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