Partial Differential Equations II

Department: 
MATH
Course Number: 
6342
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Usually every spring semester

This course covers the general mathematical theory of linear stationary and evolution problems plus selected topics chosen on the instructor's interests.

Prerequisites: 

MATH 6341

Course Text: 

At the level of Partial Differential Equations, Evans, Vol. 19, AMS/Providence

Topic Outline: 
  • Sobolev Spaces and Basic Functional Analysis
    • Sobolev spaces
    • Compactness
    • Sobolev inequalities
    • Trace theorems
  • Linear Elliptic Equations
    • Weak solutions, distributions
    • Variational methods
    • Existence, Lax-Milgram lemma, Garding inequality
    • Interior and boundary regularity
    • Maximum principles
    • Fredholm alternative
    • Eigenvalue problems
    • Integral transforms
  • Linear Evolution Equations
    • Weak solutions, distributions
    • Existence and regularity
    • Maximum principles
    • Energy methods
    • Stability and propagation of perturbations
    • Semigroups
    • Integral transforms
  • Miscellaneous Topics
    • Quasilinear equations
    • Fixed point methods
    • Methods based on monotonicity and comparison principles
    • Geometric properties of solutions
    • Conservation laws