Viscosity Solutions: Beyond Well-Posedness Theory

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This project concerns some nonlinear partial differential equations that appear naturally in physics, the social sciences, and engineering and that arise, for example, in the study of composite materials, combustion, game theory, traffic flow, and optimization. The equations considered have deep connections with a host of other areas of mathematics, including the calculus of variations, differential games, dynamical systems, geometry, homogenization theory, inverse problems, and optimal control theory. The main goal of the project is to discover new underlying principles and general methods to understand the properties of solutions of the differential equations under investigation. One of the key objects of the research is homogenization theory, in which the models (say, of physical or social phenomena) are set in heterogeneous media and have many parameters varying on a small scale. If one zooms out and looks at the macroscopic scale, one often sees a simple effective (averaging) behavior. To make practical use of the models, it is extremely important to understand deeply the qualitative and quantitative aspects of this effective behavior. The project has two overarching themes: (i) going beyond mere well-posedness in order to understand fine properties of both the limiting process and the effective equation in homogenization theory (e.g., shape of the effective Hamiltonian, optimal rate of convergence), and (ii) advancing knowledge of the duality method and fully nonlinear elliptic equations (e.g., representation formulas of solutions, the vanishing discount problem, regularity). The principal investigator and his collaborators have recently developed new approaches in these (and related) topical areas, approaches that have already provided solutions to some open problems. These methods are expected to be developed further in this project, thereby bringing fresh perspectives on and insights into the field of viscosity solutions.