Partial Differential Equations

       Instructor:  Michael Loss, Skiles 214, Tel: (404) 894 2717.  Contact me here.
                         Time and location: MW 4:35-5:55 pm, Skiles 243
                         Office hours: MW 2-3 pm.

      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.  They will be posted as needed.
 

     Topics:         This is a two semester course.  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.

                          One of the most important ones is 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 continue with first order equations and the method of characteristics. It turns out that first order
                           partial differential equations can be reduced to solving ordinary differential equations. This is a beautiful and
                           classical theory going back to Hamilton and Jacobi. In fact the Hamilton-Jacobi equation will help us to
                          get a first glimpse of conservation laws which we will deal with in some detail.

                          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.  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 uses 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.  It will be posted in due
                         time here:  Homework.

    Here is some old homework including the solutions.
                        

    Grades:         The course grade will be entirely based on homework.