*Information about the course*

**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**

**Instructions for the presentations**

- 30 minutes per presentation
- understand the material you are going to present
- if you have questions, google them, ask other students in the class
- ask yourself: is the notation clear? are all definitions clear? what does the theorem mean? what questions could be asked?
- present the theorems and the main ideas of the proof
- 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.
- use your own hand-written notes
- practice your presentation

**Office hours **

**Office hours**

Please make an appointment by email at least 48 hours before

**Feedback**

**Feedback**

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

*Useful Links*

- Multi-index
- General Leibniz rule
- $L^p$-spaces
- Step functions dense in $L^p$
- Translation continuous in $L^p$
- Arzela-Ascoli theorem
- Kronecker Produkt of Matrices (used in A. A. presentation on Hopf’s Lemma and Harnack’s inequality)