Geometric Variational Problems for Surface Tension Driven Systems

该项目将使用分析工具来研究由表面张力效应驱动到平衡的物理系统的基本数学模型。一个完整的数学理解这些系统是有用的，在解决生物和工程问题的机械性质，因为它允许研究人员获得分析预测的隐含后果的物理理论的动机，这样的系统。与此同时，获得这样的预测需要一个解决困难的数学挑战，这刺激了新的和有用的数学工具和方法的增长和发展，从而推进数学领域。该项目还将为研究生提供研究培训机会。在这个项目中解决的第一系列问题涉及使用最小的表面作为平衡液膜的模型。作为二维表面的液体膜的经典理想化不能解释其中膜的厚度起关键作用的液体膜性质（例如，给定的几何膜结构的稳定性与膜本身的直径大小之间的关系）。该项目旨在开发一种液体膜模型，作为具有小体积的三维区域，这是最近由主要研究者及其合作者提出的，并且能够解释经典方法无法获得的物理特征。第二个方向的调查包括刚性定理的最小和常数平均曲率曲面具有物理意义的奇点。这个项目调查的可能性，系统地发展成定量的几乎刚性报表，并将其适用性扩展到非光滑设置，在毛细理论的平衡配置的描述和外在曲率流的渐近行为的动机。最后，这个思想圈的广泛影响自然导致了对等周簇、分数周长和分数平均曲率以及扩散界面模型中的几何流动和平衡形状的相关问题的调查。该奖项反映了NSF的法定使命，并被认为值得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来支持。

This project will use tools from analysis to investigate fundamental mathematical models for physical systems driven to equilibrium by surface tension effects. A complete mathematical understanding of these systems is useful in addressing biological and engineering problems of mechanical nature, as it allows investigators to obtain analytical predictions on the implied consequences of physical theories motivated by such systems. At the same time, obtaining such predictions requires one to address hard mathematical challenges, which stimulates the growth and development of new and useful mathematical tools and methods, thus advancing the field of mathematics as well. The project will also provide research training opportunities for graduate students. A first series of questions addressed in this project concern the use of minimal surfaces as models for liquid films at equilibrium. The classical idealization of liquid films as two-dimensional surfaces cannot account for liquid films properties where the thickness of the film plays a crucial role (e.g., the relation between the stability of a given geometric film configuration and the size of the diameter of the film itself). This project intends to develop a model for liquid films as three-dimensional regions with a small volume, which was recently proposed by the principal investigator and his collaborators, and which is capable of explaining physical features not accessible by classical approaches. A second direction of investigation includes rigidity theorems for minimal and constant mean curvature surfaces possessing physically meaningful singularities. This project investigates the possibility of systematically developing them into quantitative almost-rigidity statements, and of extending their applicability to non-smooth settings, with motivations in the description of equilibrium configurations in capillarity theory and in the asymptotic behavior of extrinsic curvature flows. Finally, the vast reach of this circle of ideas naturally leads to an investigation of related problems for isoperimetric clusters, fractional perimeters and fractional mean curvatures, and for geometric flows and equilibrium shapes in diffused interface models.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.