Partial Differential Equations

       Instructor:  Michael Loss, Skiles 214, Tel: (404) 894 2717.  Contact me here.
                         Time and location: MWF 10:05-10:55, Skiles 243
                         Office hours: MW 11-12.

      Text:           Partial Differential Equations, by L.C. Evans,  Graduate Studies in Mathematics,
                          American Mathematical Society, Providence, RI, 1998.
                          ISBN Number: 0-8218-0772-2

                          Here are additional  notes of my own.
 

     Topics:         This is a two semester course.  Although the course is quite mathematical, it should be accessible for an engineer with
                          a mathematical inclination. The first semester is devoted to the study of examples of PDE that
                          can either be solved explicitely and if not, they can be described by other means that makes them accessible.
                          The main examples are first order PDE, Laplace's and Poisson's equation, the wave equation and the heat equation.

                          We start first with simple examples of first order PDE and introduce the method of characteristics
                          for studying those.

                          In the section on Laplace's equation we will  discuss harmonic functions, the mean value theorem
                          Harnack's inequality and energy methods. The Green's function for simple domains will be
                          computed. A similar program will be worked out for the heat equation and the wave equation.
                          We shall be explicit whenever possible.

                          We then return to first order PDE in general and treat as examples the Hamilton-Jacobi equation
                          and get a first glimpse of conservation laws.

                          If time permits I will explain other topics such as homogenization, geometric optics and stationary
                          phase methods.
  
                          Emphasis is placed on the correct mathematical formulation of the problems.  We call a PDE
                         WELL POSED if
                         a) The solution exists
                         b) It is unique
                         c) It depends continuously on the data, i.e., initial conditions or boundary conditions.

                          Note that the word `solution' is not defined. A classical solution, i.e., one where each derivative
                          in the PDE is continuous might exist or it might not. Thus, not every formulation       
                          of a PDE makes sense.  This will be especially important when discussing conservation laws with
                          shocks and entropy conditions,  where we have to formulate the problem in a `weak' fashion in order
                          to allow for non-smooth solutions.

                          Existence, uniqueness and regularity of solutions of linear second order partial differential
                          equations will be discussed in the second semester. This will invlove the machinery of
                          Sobolev spaces and Sobolev inequalities and will be quite involved.
 

    Homework:  You will be asked to solve  some homework problems which will be graded.  Here is the link to the Homework

    Grades:         There will be one test and a final exam. The test counts 20%, the homework will count 40% towards
                          the final grade and the final exam will count 40% towards the final grade.