4202  Partial Differential Equations
Monday, Wednesday, Friday 12:00pm  12:50pm, Lunt 102
Course website:
www.math.northwestern.edu/~tosatti/pde.html
Instructor:
Valentino Tosatti
Email: tosatti@math.northwestern.edu
Office: Lunt 225
Office hours:
Mondays 11:00am12:00pm
Course Contents:
This course is an introduction to the study of secondorder 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
 L^{p} estimates
 The De GiorgiNashMoser theorem
 Fully nonlinear equations: the EvansKrylov theorem
 PDE on closed Riemannian manifolds
 The real MongeAmpère equation
Prerequisites:
Familiarity with real analysis and some functional analysis.
Textbook:
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. SpringerVerlag, 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.
Grading:
There will be biweekly homework.
Daily Schedule:
This is a tentative syllabus and it is likely to change as the course progresses.
Date 
Topics Covered 
Remarks 
Jan. 3, 4, 6 
Basic theory of harmonic functions 
Homework 1 Due January 18 in class 
Jan. 9, 11, 13 
Sobolev and Poincaré inequalities 

Jan. 18, 20 
Sobolev spaces and weak solutions 
Homework 2 Due January 30 in class 
Jan. 23, 25, 27 
Energy estimates 

Jan. 30, Feb. 1, 3 
Schauder estimates 
Homework 3 Due February 13 in class 
Feb. 6, 8, 10 
L^{p} estimates 

Feb. 13, 15, 17 
Interpolation, BMO and JohnNirenberg 
Homework 4 Due February 27 in class 
Feb. 20, 22, 24 
The De GiorgiNashMoser theorem and the EvansKrylov theorem 

Feb. 27, Mar. 1, 3 
The real MongeAmpère equation 
Homework 5 Due March 10 in class 
Mar. 6, 8, 10 
The Hardy space H^{1} and compensated compactness 
