Viscosity Solutions (WS 24/25)

Basic information

Lecture: Dietmar Gallistl
Friedolin: Lecture in Friedolin
There will be an oral exam over 3 ECTS points.
The course can be booked as either "applied" or "pure" mathematics module for the MSc.

Topic: Theory and (if time allows) numerical methods for viscosity solutions to elliptic PDEs
This is a course for students interested in PDE theory and numerical methods. There are connections to stochastic control theory. We will focus on second-order elliptic linear or nonlinear problems in nondivergence form. In the simplest example of a linear equation
∑a_jk∂²_jku=f
the theory of weak solutions in Sobolev spaces cannot be employed when the coefficient a is not differentiable because in this case the intergration-by-parts formula is not available. In the course we will explore an alternative solution theory based on the notion of viscosity solutions.

[CC] L.A. Caffarelli, X. Cabré. Fully nonlinear elliptic equations Amer. Math. Soc. Colloq. Publ., 43 American Mathematical Society, Providence, RI, 1995.
[CIL] M.G. Crandall, H. Ishii, P.-L. Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., New Ser. 27, No. 1, 1-67 (1992). pdf file
[EG] L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. CRC Press, Boca Raton, 2015.
[GT] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
[K04] Shigeaki Koike. A beginner’s guide to the theory of viscosity solutions. Mathematical Society of Japan, Tokyo, 2004. pdf file
[K15] Nikos Katzourakis. An introduction to viscosity solutions for fully nonlinear PDE with applications to calculus of variations in L∞ . SpringerBriefs in Mathematics. Springer, Cham, 2015. free preprint copy from arXiv
[G] Chapter 2 of my old lecture notes on a (different) course on viscosity solutions pdf file of the lecture notes

Problem solving: Exercises are issued each Tuesday. The table below displays the exercises to be discussed in the respective week (e.g., the exercises formulated in week 15 can be found in the row for week 16).
Legend:
cyan=lecture

yellow=discussion of a problem sheet (and possibly lecture)

magenta=self-study

grey=no class

Week	Friday	Exercises	Syllabus	References
42	18 Oct		definition of viscosity solutions	[G] §1 and §4
43	25 Oct		jets and semicontinuous envelopes	[G] §5
44	1 Nov		jet closures; Dirichlet problem	[G] Defs. 2.32, 2.38, 2.39
45	8 Nov	Sheet 1
46	15 Nov	Sheet 2
47	22 Nov		comparison principle, Perron's method	[G] §6
48	29 Nov 8-10		sufficient criteria for comparison	[G] §7
48	29 Nov 10-12		semiconvex functions	[G] §8
49	~~6 Dec~~			no class; shifted to week 48
50	13 Dec 8-10		proof of Ishii's lemma	[G] §9
50	13 Dec 10-12	Sheet 3
51	~~20 Dec~~			no class; shifted to week 50
2	10 Jan	Sheet 4
3	17 Jan		Alexandrov's thm	Evans&Gariepy Thm. 6.9
4	24 Jan		Alexandrov's thm
5	31 Jan		Alexandrov's thm
6	7 Feb		Q&A