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Math 295A - Partial Differential Equations

Class is scheduled for MW 2:30-3:50 pm in RH 340N
Office Hours: MW 10-10:50 am.

Course Description
Partial Differential Equations are a multifaceted subject with several and deep connections to other areas of mathematics, such as applied mathematics, functional analysis, harmonic analysis, differential geometry, mathematical physics, ... It shouldn't therefore come as a surprise that a wide range of methods and techniques have been developed for their treatment. This course is intended to be an introduction and an overview highlighting the diverse aspects of PDEs. The following topics will be covered during the first quarter:

  • Introduction and Motivation.
  • Basics of the theory of Distributions.
  • Fundamental Solutions.
  • Sobolev Spaces and Trace Theorems.
  • Weak Solvability Theory of Uniformly Elliptic Boundary Value Problems.
  • Elliptic Regularity Theory.

Assignments in PDF format
Assignment 1, Assignment 2, Assignment 3, Assignment 4, Assignment 5,
Assignment 6.

Literature
As we progress through the material, I will send out typed notes. All topics covered in class are also contained in

  • L. C. Evans, Partial Differential Equations , Graduate Studies in Mathematics Vol. 19, AMS 1998.
  • M. E. Taylor, Partial Differential Equations, Volumes 1 and 3, Springer 1996.
but other references can be more informative for specific topics:
  • R. A. Adams, Sobolev Spaces, Academic Perss 1975.
  • R. Dautray, J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. II, Springer 1990.
  • D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.
  • M. F. John, Partial Differential Equations, Springer IV Ed 1982.
  • J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, II, III, Springer 1972/73.
  • D. Mitrovic, D. Zubrinic, Fundamentals of Applied Functional Analysis , Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman 1998.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.
  • K. Yosida, Functional Analysis, Springer 1980.

Exams and Grading
Homework will be assigned weekly in the form of written assignments. Your final grade will be based on your participation in class and in the homework.