Partial Differential Equations I (nmma405) --- lecture and exercises 2016, winter term

Exercises

Conditions for zápočet: everyone who delivers a 20 minutes presentation during the course and solves one homework obtains zapocet.


List of presentations:

(20.1.)
1) W^{2,2}_{loc} regularity for a linear PDE [E, Section 6.3.1] --- mainly Theorem 1 (Betül ÖZBAY, Martin Sýkora)
2) W^{2,2}_{loc} regularity for minimizers of variational functionals [E, Section 8.3.1] --- mainly Theorem 1, part i) (Javier Bosch)
3) W^{2,2}_[loc} regularity for weak solutions of nonlinear PDE's [E, Section 9.1, Remark below Theorem 4] (Marta Kossaczka)
4) Tr(u-k)^+=0 on \partial \Omega if k is large enough and u has a bounded trace (Rondoš, )
5) if L is convex in z and p, then any solution of EL equation is a minimizer (Cechlovsky)
(13.12.) Sommerova: Fredholm Theory
(6.12.) Matajova, Vach: Serrin's example of nonuniqueness of a solution to the elliptic problem
(29.11.) Hruška: Jensenova nerovnost v Lebesgueových prostorech
(29.11.) Doležalová: [Evans, str. 266, Section 2.6.1, Remark ii]
(22.11.) Vacek: [Evans, Section 5.8.2, Theorem 3, Part (i)]
(22.11.) Silber: [Evans, Section 5.8.2, Theorem 3, Part (ii)]
(8.11.) Stará: [Gilbarg, Trudinger, Lemma 7.14] s důkazem věty o vnoření za ní
(8.11.) Malik: [Gilbarg, Trudinger, Lemma 7.12]
(1.11.) Bárta: [Evans, Remark at the end of Section 5.4], extensions of W^{2,p} functions
(25.10.) Preradová: [Adams, Theorem 3.8] dual spaces to Sobolev spaces
(11.10.) Outrata: [Evans, Section~5.2.2, Example 3], in case of sufficient time also [Example 4]
(11.10.) Kurnas: [Ziemer, 1.6] repetition of regularization without proofs, [Ziemer, Lemma 2.1.3]


List of homeworks:
... is here.

Please prepare solutions of the problems carefully with as many details as you can. I prefer output in LaTeX but hand written notes are also possible if I am able to read it. I do not require that you finish your homework before you pass the exam but please note that without the homework you cannot obtain credits for the lecture. I also want to point out that preparation of the homework probably takes some time.

In case of problems with solving the homeworks please do not hesitate to contact me.

The homeworks were associated to students by a generator of random numbers in OpenOffice.org Calc.

Lecture

Official description of the lecture is here. Here is a more detailed sylabus


Exams:

Exam terms are January 17, 24, February 1, 7. There will be two more exam terms in summer term. The first at its beginning and the second at its end. The exact dates will be fixed according to the summer term schedule. I will not organize any further exam terms. You need to subscribe for the exam in the Student Information System (SIS).
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. I try to prepare a script of the lecture as soon as possible. It is here. Please use it with caution. It may contain mistakes.


Recommended literature:

L.C.Evans: Partial Differential Equations, AMS, 2010.
D. Gilbarg, N.S.Trudinger: Elliptic Partial Differential Equations of Second Order, Springer, 2001.
M. Renardy, R.C. Rogers: An introduction to Partial Differential Equations, Springer, 1993.
R.A. Adams, J.J.F. Fournier, Sobolev spaces, Elsevier, 2005.
W.Z. Ziemer, Weakly differentiable functions, Springer, 1989.
Notes of lecturer are here. Please be carefull when reading it. It may contain misprints. I t is safer to use textbooks mentioned above.


Preliminary knowledge:

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,