EVENT DETAILS AND ABSTRACT

Title: Potential Theory for Nonlinear PDEs
Speaker: Blaine Lawson
Speaker Info: Stony Brook University
Brief Description:
Special Note:
Abstract:

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation of the form f(D^2u) = 0. For the standard complex Monge-Ampere equation, it is just the classical pluripotential theory. I will explain how these theories are defined in general. Fundamental to the analysis is a new invariant of such equations, called the Riesz characteristic, which governs asymptotic structures. The notions of tangents to subsolutions and densities will be introduced. Results concerning existence and uniqueness of tangents, the structure of sets of high density points, and the regularity of subsolutions for certain Riesz characteristics, will be discussed. The Dirichlet problem with prescribed asymptotics will be treated. I will also touch upon the question of removable singularities. Examples include real, complex and quaternionic Hessian equations, the p-convexity equation, and equations from calibrated geometry. In particular, this establishes a potential theory on every calibrated manifold.