# Teaching 2017 / 2018

January to May, 2018, Wednesday / Thursday, 14:30 – 16:00, room 2122, building 9:
I am giving a course on Weak Solutions of PDE (AMCS 232). This will be a hard and difficult course.

The course begins with an introduction to  weak derivatives. Next, we consider Sobolev spaces and fundamental results: extension and trace theorems, Poincare’s inequality and Rellich-Kondrachov theorem. Then, we examine weak solutions of elliptic equations through Lax-Milgram theorem.

A substantial part of the course will involve the students presenting individually some of the lecture material and papers.

Please get a copy of the book Partial Differential Equations by L. C. Evans. You will need it on Wednesday, January 31st.

### Schedule of the presentations

Sections of the book Partial Differential Equations assigned / to be assigned:

Week 1, 04.02:
5.2 – 5.2.2: A. A.
5.2.2 – 5.2.3: W. A.

Week 2, 11.02:
5.2.3 – 5.3: R. C.
5.3 – 5.3.3: M. S.

Week 3, 18.02:
5.3.3 – 5.4 Theorem 1 proof: F. A.
5.4 Theorem 1 proof – 5.5: M. K.

Week 4, 04.03:
5.5 – 5.6: G. S.

After hearing everyone present once, let me say that you can be proud of what you have achieved! Keep up the good work!

5.6 – 5.6 Theorem 2: W. A.
5.6 Theorem 2 – 5.6.2 Theorem 5: R. C.

Week 5, 11.03:
5.6.2 Theorem 5 – 6.1: A. L.

I heard that you are having the mid-term in Athanasios Tzavaras’ class on Thursday, 13:00 – 14:30.
BEST OF LUCK FOR THE EXAM!

Week 6, 18.03:
6.1 – 6.2.1 Theorem 1 proof: M. S.
6.2.1 Theorem 1 proof – 6.2.2 Theorem 3: M. K.
6.2.2 Theorem 3 – 6.2.3 Theorem 5: A. A.

Huge Thank You to Myoungkeun, Muhannad  and  AbdulRahman for the EXTRA EFFORT !

Week 7, 25.03:
For this week ONLY the lecture on Thursday is from 13:00 – 14:30.
6.2.3 Theorem 5 – 6.3.1: G. S.
6.3.1. – 6.3.1. Theorem 2: F. A.

Week 8, 08.04:
6.3.1. – 6.3.1. Theorem 2: F. A.
6.3.2 Boundary Regularity Theorem 4 – 6.3 Theorem 5:  R. C.

Week 9, 15.04:
6.3.2 Boundary Regularity Theorem 4 – 6.3 Theorem 5:  R. C.
6.4  – 6.4.2 Lemma (Hopf’s Lemma): G. S.
6.4.2 Lemma (Hopf’s Lemma) proof – 6.5 (this section includes the proof of Harnack’s inequality, second edition): A. A.

Week 10, 22.04:
6.4.2 Lemma (Hopf’s Lemma) proof – 6.5 (this section includes the proof of Harnack’s inequality, second edition): A. A.
7 – 7.1.2 Theorem 2: F. A.

Week 11, 29.04:
6.4.2 Lemma (Hopf’s Lemma) proof – 6.5 (this section includes the proof of Harnack’s inequality, second edition): A. A.
6.5 – 6.5.2: M. K.
7.1.2 Theorem 2 – 7.1.2 Theorem 5: W. A.
7.1.4 – 7.2 (try to find an intuition for Harnack inequality, omit the proof): M. S.

Week 12, 06.05:
A. A.

For sections longer than 5 pages, be selective

### Instructions for the presentations

1. 30 minutes per presentation
2. understand the material you are going to present
3. if you have questions, google them, ask other students in the class
4. ask yourself: is the notation clear? are all definitions clear? what does the theorem mean? what questions could be asked?
5. present the theorems and the main ideas of the proof
6. only use the whiteboard; you can put information you will refer to repeatedly on a few slides, e.g. you can put a theorem on a slide and refer to it while presenting the proof on the whiteboard. Bring your own laptop if you use slides.
7. use your own hand-written notes
8. practice your presentation

### Office hours

Please make an appointment by email at least 48 hours before

### Feedback

Students can leave anonymous feedback and suggestions for improvements on this google form.