Geometry of measures and applications

把钢丝架浸在肥皂泡沫溶液中，就会产生一层薄薄的肥皂膜。从数学上讲，这是一个非常有趣的物体。它与高原问题的解密切相关，高原问题要求在空间中找到跨越给定轮廓的最小面积曲面。（这个经典问题是人们在变分学中遇到的一个例子。）表面的面积可以理解为能量的量度。基本的指导原则是最小化它的能量将导致任何物理系统的稳定配置。在这个项目中，首席研究员解决了有关能量最小化的问题，考虑到正在建模的现象的噪声和小波动的问题。希望这一理论将比现有的理论更适合于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表明“几乎最小值”，即噪声变分问题的最小值，继承了相同泛函减去噪声的最小值的一些性质。这项研究需要使用的工具从变分法，谐波分析，偏微分方程，势理论，几何测量理论。该项目将在上述地区之间架起桥梁，希望在这个过程中通过新思想的涌入来改变它们。我们的期望是，这些新思想将在其他具有自由边界的变分问题中得到特别的应用。

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.