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。对这些系统的完整的数学理解对于解决机械性质的生物和工程问题是有用的，因为它允许研究人员对由这些系统驱动的物理理论的隐含结果进行分析预测。与此同时，获得这样的预测需要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。在这个项目中解决的第一个系列问题是关于在平衡状态下使用最小表面作为液膜模型。液体膜作为二维表面的经典理想化不能解释液体膜的特性，其中膜的厚度起着至关重要的作用（例如，给定几何膜结构的稳定性与膜本身直径大小之间的关系）。这个项目打算开发一个小体积的三维液体膜模型，这是最近由首席研究员和他的合作者提出的，它能够解释经典方法无法获得的物理特征。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。该项目研究了系统地将它们发展成定量的几乎刚性陈述的可能性，并将它们的适用性扩展到非光滑设置，其动机是描述毛细理论中的平衡构型和外在曲率流动的渐近行为。最后，这个思想圈子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研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.