Calculus of Variations Math 3020 Fall 2014

Instructor

Piotr Hajlasz
Office: Thaceray Hall 420
Office hours: MWF 1:00-2:00 pm + by appointment.
E-mail: hajlasz@pitt.edu or hajlasz@gmail.com (preferred one)

Textbook

The main material for the course will be contained in my notes.

Calculus of Variations

Calculus of Variations 2

Other useful notes:

Additional reading:

L. C. Evans, Partial differential equations. Second edition.
Graduate Studies in Mathematics, 19. American Mathematical Society.

L. C. Evans, Weak convergence methods for nonlinear partial differential equations.
CBMS Regional Conference Series in Mathematics, 74.

B. Dacorogna, Direct methods in the calculus of variations. Second edition.
Applied Mathematical Sciences, 78. Springer.

We will cover the following topics:

Sobolev spaces
Direct methods of calculus of variations: lower semicontinuity, convexity, quasiconvexity, polyconvexity, rank-one convexity,
Monotone operators,
Ekeland variational Principle,
Mountain Pass Theorem,
sub-supersolution method,
Relaxation,
Compensated compactness,
Concentrated compactness,
Gamma convergence,
Moving plane method,
Linear elliptic equations - existence and regularity up to the boundary.

If the time permits I will also briefly discuss the regularity theory: (Schauder estimates, $L^p$ estimates, Moser and De Giorgi iteration) which leads to solution of one of Hilbert's problems.

Grades

Homework 40% (no late homework is accepted).

Two exams 30% each (dates to be fixed later).

Bonus: optional presentation 20%. The presentation will be held outside the class time.

Bonus: attendance 20%.

Homework

Homework #1 Due date: Monday September 8.

Homework #2 Due date: Monday September 15.

Homework #3 Due date: Monday September 22.

Homework #4 Due date: Monday September 29.