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Nonlinear Parabolic Partial Differential Equations
| Professor: | Prof. Dr. H. Mete Soner | Lectures: |
Tuesday 15-17 HG F 5 Thursday 15-17 HG E 3 |
| Coordinator: | Albert Altarovici | Exercises: | Monday 16-17 HG G 26.1 |
First Lecture: Tuesday, February 28, 2012, Week 2 of the semester.
Start of Exercises: March 05, 2012
Conditions to obtain the certificate (Testat): 2/3 of all exercises reasonably tackled.
Syllabus: Click here for a copy of the syllabus
Content
We will first study linear elliptic and parabolic equations and develop a concise existence, uniqueness and regularity theory. Then, weak-viscosity solutions for fully nonlinear equations will be developed. Connections to stochastic processes will also be developed. The tentative lecture schedule given below provides a list of the topics that will be covered during the semester.
References
We will use parts of the following book quite closely:
- "Partial Differential Equations", L.C.Evans, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society
The following books are also useful:
- "Elliptic Partial Differential Equations of Second Order", D. Gilbarg & N.S. Trudinger, Springer
- "Linear and Quasilinear Equations of Parabolic Type", O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Translations of Mathematical Monographs, Volume 23, American Mathematical Society
- "Nonlinear Elliptic and Parabolic Equations of the Second Order", N.V. Krylov, Kluwer
Examination and ECTS Points
Session examination, oral 30 minutes. 10 ECTS Points.
Tentative Schedule
| Lecture Number | Date | Topic |
| Lecture 1 | 21.02 | No Class |
| Lecture 2 | 23.02 | No Class |
| Lecture 3 | 28.02 | 2.2 Ev. Laplace Equation, Fundamental Solution, Maximum Principle |
| Lecture 4 | 01.03 | 2.2 Ev. Laplace Equation, Green's function, Representation, Spectrum |
| Lecture 5 | 06.03 | 2.3 Ev. Heat Equation, Introduction, Existence, Maximum Principle, Mean Value |
| Lecture 6 | 08.03 | 2.3 Ev. Heat Equation, Smoothness, Estimates |
| Lecture 7 | 13.03 | 6.2.1 Ev. Elliptic Equations, Lax-Milgram |
| Lecture 8 | 15.03 | 6.2.3, 6.2.5 Ev. Elliptic Equations, Fredholm Alternative, Spectrum |
| Lecture 9 | 20.03 | 7.1.1 Ev. Parabolic Equations, Weak Existence |
| Lecture 10 | 22.03 | 7.1.2 Ev. Parabolic Equations, Galerkin Method |
| Lecture 11 | 27.03 | 7.1.3 Ev. Parabolic Equations, Regularity |
| Lecture 12 | 29.03 | Notes: Probabilistic representation |
| Lecture 13 | 03.04 | Notes: Probabilistic representation |
| Lecture 14 | 05.04 | Viscosity solutions |
| Easter Break | 06.06-15.04 | |
| Lecture 15 | 17.04 | Generalized viscosity solutions, Barles-Perthame procedure |
| Lecture 16 | 19.04 | Uniqueness of viscosity solutions, comparison principle |
| Lecture 17 | 24.04 | No class |
| Lecture 18 | 26.04 | No class |
| Labor Day | 01.05 | No class |
| Lecture 19 | 03.05 | Comparison principle cont'd, viscosity framework for 2nd order equations |
| Lecture 20 | 08.05 | Comparison for 2nd Order Equations, Crandall-Ishii Lemma. |
| Lecture 21 | 10.05 | Jensen's Lemma, Aleksandrov's Maximum Principle |
| Lecture 22 | 15.05 | Convergence of approximation schemes for fully nonlinear 2nd order equations |
| Ascension Day | 17.05 | No class |
| Lecture 23 | 22.05 | 10 Ev. HJB Equations |
| Lecture 24 | 24.05 | 10 Ev. HJB Equations |
| Lecture 25 | 29.05 | 10 Ev. HJB Equations |
| Lecture 26 | 31.05 | 10 Ev. HJB Equations |
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