420-2 - 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 11:00am-12:00pm

Course Contents:

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:

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:

Grading:

There will be bi-weekly 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 Lp estimates
Feb. 13, 15, 17 Interpolation, BMO and John-Nirenberg Homework 4
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 Homework 5
Due March 10 in class
Mar. 6, 8, 10 The Hardy space H1 and compensated compactness