Tutorial: Partial Differential Equations (SS 2004)

Tutorial Times

Day	Time	Room	Start	End
Friday	10:00-11:30	48-210	May, 7th	July, 23rd
Monday	8:15-9:45	48-208	May, 10th	July, 26th

There are two tutorial groups. Each participant should visit one.

Organization of the Tutorials

Tutorials and exercise sheets consist of a homework part and a tutorial/discussion part. The homework part (exercises 1,2,3,...) should be worked on in groups and submitted to office 48-558. Solutions will be corrected and discussed in the tutorials. The tutorial/discussion part (exercises A,B,C,...) should not be submitted. It is to be worked on in the tutorials and for private practice outside of any lectures and tutorials.

Exercise Sheets

Exercise Sheet 1 (return till 12.5.)
Exercise Sheet 2 (return till 18.5.)
Exercise Sheet 3 (return till 25.5.)
Exercise Sheet 4 (return till 8.6.)
Exercise Sheet 5 (return till 15.6.)
Exercise Sheet 6 (return till 28.6.)
Exercise Sheet 7 (return till 29.6.)
Exercise Sheet 8 (return till 7.7.)
Exercise Sheet 9 (return till 14.7.)
Exercise Sheet 10 (return till 22.7.)

Material

We recommend to use the Lecture Notes on Analytical Methods for Partial Differential Equations (by H. Neunzert and J. Struckmeier) to accompany the lecture (you should also use many other good books on partial differential equations). The notes can substitute neither lecture nor tutorials. The lecture might vary from the notes.
A brief writeup on the Method of Characteristics, which motivates and derives the method for general nonlinear first-order PDE. The method is then applied to the special case of quasilinear systems. The cooking recipe how to apply the method step by step is given along an example.

Comments on the Exercises

Ex 2: Note that appropriate boundary conditions have to be provided such that conservation of energy is valid.
Ex D (iv): In order to guarantee unqiueness, the domain can be restriced in order to obtain a solution which is continuous w.r.t. the initial data.
Ex 7: The case ρ≥1 is slightly tricky. It is perfectly fine to restrict to the case ρ<1 in the first place.
Ex 16 (iii): Here is a plot of the one dimensional Green function on the interval (0,1):
Ex 19: Here you can see a two dimensional H¹ function which has a pole at the origin:
Ex 21: Here is a function which is harmonic inside the 2d unit ball apart from the origin. It is obtained when applying the Kelvin transform to the function u(x,y)=x outside of the unit ball.
Ex 22: The Tikhonov example of non uniqueness of the heat equation on the whole IR. Plotted is a solution to zero initial data, which is not analytic in the origin.