Math 295 - Partial Differential Equations

Class is scheduled for MWF 9:00-9:50 am in RH 340N
Office Hours: By appointment.

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 two quarters:

  • 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.
  • Linear Evolution Equations.
  • First Order Nonlinear Equations.
  • Hamilton-Jacobi Equations.
  • Conservation Laws.
  • Fixed-Point Theorems.
  • The Maximum Principle.
  • Spectral Theory of Symmetric Compact Operators
  • Nonlinear Elliptic Equations (sub- and supersolutions, Galerkin method).
  • Variational Methods
  • Viscosity Solutions for Hamilton-Jacobi Equations

Assignments in PDF format
Homework 16, Homework 15, Homework 14, Homework 13, Homework 12, Homework 11,
Homework 10, Homework 9, Homework 8, Homework 7, Homework 6,
Homework 5, Homework 4, Homework 3, Homework 2, Homework 1.

Literature
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 homework and YAQ (You Ask a Question). The latter requires you to send me a question at the end of each week regarding the topics just covered in class. Your final grade will be based on your homework scores and your active participation in YAQ and a research project due by the Spring quater.