Course description

This course is trying to explore some essential theories and techiniques in nonlinear analysis. We will also discuss the application in partial and ordinary differential equations, differential geometry and mathematical physics. The main topics will cover linearization, topological degree theory and variational method.

Text book: Methods in Nonlinear Analysis, by Kung-Ching Chang, Springer, ISBN:3-540-24133-7


Time: T & Th 12:00-1:15PM

Location: Gilman 10, Homewood Campus

Office Hour:  T 10:00am-11:30am and Th 3:30-5:00pm or by appointment


Week 1

Differential Calculus in Banach spaces: Frechet derivative and Gateaux derivative.
Implicit function theorem on Banach spaces.

Week 2

Continuity method and its application to nonlinear ellptic equation.

Week 3

Lyapunov Schmidt Reduction and bifurcation, Crandall Rabinowitz theorem, application.

Week 4

Lyapunov Schmidt Reduction, Łojasiewicz-Simon inequality.

Week 5

Brouwer degree. Definition and property.

Week 6

Brouwer degree and its application.

Week 7

Leray-Schauder degree

Week 8

Spring break

Week 9

Global Bifurcation

Week 10

Direct method for variational principle

Week 11

Direct method for variational principle

Week 12

Quasi convexity

Week 13

BV functions and Hardy Space

Week 14

Minimax Method and Mountain Pass Theorem