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.

各向异性能量是由吉布斯在19世纪引入的，用来模拟晶体的平衡形状，更一般地说，用来模拟任意两种不同材料界面的表面张力。数学中相应的几何变分问题日益引起人们的兴趣。最小化面积泛函，最简单的各向异性能量，是这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，对物理学和数学都产生了重大影响。1936年，杰西·道格拉斯（Jesse Douglas）因在这一课题上的研究成果获得了第一枚菲尔兹奖；然而，各种基本问题仍未解决，特别是在更普遍的各向异性环境中。本计划的目标是发展新的理论和技术，以改善各向异性几何变分问题的现状。本项目旨在增加我们对一般椭圆积分的临界点的理解。相对于最小曲面的到达理论，即功能区的临界点，在各向异性框架中很少被理解。本课题的主题之一是闭黎曼流形中各向异性极小超曲面的存在性。这是几何分析中一个引人入胜的核心问题，在80年代，Pitts、Schoen、Simon和Yau已经回答了这个问题。研究者将解决更一般的各向异性设置，扩大最小-最大理论的范围。这需要对变量的结构进行精细的分析，这也将在几何流的构造中得到应用。本课题的另一个目标是研究一般度量空间中集论平台问题能量极小值的存在性和正则性结果。这项研究将进一步细化，以减小可整流电流的尺寸。此外，研究者将研究具有几乎恒定各向异性平均曲率的超曲面的行为。最后，讨论了最优分支输运的稳定性和正则性猜想。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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