Nonlocal Variational Problems from Physical and Biological Models

Nonlocal models are used to describe a wide array of physical phenomena in material science, quantum mechanics, mathematical physics, and biology. The pertinence of these models is that they introduce scales used to investigate microstructures in macroscopic domains, as in the case of the nonlocal variational models characterizing the collective behavior in biological systems. This project involves the mathematical analysis of nonlocal attractive-repulsive interaction energies that are directly connected to aggregations models for biological and robotic swarming, granular media, and self-assembly of nanoparticles. The emphasis is on understanding the competing effects of interactions on the physical systems considered. This will yield insight into the general phenomenology of nonlocality and ultimately provide predictions of collective behavior in these systems and guide the design of improved devices. Training of undergraduate and graduate students will be integrated in the research project, which will lead to the discovery of new results through student research projects. The project has three main mathematical aims: (1) Develop new tools to study symmetry of optimizers of a model describing the distribution of oppositely charged phases; (2) study the effect of regularization via an interfacial free energy in swarming models described by attractive-repulsive nonlocal energies; and, finally, (3) introduce an extension of Gamow’s liquid drop model to investigate the effect of neutrons in determining the shape of the nucleus of an atom and the threshold of nuclear fission. To pursue these goals, techniques varying from nonlinear to geometric analysis, optimization and partial differential equations will be implemented. These techniques will be combined with computer-based experimentation. The common aim of all three directions is understanding structures of optimizers in a variety of physical systems of practical interest. 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.