Yanyan Li, Rutgers University
Title: Liouville Theorems and Conformally Invariant Fully Nonlinear Equations
Abstract: The methods of moving planes and moving spheres, together with conformal invariance, have played a significant role in the study of nonlinear elliptic equations and conformal geometry. Beginning with the classical Liouville theorem for positive harmonic functions and the celebrated classification theorem of Caffarelli, Gidas, and Spruck, the theory has evolved to encompass fully nonlinear conformally invariant equations involving the conformal Hessian (or, equivalently, the Schouten tensor).
This mini-course introduces the basic structure of these equations, their conformal invariance, and their connections with the fully nonlinear Yamabe problem. We will discuss recent progress on Liouville-type theorems for general conformally invariant fully nonlinear equations, including sharp necessary and sufficient conditions for their validity. Particular emphasis will be placed on isolated singularities and the optimal geometric conditions on admissible cones. Time permitting, we will also discuss applications to local gradient estimates and problems in conformal geometry.
Hung V. Tran, University of Wisconsin–Madison
Title: Homogenization and Selection Problems for Hamilton-Jacobi equations
Abstract: I will give an introduction to periodic homogenization of first-order and second-order Hamilton-Jacobi PDEs. I will then describe some recent progress on the optimal convergence rates for both cases. Finally, I will show the nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic (cell) problems.
Tarek M. Elgindi, Duke University
Title: Some Problems Related to Incompressible Fluids
Abstract: The plan will be to discuss a number of problems related to the incompressible Euler equations. Time permitting, we will discuss steady solutions and their stability properties, long-time behavior in 2D, and issues related to finite-time singularity formation.