MA8105 Nonlinear Partial Differential Equations and Sobolev Spaces
Spring 2019
Welcome to MA8105 !
 What is this course about?
 Mathematical methods and structures that are fundamental for the study of partial differential equations (PDEs), variational calculus, numerical methods etc.
 Main focus on analytical tools: Functional analysis, \(L^p\) and Sobolov spaces, compactness, modes of convergence, distributions, error estimates.
 13 weeks on applications to linear and nonlinear PDEs.
 Who can take this course?
 Interested students at Master or PhD level.
 The level should be suitable for good 4th year students in the industrial mathematics program.
 It can be taken as a regular course or a 'fordypningsemne'.
 Very relevant for students specialising in PDEs, optimization, analysis, numerical analysis, or probability.
 Prerequisites:
 Elementary functional analysis, equivalent to TMA4145 Linear Methods.
 Some real analysis (Lebesgue integration theory).
 It is advantageous, but not necessary, to have some background in partial differential equations.
Messages
08.04.2019  Exam dates  See "General information" in the left menu 
Office hours  See table below 

My notes  Some of them are posted under "Lectures" in the left menu. These are the notes for the lectures that deviated substantially from Holden. I strongly recommend you look at them. 
Office hours
Date  Time  Room 

Monday 13.05.  14:1515:00  Room 1148, SB2 
Friday 24.05.  14:1516:00  Room 1148, SB2 
Tuesday 28.05.  09:3011:00  Room 1148, SB2 
05.04.2019  Last lecture  Monday April 8 No lecture on Friday April 12 
Exam  We fix a date or two for the oral exam in the last lecture. Updated info under "General information". 

Reference group  We take a last short meeting on Monday. I need a volunteer to write a (possibly very!) short report on behalf of the students and send it to me. 

Final curriculum  See "General information" in the left menu. 

Info about lectures  What did we do + several copies of my lecture notes, see "Lectures" in the left menu. 

Solutions  Solutions to all exercises available at under "Exercises" in the left menu 

03.04.2019  Application  The porous medium equation, existence of weak solutions Convergence of a finite difference approximation Apriori estimates and compactness. 
Exercise 12 and 13  Available at under "Exercises" in the left menu.  
Solutions 10 and 11  Available at under "Exercises" in the left menu.  
22.03.2019  No lecture on Friday March 29  I am traveling. 
Exercise 11  Available at under "Exercises" in the left menu. Volunteer wanted… 

Solutions 8 and 9  Available at under "Exercises" in the left menu.  
15.03.2019  3 lectures Week 12  One extra lecture since the week after will only have one lecture (Monday) 
Extra lecture Thursday March 21  Room 656 (SIMA stuen) Sentralbygg 2 One of 3 lectures Week 12. Replaces the Friday lecture in the week after. 

Exercise 10  Available at under "Exercises" in the left menu. Volunteer wanted… 

Solutions 7  Available at under "Exercises" in the left menu.  
06.03.2019  No lecture Friday March 8  I will be travelling 
Exercise 9  Available at under "Exercises" in the left menu.  
Solutions 6  Available at under "Exercises" in the left menu.  
Voluteers  Exercise 8: Alexandra, Exercise 9: Max  
25.02.2019  Lecture Thursday March 7  Room 656 sentralbygg 2 (SIMA stuen) Instead of the lecture Friday March 8 
Exercise 8  Updated! Available at under "Exercises" in the left menu.  
Solutions 5  Available at under "Exercises" in the left menu.  
20.02.2019  Exercise 7  Available at under "Exercises" in the left menu. Volunteer wanted… 
Solutions 4  Available at under "Exercises" in the left menu.  
11.02.2019  No lecture Monday 18.02.  I will be away that day. 
Extra lecture Thursday 14.02  Auditorium F4 (2nd floor Gamle Fysikk) To catch up for missing Monday lecture. 

Exercise 6  Available at under "Exercises" in the left menu. Volunteer wanted…  
Solutions 3  Available at under "Exercises" in the left menu.  
Volunteer HW 5  Helge  
Reference group  = the whole class. First meeting Monday 11.02. in the break. Conclusions, see "Reference group" in left menu. 

04.02.2019  Alternative lecture time  Thursday 14:1516:00 Some weeks I will travel, then we will use this time. 
Exercise 5  Available at under "Exercises" in the left menu.  
Solutions 2  Available at under "Exercises" in the left menu.  
Voluteers  Exercise 3: Morten, Exercise 4: Max  
28.01.2019  Exercise 3 and 4  Available at under "Exercises" in the left menu. 
Solutions 1  Available at under "Exercises" in the left menu.  
19.01.2019  New room Mondays  New room: B23, Berg, 4th floor. Directions can be found here: Maze Map 
Monday Jan 21  We meet in room 822 at 14:05, to go all together to B23. 

Exercise 2  Available at under "Exercises" in the left menu.  
Problem sessions  Fridays 09:1510:00, room 656 SB2 Student run. One student take special responsibility to look at all problems in advance. You may ask me for help in the breaks in the lectures or at my office hours. 

Office hours  Monday 16:1517:00 room 1148  
15.01.2019  Corrections  Here is the corrected proof of Proposition 23 in Lecture Monday Jan 14. 
Exercise 1  Available at under "Exercises" in the left menu.  
Lectures  Monday 14:1516:00, room B23 (Berg, 4th floor) Friday 10:1512:00, room S21 (SB2, 2nd floor) 

11.01.2019  First Lecture  
Lecture note  Helge Holden: Tools from the Toolbox. Functional Analysis for PDEs You will get a free copy on the condition that you do not copy it or pass it on. 

12.11.2018  News  Lecturer 2019: Espen R. Jakobsen Curriculum and lecture plan: Similar to 2017, but some changes 
Info meeting Tuesday 08/01/2019 15:1516:00 room 734, SB2 (7th floor)  If you are interested in this course, you should participate:  Outline of course.  Determine lecture times.  Register for course email list (compulsory).  Can not attend? Send me an email with your preferred lecture times. 
Mon  Tue  Wed  Thur  Fri  

08:15  09:00  
09:15  10:00  
10:15  11:00  Lecture Room S21  
11:15  12:00  
12:15  13:00  
13:15  14:00  
14:15  15:00  Lecture Room 822 (SB2)  
15:15  16:00 