Geometry of measures and applications

测量几何和应用

基本信息

批准号：
1664867
负责人：
Tatiana Toro
金额：
$ 19.2万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-15 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664867&HistoricalAwards=false
关键词：
Geometry measures applications

项目摘要

When dipping a wire frame in a solution of soap suds one produces a thin soap film. Mathematically this is a very interesting object. It is closely related to the solution of the Plateau problem, which requires one to find a surface of minimal area that spans a given contour in space. (This classical problem is an example of those that one encounters in the calculus of variations.) The area of a surface can be understood as a measurement of energy. The basic guiding principle is that minimizing its energy will lead to a stable configuration in any physical system. In this project the principal investigator addresses questions concerning the minimization of energies, questions that take into account noise and small fluctuations of the phenomena being modelled. The hope is that this theory will be better suited than existing ones to address minimization questions that arise in nature.The principal investigator's goal in the project is to show that "almost minimizers," which are minimizers to noisy variational problems, inherit some of the properties of minimizers of the same functional minus the noise. This research requires the use of tools from the calculus of variations, harmonic analysis, partial differential equations, potential theory, and geometric measure theory. The project will build bridges between the aforementioned areas, hopefully transforming them in the process through the influx of new ideas. The expectation is that these new ideas will, in particular, find applications in other variational problems with free boundaries.

当将金属丝框架浸入肥皂泡沫溶液中时，会产生一层薄薄的肥皂膜。从数学上来说，这是一个非常有趣的物体。它与高原问题的解决密切相关，高原问题要求人们找到一个跨越给定空间轮廓的最小面积的表面。（这个经典问题是人们在变分法中遇到的问题的一个例子。）表面的面积可以理解为能量的测量。基本指导原则是最小化其能量将导致任何物理系统的稳定配置。在这个项目中，主要研究者解决了有关能量最小化的问题，这些问题考虑了噪声和所建模现象的小波动。希望该理论比现有理论更适合解决自然界中出现的最小化问题。该项目的主要研究者的目标是证明“几乎最小化”，即噪声变分问题的最小化，继承了一些相同函数减去噪声的最小化器的属性。这项研究需要使用变分法、调和分析、偏微分方程、势论和几何测度论等工具。该项目将在上述领域之间架起桥梁，希望通过新想法的涌入来改变它们。人们期望这些新想法尤其能够在其他具有自由边界的变分问题中找到应用。