Anisotropic Energy Functionals in Geometric Analysis

几何分析中的各向异性能量泛函

基本信息

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

项目摘要

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.

吉布斯在 19 世纪引入各向异性能量来模拟晶体的平衡形状，更一般地说，模拟任何两种不同材料界面处的表面张力。人们对数学中相应的几何变分问题越来越感兴趣。最小化泛函面积（最简单的各向异性能量）是此类问题中最著名的例子之一。它的历史可以追溯到 1760 年拉格朗日的工作，对物理学和数学都产生了重大影响。杰西·道格拉斯 (Jesse Douglas) 因其在这一主题上的成就于 1936 年荣获第一届菲尔兹奖；然而，各种基本问题仍然悬而未决，特别是在更一般的各向异性环境中。该项目的目标是开发新的理论和技术来提高各向异性几何变分问题的最新水平。该项目旨在增进我们对一般椭圆被积函数临界点的理解。与最小曲面的到达理论（即面积泛函的临界点）相反，在各向异性框架中我们了解甚少。该项目的主题之一是闭黎曼流形中各向异性最小超曲面的存在。这是几何分析中一个令人着迷的核心问题，皮茨、舍恩、西蒙和丘在八十年代就泛函面积回答了这个问题。研究人员将解决更一般的各向异性设置，扩大最小-最大理论的范围。这将需要对varifolds的结构进行精细分析，这也将在几何流的构造中找到应用。该项目的另一个目标是研究一般度量空间中集合论高原问题的能量最小化器的存在性和规律性结果。然后，这项研究将进一步细化，以实现尺寸最小化的可整流电流。此外，研究人员将研究具有几乎恒定的各向异性平均曲率的超曲面的行为。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 的法定使命，并通过使用基金会的智力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

期刊论文数量（8）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

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