The investigator develops new methods needed to study key issues in the emerging area of Calculus of Variations in L-infinity, for the case where the functionals to be minimized are the essential supremum of some function of fields that are subject to constant rank differential constraints. This general framework includes the treatment of curl-free fields, divergence-free fields, symmetrized and higher order gradients, and fields that satisfy Maxwell's equations. He studies necessary and sufficient conditions under which these functionals possess a minimizer, relaxation, supremal representation, homogenization, approximation via Gamma-convergence, and he investigates selected systems of partial differential equations that arise as the Aronsson equations associated to such functionals. Previous work in this direction has been focused on the particular case where the admissible fields are gradients (curl-free); the corresponding issues for supremal functionals acting on fields subject to other differential constraints that appear naturally as balance laws have remained largely unaddressed. An important goal of the project is to bridge this gap by pursuing a systematic study of supremal functionals in this generality. The project aims to provide a unified view of the subject and, in particular, to reveal how the known properties of supremal functionals depending on gradients can be understood as a particular case in the broader context considered. The analytical methods developed by the investigator and his collaborators are used to study selected problems in materials sciences that are described by means of explicit variational principles involving supremal functionals. In particular, the project seeks to answer certain questions regarding the macroscopic behavior of polycrystals, and to elucidate several issues that come up in the study of granular materials, where new models are analyzed and used to study the flow of heterogeneous sand piles over given landscapes in the context of related Monge-Kantorovich mass transfer problems.
The project is motivated by specific questions about the behavior of polycrystalline and granular materials. Such materials are very common: most metals found in nature are polycrystals; grains, coal, plastic, building materials (sand, gravel), and various powders and chemical compounds, are all examples of granular materials that are handled and stored on a daily basis. Thus, understanding how to mix and transport them in an optimal fashion is an issue of natural interest. Problems of practical importance where the more realistic approach is to minimize the supremum rather than the average of certain quantities are ubiquitous in many other areas, such as nonlinear elasticity, image processing, damage and fracture mechanics, plasticity, and semiconductor design. It is expected that the methods developed within this project will be relevant to an extensive range of applications, beyond those considered here explicitly. The broader impacts of the project are also achieved through training of graduate and undergraduate students, who are exposed to modern developments in applied analysis and are involved in projects related to the proposal, as well as through other educational and outreach activities.
