Energy-Driven Geometry of Interfaces and Free Boundaries in Complex and Multiphase Systems

This project aims to advance the mathematical theory of complex and multi-phase media including films, foams, and liquid crystals. Such systems arise in applications ranging from foods, pharmaceuticals, water treatment, and the design of soft and bio-inspired materials. The first goal is to characterize through rigorous analysis the interfacial structures in films and foams that govern processes such as drainage in the above applications. The second aim is to analytically predict optimal configurations in mathematical models for phase transitions, fluid interfaces, and thin film liquid crystals. This project also incorporates the training and mentoring of graduate and undergraduate students and conference organization. The first portion of the project concerns the mathematical analysis of thin films and foams driven by surface tension, with a particular focus on understanding Plateau borders. These are thickened tubes of liquid surrounding the junctions between films. Using tools from geometric measure theory, partial differential equations, and the calculus of variations, the project seeks to predict the shape, size and location of the Plateau borders. These results will lay the groundwork for further investigation into these phenomena, as the fluid equations used to study foam drainage depend on scaling laws for the Plateau borders. The second part of the project builds upon the first by characterizing geometric properties of interfaces and free boundaries in a variety of physically motivated settings including free boundary problems arising from spectral partitioning problems, cluster energies modeling immiscible fluids, and variational models for liquid crystals. 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.