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401-3353-00L 4 Credits DR , MSC D-USYS , D-BAUG , D-MAVT , D-INFK , D-MTEC , D-MATH , D-BIOL , D-GESS , D-ITET , D-ARCH , D-CHAB

Introduction to Viscosity Solutions: Methods and Applications

Lecturers & Examiners: Prof. Dr. Francesca Da Lio
VVZ CR n/a

Last Updated: 2026-02-05 15:14:57

Abstract

We will start by showing some examples of the different applications of the theory. We will then describe the properties of viscosity solutions and explain the methods to get existence and uniqueness results. We shall finally consider in more detail the application of the theory to study ergodic and homogenization problems for fully nonlinear first and second order pde's.

Objective

The aim of the course is to present the basic ideas and the main results of the theory of viscosity solutions.

Content

The aim of the course is to present the basic ideas and the main results of the theory of viscosity solutions. The notion of viscosity solution has been introduced in 1981 by M.G. Crandall and P.L. Lions to solve some problems related to first order Hamilton-Jacobi-Bellman equations and then it was extended to second-order fully nonlinear elliptic (and possibly degenerate) equations. We will start by showing some examples of the different applications of the theory (deterministic and stochastic optimal control problems, front propagations problems, homogenization...). We will then describe the properties of viscosity solutions and explain the methods to get existence and uniqueness results. We shall finally consider in more detail the application of the theory to study ergodic and homogenization problems for fully nonlinear first and second order pde's.

Resources

Literature

[1] M.Bardi, I. Capuzzo Dolcetta: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser, Boston, 1997. [2] Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. Collection "Mathématiques et Applications" de la SMAI, no. 17, Springer-Verlag (1994). [3] Crandall M.G., Ishii, H. and Lions, P.L.: User's guide to viscosity solutions of second order Partial differential equations. Bull. Amer. Soc. 27 (1992), pp 1-67. [4] Evans, L.C: Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. [5] Fleming, Wendell H.; Soner, H. Mete: Controlled Markov processes and viscosity solutions. Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 20 minutes

Course Components

Type Title Time & Place Hours
lecture Introduction to Viscosity Solutions: Methods and Applications. Part 1
  • Mon 15:15-17:00 (HG G 26.1)
2 h weekly

Offered In