Geometric Variational Problems for Surface Tension Driven Systems

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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.