Geometric Variational Problems for Surface Tension Driven Systems

这个项目将使用分析中的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。对这些系统的完整的数学理解有助于解决机械性质的生物和工程问题，因为它使研究人员能够获得对这些系统所驱动的物理理论的隐含后果的分析预测。与此同时，要获得这样的预测，就需要应对严峻的数学挑战，这刺激了新的有用的数学工具和方法的增长和发展，从而也推动了数学领域的发展。该项目还将为研究生提供研究培训机会。本项目涉及的第一系列问题涉及使用极小表面作为平衡状态下的液膜模型。经典的将液膜理想化为二维表面不能解释液膜的性质，其中液膜的厚度起着至关重要的作用(例如，给定几何膜构型的稳定性与膜本身直径的大小之间的关系)。该项目旨在开发一个由主要研究人员和他的合作者最近提出的、能够解释经典方法无法获得的物理特征的三维小体积液膜模型。第二个研究方向包括具有物理意义奇点的最小和常平均曲率曲面的刚性定理。这个项目研究了系统地将它们发展成定量的几乎刚性的陈述的可能性，以及将它们的适用性扩展到非光滑环境的可能性，动机是描述毛细理论中的平衡构型和外部曲率流的渐近行为。最后，这一思想圈的广泛覆盖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，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.