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401-3353-08L 4 Credits DR , MSC D-USYS , D-MTEC , D-BAUG , D-MAVT , D-INFK , D-MATH , D-BIOL , D-ERDW , D-GESS , D-ITET , D-CHAB

Introduction to Viscosity Solutions: Methods and Applications. Part 2

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

Last Updated: 2026-02-05 15:29:50

Abstract

In this course I introduce the notion of viscosity solution for second order fully nonlinear elliptic and parabolic PDEs. We present the techniques and methods to get uniqueness and existence results in the second order case. In particular we give the proof of two fundamental results: the Jensen's Maximum Principle and the so called Ishii's Lemma.

Content

In this course I introduce the notion of viscosity solution for second order fully nonlinear elliptic and parabolic PDEs. We present the techniques and methods to get uniqueness and existence results in the second order case. In particular we give the proof of two fundamental results: the Jensen's Maximum Principle and the so called Ishii's Lemma. As an application we describe the level set approach to geometric evolution of hypersurfaces in the all space. We prove some properties of this weak evolution and we show the agreement with the classical flow when the fronts are smooth.

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 2
Monday, March 3 according to the old schedule 15:15 to 17:00 in HG D 1.1
  • Mon 13:15-15:00 (CAB H 52)
  • 03.03 Date 15:15-17:00 (HG D 1.1)
2 h weekly

Offered In