## MATS424 Viskositeettiteoria, 9 op, kevät 2015

Viscosity theory, 9 cr, spring 2015

This course can be considered as 'Partial differential equations 3'. The theory of viscosity solutions provides a **modern approach to partial differential equations** and extends the classical concept of a solution. The course deals with existence, uniqueness and regularity of viscosity solutions, and introduces connections to optimal control and games.
### Announcements

The course has been graded. To get back your exercises, visit MaD306.
### Schedule

Lectures on Thursdays 10.15-12.00 at MaD381 and Fridays 10.15-12.00 at MaD380. The first lecture is on Thursday 22.1.
### Korppi

Korppi
### Lectures

Lecture notes. (Further material and details during the lectures. Please email typos in the material to the lecturer.)
- Week 1: Vanishing viscosity, definition, examples.
- Week 2: sub- and supersolutions, equivalent definitions, examples
in comparison and uniqueness
- Week 3: comparison, Thm of sums, parabolic case
- Week 4: comparison for p-Laplacian and Perron's existence method
- Week 5: Perron examples, solutions for discontinuous operator, stability, existence through stability principle, uniformly elliptic operators
- Week 6: Pucci operators, Examples of uniformly elliptic operators, ABP-max principle
- Week 7: Proof of Harnack, C^a, C^( 1,a)
- Week 8: Finished C^(1,a), started one controller deterministic optimal control, group discussion on infinity harmonic functions and connections between weak and viscosity theory for div(A(x,Du))=0.
- Week 9: End of one controller deterministic optimal control, started Evans-Krylov C^(2,a)
- Week 10: C^(1,1), C^(2,a)
- Week 11: Ishii-Lions method for Hölder continuity, started Bernstein method
- Week 12: Bernstein method, two player deterministic game

Lecturer
Mikko Parviainen,
mikko.j.parviainen@jyu.fi, MaD306

### Course requirements

Course is passed by solving a sufficient number of exercises, and returning solutions to the lecturer. There will be three exercise sets (about 20-25 problems each). You can return exercises to the lecturer at the lectures, at lecturer's office MaD306 (mailbox outside the office), or you can scan exercises and send them via email.

The course will be graded as follows

50% problems solved -> grade 1

....

90% problems solved -> grade 5

**In addition, at least 5 problems of each set must be solved.**
Prerequisites: PDE1. Also some linear algebra and differential calculus will be required.

### Exercise sets

### Literature

Outline of lectures will appear on the website.
Additional reading

- Koike: Beginners guide to the theory of viscosity solutions
- Caffarelli, Cabre: Fully nonlinear elliptic equations
- Fleming, Soner: Controlled Markov processes and viscosity solutions