Webrelaunch 2020

The exercises will be solved on Wednesday from 14:30. They will be uploaded on the Friday, the week before they will be solved.

Exercise sheet 1
Subjects: isolated systems, properties of solutions of initial boundary value problems and rotational invariance of the Laplace operator.
Solution sheet 1

Exercise sheet 2
Subjects: harmonic functions and Newtonian potential
Solution sheet 2

Exercise sheet 3
Subjects: spherical symmetric solutions of the poisson equation and the Green's function in the first quadrant.
Solution sheet 3
Correction: In exercise 4 it is not possible to get the uniqueness of the solution for the given assumptions. Therefore I changed the exercise and one now just has to show that u is a solution.

Exercise sheet 4
Subjects: Newtons theorem and an application, a part of the proof of Thm 1.17 and uniqueness of the boundary value problem for specific elliptic differential operators.
Solution sheet 4

Exercise sheet 5
Subjects: Uniqueness of Green's function, a modification of Theorem 2.4 and applications of the weak maximums principle.
Solution sheet 5

Exercise sheet 6
Subjects: A property of spherical symmetric functions, the Heat equation with L^1 initial condition and the Heat equation on the half line with Neumann boundary conditions.
Correction: The integration domain in Exercise 3 was changed, now it makes sense.
Solution sheet 6
Remark: I may have said Dirichlet boundary conditions in the exercise session for Exercise 3, that was wrong.

Exercise sheet 7
Subjects: Heat equation on half-line with Dirichlet boundary conditions, Parts of the proof of Theorem 3.7, non-uniqueness of the Heat equation.
Solution sheet 7

Exercise sheet 8
Subjects: The damped string, the clamped plucked string and another problem that can be solved using seperation of variables
Solution sheet 8
Remark: I was made aware of a BIG mistake in Exercise 1, see Exercise sheet 14 for more details

Exercise sheet 9
Subjects: The solution formula for the transport equation and for the homogeneous and inhomogeneous wave equation
Solution sheet 9
Remark: Due to time constraints we did not discuss the solution of Exercise 4 and 5, this will be done next time.

Exercise sheet 10
Subjects: Inhomogeneous 1-dim and homogeneous 2-dim wave equation plus concrete calculations of solutions
Correction: There were some typos in the problem sheet.
Solution sheet 10
Remark: Disregard the incomplete solution for Exercise 3 from the exercise session. A complete and corrected solution is given here.

Exercise sheet 11
Subjects: Fourier transform on Schwartz space
Solution sheet 11
Remark: A corrected Version has been uploaded.

Exercise sheet 12
Subjects: Seminorm and metric on Schwartz space and Classification of PDEs
Solution sheet 12

Exercise sheet 13
Subjects: Uniqueness of weak limits and Minkowski's integral equation
Solution sheet 13

Exercise sheet 14
Subjects: Damped string again, Telegraph equation, Non-compact identity map and assertions about W_0^{1,p}
Solution sheet 14