most famous examples in this class of problems. It dates back to the work of Lagrange in 1760 and has had a major impact both in physics and in mathematics. Jesse Douglas was awarded the first Fields Medal in 1936 for his results on this topic; yet, a variety of basic questions remain open, especially in the more general anisotropic setting. The goal of this project is to develop new theories and techniques to improve the state of the art of anisotropic geometric variational problems. This project aims to increase our understanding of critical points for general elliptic integrands. In contrast to the reach theory of minimal surfaces, i.e., critical points of the area functional, very little is understood in the anisotropic framework. One of the main themes of this project is the existence of anisotropic minimal hypersurfaces in closed Riemannian manifolds. This is a fascinating and central question in geometric analysis, which has been answered for the area functional by Pitts, Schoen, Simon and Yau in the eighties. The investigator will address the more general anisotropic setting, broadening the reach of the min-max theory. This will require a refined analysis of the structure of varifolds, which will find applications also in the construction of geometric flows. Another goal of this project is to study existence and regularity results for energy minimizers of the set-theoretic Plateau problem in general metric spaces. This investigation will be then further refined for size minimizer rectifiable currents. Moreover, the investigator will study the behavior of hypersurfaces with almost constant anisotropic mean curvature. Finally, the stability and regularity conjectures in optimal branched transport will be addressed.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.

吉布斯（Gibbs）在19世纪引入了各向异性能量，以建模晶体的平衡形状，更普遍地是在任何两种不同材料的界面处的表面张力。越来越多的兴趣致力于数学中相应的几何变异问题。最小化该区域功能，最简单的各向异性能量是此类问题中最著名的例子之一。它可以追溯到1760年拉格朗日的工作，并且在物理和数学方面都产生了重大影响。杰西·道格拉斯（Jesse Douglas）在1936年因在这个话题上的成绩而获得了第一枚奖牌。然而，各种基本问题仍然开放，尤其是在更普遍的各向异性环境中。该项目的目的是开发新的理论和技术，以改善各向异性几何变异问题的艺术状态。该项目旨在提高我们对一般椭圆集成的关键要点的理解。与最小表面的范围理论（即区域功能的临界点）相反，在各向异性框架中几乎没有理解。该项目的主要主题之一是在封闭的Riemannian歧管中存在各向异性最小的超曲面。这是几何分析中的一个引人入胜且中心的问题，在八十年代，皮茨，Schoen，Simon和Yau为该地区的功能回答了这一问题。研究者将解决更通用的各向异性环境，从而扩大了最小值理论的影响力。这将需要对Varifolds结构进行完善的分析，这也将在几何流量的构建中找到应用。该项目的另一个目标是研究一般度量空间中固定理论高原问题的能量最小化的生存和规律性结果。然后，将进一步完善该研究，以使尺寸最小化的电流最小化。此外，研究者将研究具有几乎恒定各向异性平均曲率的高曲面的行为。最后，将解决最佳分支运输中的稳定性和规律性猜想。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。

项目成果

On the anisotropic Kirchhoff-Plateau problem

关于各向异性基尔霍夫-高原问题

• DOI: 10.3934/mine.2022011
• 发表时间: 2021
• 期刊: Mathematics in Engineering
• 影响因子: 1
• 作者: De Rosa, Antonio;Lussardi, Luca
• 通讯作者: Lussardi, Luca

Stability of optimal traffic plans in the irrigation problem

灌溉问题中最优交通计划的稳定性

• DOI: 10.3934/dcds.2021167
• 发表时间: 2022
• 期刊: Discrete & Continuous Dynamical Systems
• 影响因子: 1.1
• 作者: Colombo, Maria;De Rosa, Antonio;Marchese, Andrea;Pegon, Paul;Prouff, Antoine
• 通讯作者: Prouff, Antoine

Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets

• DOI: 10.1007/s00205-020-01562-y
• 发表时间: 2019-08
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: A. Rosa;Sławomir Kolasiński;Mario Santilli

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

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