Schedule
The class schedule below will be updated on a weekly basis. The class will mainly follow the lecture notes below, which will be periodically updated.
Calder, Jeff. Lecture notes on viscosity solutions. [pdf]. (Updated Nov 2018)
Lecture | Date | Topic | Notes |
1 | Sep 5 | Motivational examples | Slides |
2 | Sep 7 | Motivational examples | |
3 | Sep 10 | Motivation and definitions | Slides |
4 | Sep 12 | Motivation and definitions | |
5 | Sep 14 | Comparison principle | Slides |
6 | Sep 17 | Comparison principle | |
7 | Sep 19 | Hamilton-Jacobi-Bellman Equation | Slides |
8 | Sep 21 | Hamilton-Jacobi-Bellman Equation | HW 1 due |
9 | Sep 24 | Method of vanishing viscosity | Slides |
10 | Sep 26 | Method of vanishing viscosity | |
11 | Sep 28 | Semiconcavity | |
12 | Oct 1 | Boundary conditions in viscosity sense | Slides |
13 | Oct 3 | Boundary conditions in viscosity sense | |
14 | Oct 5 | The Perron method | HW 2 due |
15 | Oct 8 | The Perron method | Slides |
16 | Oct 10 | Inf/Sup Convolutions | Slides |
17 | Oct 12 | Inf/Sup Convolutions | |
18 | Oct 15 | Finite difference schemes | Slides |
19 | Oct 17 | Finite difference schemes | |
20 | Oct 19 | Finite difference schemes | HW 3 due |
21 | Oct 22 | Finite difference schemes | |
22 | Oct 24 | Finite difference schemes | |
23 | Oct 26 | Homogenization | Slides |
24 | Oct 29 | Homogenization | |
25 | Oct 31 | Discontinuous coefficients | Slides |
26 | Nov 2 | Discontinuous coefficients | HW 4 due |
27 | Nov 5 | Second order equations | Slides |
28 | Nov 7 | Second order equations | |
29 | Nov 9 | Second order equations | |
30 | Nov 12 | Second order equations | |
Nov 14 | No class | ||
31 | Nov 16 | Second order equations | HW 5 due |
32 | Nov 19 | Second order equations | |
33 | Nov 21 | Second order equations | |
Nov 23 | Thanksgiving | ||
34 | Nov 26 | Carter Chain: Pegden, Wesley, and Charles K. Smart. "Convergence of the Abelian sandpile." Duke Mathematical Journal 162.4 (2013): 627-642. | |
35 | Nov 28 | Drisana Mosaphir: Zhao, Hongkai. "A fast sweeping method for eikonal equations." Mathematics of computation 74.250 (2005): 603-627. | |
36 | Nov 30 | Shiqiang Xia: Oberman, Adam M. "A convergent monotone difference scheme for motion of level sets by mean curvature." Numerische Mathematik 99.2 (2004): 365-379. | |
37 | Dec 3 | Zanbing Dai: Manfredi, Juan, Mikko Parviainen, and Julio Rossi. "An asymptotic mean value characterization for p-harmonic functions." Proceedings of the American Mathematical Society 138.3 (2010): 881-889. | |
38 | Dec 5 | Brendan Cook: Levine, Lionel, Wesley Pegden, and Charles K. Smart. "Apollonian structure in the Abelian sandpile." Geometric and functional analysis 26.1 (2016): 306-336. | |
39 | Dec 7 | Amber Yuan: Ishii, Hitoshi, and Paola Loreti. "Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems." SIAM Journal on Mathematical Analysis 37.2 (2005): 411-437. | |
40 | Dec 10 | Wenjie Lu: Evans, Lawrence C., H. Mete Soner, and Panagiotis E. Souganidis. "Phase transitions and generalized motion by mean curvature." Communications on Pure and Applied Mathematics 45.9 (1992): 1097-1123. | |
41 | Dec 12 | Montie Avery: Evans, Lawrence C., H. Mete Soner, and Panagiotis E. Souganidis. "Phase transitions and generalized motion by mean curvature." Communications on Pure and Applied Mathematics 45.9 (1992): 1097-1123. |