Geometry of Measures and Applications

当将相框浸泡在肥皂溶液中时，会产生一层很薄的肥皂膜。从数学上讲，这个物体是一个恒定的平均曲率曲面。它与高原问题的求解密切相关，该问题要求在空间中寻找一个跨越给定形状的面积最小的曲面。这个问题是变分中的一个经典问题。该区域是一个能量泛函，预计将其最小化将导致稳定的配置。在这个项目中，PI解决了某些能量泛函的最小化问题，这些泛函考虑了被建模现象的噪声和小的随机波动。期望这一理论能更好地反映自然界中出现的实际最小化问题。面积泛函的一个基本特征是它在空间的旋转下是不变的(如果该空间是齐次的)。PI将解决不均匀和晶状空间中的几何和分析问题，提供一个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，为美国劳动力发展做出贡献。PI的目标之一是证明“几乎极小化”，即有噪声的变分问题的极小化继承了相同泛函的无噪声的极小化的一些性质。本研究需要运用变分、调和分析和几何测度论等工具。人们的期望是，在此过程中发展出的新想法将在其他自由边界的变分问题中得到应用。关于非光滑区域的进一步分析的项目的目的是根据正则(各向异性)算子解的性质来刻画欧氏空间中区域的几何。关于措施可纠正性的项目有望揭示定义在晶状空间上的措施的精细结构。这个项目的总体主题汇集了几何测量理论、偏微分方程、势能理论、调和分析和变分演算的工具，在这些领域之间架起了桥梁，同时通过新思想的涌入改变了它们。这个奖项反映了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。

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.