Geometry of Measures and Applications

测量几何与应用

• 批准号: 1954545
• 负责人: Tatiana Toro
• 金额: $ 22.81万
• 依托单位: University of Washington
• 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: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954545&HistoricalAwards=false
• 关键词: Geometry; Measures; Applications

When dipping a frame in a solution of soap suds one produces a thin soap film. Mathematically this object is a constant mean curvature surface. It is closely related to the solution of the Plateau problem, which requires finding a surface of minimal area that spans a given shape in space. This problem is a classical question in the Calculus of Variations. The area is an energy functional, and the expectation is that minimizing it will lead to a stable configuration. In this project the PI addresses questions concerning the minimization of certain energy functionals that take into account noise and small random fluctuations of the phenomena being modeled. The expectation is that this theory will be better suited to reflect actual minimization questions arising in nature. A fundamental feature of the area functional is that it is invariant under rotations of space (if that space is homogeneous). The PI will address geometric and analytic questions in inhomogeneous and crystal-like spaces providing a model that reflects nature more accurately. This project will contribute to US workforce development through training and mentoring of graduate students and post-docs. One of the PI’s goals is to show that “almost minimizers”, which are minimizers to noisy variational problems inherit some of the properties of minimizers of the same functional without noise. This study requires using tools from calculus of variations, harmonic analysis and geometric measure theory. The expectation is that the new ideas developed along the way will find applications in other variational problems with free boundaries. The aim of the project concerning further developing analysis on non-smooth domains is to characterize the geometry of domains in Euclidean space in terms of the properties of solutions to canonical (anisotropic) operators. The project concerning the rectifiability of measures promises to reveal the fine structure of measures defined on crystal-like spaces. The overarching theme of this project brings together tools from Geometric Measure Theory, PDE, Potential Theory, Harmonic Analysis and Calculus of Variations, building bridges between these areas while transforming them by the influx of new ideas.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.

当把镜框浸入肥皂水溶液中时，会产生一层薄薄的肥皂膜。在数学上，这个物体是一个恒定平均曲率的表面。它与Plateau问题的解决方案密切相关，Plateau问题需要找到一个在空间中跨越给定形状的最小面积曲面。该问题是变分学中的一个经典问题。面积是一个能量泛函，期望最小化它将导致稳定的配置。在这个项目中，PI解决了关于最小化某些能量泛函的问题，这些泛函考虑到了被建模现象的噪声和小的随机波动。期望的是，这个理论将更适合于反映实际的最小化问题，在性质上产生的。面积泛函的一个基本特征是它在空间旋转下是不变的（如果空间是齐次的）。PI将解决非均匀和晶体状空间中的几何和分析问题，提供更准确地反映自然的模型。该项目将通过对研究生和博士后的培训和指导，为美国的劳动力发展做出贡献。PI的目标之一是表明，“几乎极小”，这是噪声变分问题的极小继承的一些性质的极小相同的功能没有噪音。这一研究需要借助变分法、调和分析和几何测度理论等工具。我们希望沿着这种方法发展起来的新思想能在其他自由边界的变分问题中得到应用。关于进一步发展非光滑域分析的项目的目的是根据正则（各向异性）算子的解的性质来表征欧几里德空间中域的几何。关于措施的可纠正性的项目有望揭示晶体状空间上定义的措施的精细结构。该项目的首要主题汇集了几何测量理论、偏微分方程、势理论、谐波分析和变分法等工具，在这些领域之间建立桥梁，同时通过新思想的涌入进行改造。该奖项反映了NSF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。