# Lecture 12

Dear Participants,

this week we complete the analysis of Cauchy-Dirichlet problems considering the case when the halfspace is replaced by a bounded and smooth enough domain $\Omega$. The main theorem (Theorem 12.0.5) is the "natural" counterpart of Theorems 8.1.1 and 11.3.1 and its proof relies on these theorems. The strategy to attack the problem consists in strenghtening the boundary and, roughly speaking, replacing one single Cauchy problem with a finite numbers of Cauchy problems in the halfspace and an additional Cauchy problem in the whole space. To these problems we can apply the results in Theorems 8.1.1 and 11.3.1 which lead us to the proof of Theorem 12.0.5. As you will see the geometry of plays a relevant role in all the analysis. Coming back to the concept of "smooth domain" in this lecture we just give the definition but to make it much more familiar we prove in Appendix C (attached to this lecture) a different characterization which is very useful to identify easily smooth domains. Clearly, also more general (homogeneous) boundary conditions can be considered instead of Dirichlet ones as it is remarked in the Notes of this lecture. The Notes tell you also the reason why we do not address this Cauchy problems in the lectures....

This time we kindly ask the team from Wuppertal to produce the official solutions of the exercises of the lectures.

Last but not least, some important information.

1) We plan to upload an updated version of Lecture 10 on wiki in one or two days. Unfortunately that lecture contains a problem in the proof of Proposition 10.1.1 as pointed out by Juergen Voigt. In the discussion board you find a possible different proof of that proposition. We have rather preferred to follow a different (easier) strategy to prove Theorem 10.1.2 but we have taken into account the suggestions in the discussion board to simplify also another proof of the lecture.

2) As it is clear from the previous point, the discussion board is a very powerful tool for discussions/doubts and in some cases it also provides very valuable alternative and easier proofs. A hint: look at the discussion board of lecture 11.....

That's all for the moment. Enjoy the lecture. Your virtual lecturers.

Abdelaziz and Luca.