Geometry of measures and applications

当把金属丝框浸入肥皂水溶液中时，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Plateau问题的解决方案密切相关，Plateau问题需要找到一个跨越空间中给定轮廓的最小面积的表面。(This经典问题是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中的稳定配置。在这个项目中，主要研究者解决了关于能量最小化的问题，这些问题考虑到了噪声和正在建模的现象的小波动。希望这个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证明“几乎最小化者”，即噪声变分问题的最小化者，继承了相同泛函减去噪声的最小化者的一些性质。这项研究需要使用的工具，从变分法，调和分析，偏微分方程，潜在的理论和几何测量理论。该项目将在上述领域之间建立桥梁，希望通过新思想的涌入在这一过程中改变它们。我们的期望是，这些新的想法将，特别是，发现在其他变分问题的自由边界的应用。

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.