This project is to analyze the properties of solutions of certain classes of nonlinear partial differential equations (PDE). One class is "Weak KAM" theory which studies far-from-integrable Hamiltonian systems, using variational and PDE methods to identify certain classes of well-behaved trajectories, associated with action minimizing measures. New PDE approaches will try to discover, for instance, the full implications of nonresonance conditions.
Another class of problems is the regularity of solutions of the infinity Laplacian equation and Aronsson's equation. These highly degenerate, highly nonlinear equations model variational problems in the sup-norm as well as certain random-turn stochastic games. A major goal here will be to prove that weak solutions are continuously differentiable.
The project will also investigate a "lakes and rivers" PDE model. This is a singular limit of a certain nonlinear parabolic diffusion equation, which has a simple interpretation: the limiting solution either hugs a given landscape or else forms flat ``lakes'' connected by ``rivers''. The technical challenge will be to develop PDE and measure theory tools to understand and predict the location of the "lakes and rivers".
This research project will also explore the use of certain entropy type inequalities to study the asymptotic limit of a regularized nonlinear diffusion equation. We expect that for a nonmonotone linearity this asymptotic limit should develop hysteresis effects, called dynamic phase transitions. A different and quite speculative undertaking is to develop for fully nonlinear PDE some kind of analog of the important kinetic formulation for conservation laws.
These various PDE arise in a variety of settings including physics, game theory and image processing, so the results should be of importance in many application areas.
