Description

A typical case for first order fully nonlinear PDEs is the class of Hamilton--Jacobi--Bellman equations, linked to optimal control theory. In the second order settings such equations arise in stochastic control problems involving controlled volatility. These are some fascinating mathematical objects which attracted a huge number of high level mathematicians because of their rich applicability. The birth of optimal control theory dates back to the early 19th century, however its modern mathematical look started to be shaped around Word War II by the fundamental works of Isaacs, Bellman and others. This discovery had its first peak point in the 1980s by the works of Crandall, Lions and Evans. The revolutionary notion of viscosity solutions for such equations, introduced by these researchers, eventually granted the Fields Medal for Lions in 1994.

Another class of fully nonlinear PDEs is the so-called Monge--Ampère type equations which are fundamental objects arising in the modern theory of optimal transport and in geometry (such as the Minkowski problem; the prescribed Gaussian curvature problem; the reflector antenna problem, etc.). Optimal transport is a powerful theory at the crossroad of pure and applied mathematics, and it has been recognised by the Fields medals of Figalli and Villani.

Goals of the project

The main objective of the project is the fine qualitative and quantitative mathematical analysis of fully non-linear PDEs. To achieve this, we will show and motivate suitably defined weak solutions and study the regularity theory of these solutions for the two main classes of PDEs mentioned above. Regularity theory is the study of smoothness (Hölder or Sobolev) of weak solutions and the understanding of singularity formations. These will in particular involve Schauder theory, regularity theory for Alexandrov and Brenier solutions to the Monge--Ampère equations and many other potential directions.

Mode of operation and evidence of learning

This project will revolve around learning through reading with a focus on the underlying theory, great mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding of the subject matter by solving relevant problems, exploring and constructing examples, investigating theoretical applications of the material, and clearly communicating it in both written and oral formats.

This is planned to be a project in pure mathematics, so there is no plan to have associated numerical or computational component.

Pre- and co-requisites

Analysis III; Partial Differential Equations III; Functional Analysis and Applications IV. These are all essential.

Other useful modules: Stochastic Analysis IV is recommended; Riemannian Geometry IV could also be useful.

References

+ F. Santambrogio, Optimal transport for applied mathematicians, Progr. Nonlinear Differential Equations Appl., 87, Birkhäuser/Springer, Cham, 2015, xxvii+353 pp.

+ C. Villani, Optimal transport, Grundlehren Math. Wiss., 338, Fundamental Principles of Mathematical Sciences, Springer-Verlag, Berlin, 2009, xxii+973 pp.

+ A. Figalli, The Monge--Ampère equation and its applications, Zur. Lect. Adv. Math. European Mathematical Society (EMS), Z¨rich, 2017. x+200 pp.

+ P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhauser Boston, Inc., Boston, MA, 2004.

+ D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 2001. xiv+517 pp.

+ C.E. Gutiérrez, The Monge--Ampère equation, Progr. Nonlinear Differential Equations Appl., 44, Birkhäuser Boston, Inc., Boston, MA, 2001, xii+127 pp.

+ C.E. Gutiérrez, Optimal transport and applications to geometric optics, SpringerBriefs PDEs Data Sci. Springer, Singapore, 2023. x+135 pp.

+ M. Bardi, I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. With appendices by Maurizio Falcone and Pierpaolo Soravia. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1997. xviii+570 pp.

+ M. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67.