Course Description
This is a continuation of MATH 516 (elementary linear PDEs), with the focus now on nonlinear PDEs. We
will discuss variational, topological and other nonlinear methods for nonlinear differential equations,
including the direct method of calculus of variations, mountain pass lemma, concentration compactness,
method of moving planes, degree theory, Crandall-Rabinowitz bifurcation, monotonicity formula, and infinite dimensional
Liapunov-Schmidt reduction method.
Lecture summaries and some references will be available in a public owncloud folder, whose link will be
given to the audience. The link will expire on April 30, 2020.
Prerequisites
MATH 516 or equivalent.
Other relevant materials will be reviewed during the course.
Topics
Here is the
tentative outline. It can be adjusted according to audience background
and interests.
-
The direct methods in the calculus of
variations, constrained minimizations
- Concentration compactness
-
Minimax methods, mountain pass lemma
- Maximum principle and the method of moving planes
- Index theory
-
Crandall-Rabinowitz bifurcation theorems and applications
References
- M. Struwe, Variational methods, applications to nonlinear partial differential equations
and Hamiltonian systems, 4th ed.
This is our main text for the first half of the course. (in owncloud)
-
Chen, Wenxiong; Li, Congming,
Methods on nonlinear elliptic equations.
for the moving plane method. (in owncloud)
-
Lawrence C. Evans, Partial differential equations, AMS, 1998.
- Gilbarg-Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Classics in
Mathematics series. (in owncloud)
- Qing Han and Fanghua Lin, Elliptic partial differential equations, volume 1 of Courant Lecture Notes in Mathematics.
- L. Nirenberg, Topics in nonlinear functional analysis, volume 6 of Courant Lecture
Notes in Mathematics.
- Quittner-Souplet, Superlinear parabolic problems, blow-up, global existence and steady states. (in owncloud)
More references will be added during the term.
Evaluation
The evaluation is based on homework assignments and class participation.
- Homework 1, due ??
- Homework 2, due ??
- Homework 3, due ??
- Homework 4, due ??
- Homework 5, due ??
Instructor and lectures
Instructor:
Dr. Tai-Peng Tsai, Math building
room 109, phone 604-822-2591, ttsai at math.ubc.ca.
Lectures:
Tue Thu 14:00-15:15pm,
Math Annex MATX 1118
Update: Since Jan 17, we meet Mon Wed Fri 11:00-11:50am, MATH 203
Office hours:
TBA, Mon Wed 1:30-2:20pm, and by appointment
(Tsai's schedule).