Singular Points of Solutions of Hamilton-Jacobi Equations
The Hamilton-Jacobi equation is a prototypical nonlinear first-order PDE which appears in various settings including dynamical systems, optimal control, differential games, etc. where some action/value function solves this PDE. However, these action/value functions are often not smooth on the entire domain, and indeed the Hamilton-Jacobi equation often does not admit globally smooth solutions.
Overview
Thus the complete connection between the models and the equation is rigorously established with the viscosity solution concept. As such, understanding the local and the global picture of the points of singularity (non-differentiability) in the viscosity solutions of the Hamilton-Jacobi PDE is an important question for a better understanding of the models, and it is an active area of research. In this expository talk, I will introduce the Hamilton-Jacobi PDE, the viscosity solution concept, and recent results concerning the singularities.
Presenters
Brief Biography
Melih Ucer is a postdoctoral researcher in the Mean-Field Games (MFG) group of Prof. Diogo Gomes at KAUST. He obtained his bachelor's degree from MIT in physics and PhD degree from Bilkent University in mathematics, where he did research on topology of algebraic varieties. Since he joined KAUST, he has been primarily working on the weak solution concepts to MFG equations. In addition, being a former IMO medalist, he is still active in training olympiad students.