Geometry of measures and applications

当将线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，这是一个非常有趣的物体。它与高原问题的求解密切相关，该问题要求在空间中找到一个跨越给定等高线的面积最小的曲面。(这个经典问题是人们在变分法中遇到的一个例子。)表面的面积可以理解为能量的量度。基本的指导原则是，将其能量最小化将导致任何物理系统的稳定配置。在这个项目中，首席研究人员解决了关于能量最小化的问题，这些问题考虑到了被建模现象的噪音和微小波动。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。该项目的主要研究人员的目标是证明“几乎最小化”，即噪声变分问题的最小化，继承了相同函数减去噪声的最小化的一些性质。这项研究需要使用变分、调和分析、偏微分方程组、位势理论和几何测度论的工具。该项目将在上述领域之间架起桥梁，希望通过涌入新想法来改变这些领域。人们的期望是，这些新想法将特别适用于其他具有自由边界的变分问题。

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.