420-2 - Partial Differential Equations
Monday, Wednesday, Friday 12:00pm - 12:50pm, Lunt 102
Course website: www.math.northwestern.edu/~tosatti/pde.html
Office: Lunt 225
This course is an introduction to the study of second-order elliptic partial differential equations (PDE),
both in domains in Euclidean space and on Riemannian manifolds.Topics that will likely be covered include:
- Harmonic functions
- Sobolev spaces and existence of weak solutions
- Energy estimates
- Schauder estimates
- Lp estimates
- The De Giorgi-Nash-Moser theorem
- Fully nonlinear equations: the Evans-Krylov theorem
- PDE on closed Riemannian manifolds
- The real Monge-Ampère equation
Familiarity with real analysis and some functional analysis.
We will not follow any textbook directly, but the following references might be useful when studying:
- L.C. Evans Partial differential equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
- M. Giaquinta and L. Martinazzi An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Second Edition, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 2. Edizioni della Normale, Pisa, 2013.
- D. Gilbarg and N.S. Trudinger Elliptic partial differential equations of second order Classics in Mathematics. Springer-Verlag, Berlin, 2001.
- Q. Han and F.H. Lin Elliptic partial differential equations, Second Edition, Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.
There will be bi-weekly homework.
This is a tentative syllabus and it is likely to change as the course progresses.
|| Topics Covered
| Jan. 3, 4, 6
|| Basic theory of harmonic functions
Due January 18 in class
| Jan. 9, 11, 13
|| Sobolev and Poincaré inequalities
| Jan. 18, 20
|| Sobolev spaces and weak solutions
Due January 30 in class
| Jan. 23, 25, 27
|| Energy estimates
| Jan. 30, Feb. 1, 3
|| Schauder estimates
Due February 13 in class
| Feb. 6, 8, 10
|| Lp estimates
| Feb. 13, 15, 17
||Interpolation, BMO and John-Nirenberg
Due February 27 in class
|Feb. 20, 22, 24
||The De Giorgi-Nash-Moser theorem and the Evans-Krylov theorem
|Feb. 27, Mar. 1, 3
||The real Monge-Ampère equation
Due March 10 in class
|Mar. 6, 8, 10
||The Hardy space H1 and compensated compactness