Partial Differential Equations II (nmma406) --- lecture and exercises 2017, summer term
Exercises
Conditions for zápočet: everyone who delivers three written homeworks obtains zapocet. The homeworks will be set up during the lecture and will also appear on this web page. They must be solved till the following lecture.
Listof homeworks:
19.5. --- solve Homework 13 in script
12.5. --- solve Homework 12 in script (Rondos, Vach)
5.5. --- solve all these problems (Petrasova, Preradova, Outrata, Cechlovsky, Malik, Sommerova, Dolezalova, Silber, Sykora, Rondos, Vach, Kossaczka,Rohal)
28.4. --- solve Homework~10, script, page 11 - I have got no response.
21.4. --- solve this problem (Petrasova, Malik)
14.4. --- prove existence of weak solution of a hyperbolic equation of second order (Preradova, L. Vacek, Matajova, Sommerova, P. Vacek, Rohal)
7.4. --- prove that composition with a \(C^1\) Lipschitz function is continuous from \(W^{1,2}_0(\Omega)\) into itself or show approximation propertiy in \(D(I, W^{1,2}_0(\Omega))\) (Petrasova, Kossaczka, Silver, L. Vacek)
31.3. --- prove Lemma 27 or Lemma 28
24.3. --- prove Lemma (25) or Lemma (18) (Malik, Preradova, Sykora, Cechlovsky, Vach, Raumer, Sommerova, Dolezalova, Outrata)
17.3. --- prove Lemma (22) or Lemma (18) (Raumer, Outrata)
10.3. --- prove Lemma (18) --- \(\Phi\)
3.3. --- prove Theorem (11) (Outrata, Silber, Raumer, Kossaczka, Matajova, L. Vacek)
24.2. --- prove Corollary (5) and Lemma (6) (Outrata, Sykora, Matajova, Raumer, P. Vacek, Rondos)
(The number in (brackets) refers to the numbering presented in the lecture. In script of the lecture these numbers are in the same rounded brackets.)
Lecture
Official description of the lecture is here. Here is a more detailed sylabus
Exams:
The exam will have only oral part. At the beggining you will be asked to state one definition and two theorems and prepare their proofs. You will have time to prepare and write it down. Then you will present the statements and we will discuss their proofs.
The knowledge will be required in the extent presented in the lecture. When deciding about the best evaluation, the ability of proving new theorems will help.
Recommended literature:
Script of the lecture. Please use it with caution. It may contain mistakes. If you find some, please let me know.
L.C.Evans: Partial Differential Equations, AMS, 2010.
J. Diestel, J.J.Uhl: Vector Measures, AMS, 1977.
H. Gajewski, K. Groeger, K. Zacharias: nichtlineare Operator-gleichungen und Operatordifferential-gleichungen, Akademie-Verlag, Berlin, 1974.
A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
M. Renardy, R.C. Rogers: An introduction to Partial Differential Equations, Springer, 1993.
Preliminary knowledge:
The course PDE I
Basic courses of Mathematical Analysis, especially semestr 4, nmma101, nmma102, nmma201, nmma202,
Measure and Integration Theory - NMMA203,
Introduction to Functional Analysis - NMMA331, web page of doc. Kalenda,
Functional Analysis 1 - NMMA401, web page of doc. Kalenda,