SF3625 HT18-1 Partial Differential Equations

SF3625 HT18-1 Partial Differential Equations

The exam will be on Friday the 18th from 8-12 in Room 3418 (the usual room). 

Study Guide: List of the most important results in the course and what you should read carefully before the exam.

The third and final set of homework can be found here (due January 11th).

The Final lecture will be on Friday the 14th in F11 at 8:15. (sorry about the time, but it was the only I could find)

Second set of homework (due 7th December) can be found here: Homework_2_SF3625-2.pdf

First set of Homework can be found here: Homework_1_SF3625.pdf. It is due on the 18th of October (before midnight).

Examination: The role of the examination will be to make sure that you read through the course book once and thinks through the material. There will be three sets of homework during the course, these are mandatory and will be marked pass/fail.

The final exam will consist of 4 problems. Two of these problems will be taken from your homework sets (or slight variations of homework problems). This means that you will benefit from solving the more difficult homework problems (since you will get those marked). The final exam is also pass/fail and the idea is that everyone that tries hard and reads the book will pass - however no one will pass without knowing some basic PDE theory.



Time: Thursdays 13:15-15:00 from September 6th to December 20th (except October 25th).

Place: Room 3418 in the mathematics department.

Course book: Lawrence C. Evans "Partial Differential Equations".

Brief course description: We will cover the basics of the theory of partial differential equations (PDE) at graduate level. The course will more or less be split into five parts. The first part will cover potential theory methods (Newtonian kernels, Green's functions et.c) for the three big constant coefficient partial differential equations: the Laplace, heat and wave equations. These equations are also covered in most masters courses in PDE - so you may view them as repetition or you may feel confident that you can follow this course even if you haven't studied PDE at masters level.

In the second part of the course we will cover the fundamentals of Sobolev spaces. Sobolev spaces are spaces of functions that have derivatives in an appropriate weak sense which makes them ideal for applying functional analytic methods to study PDE.

In the third and fourth part of the course we will use Sobolev spaces to study more general elliptic and evolution equations.

The final part of the course will be a brief introduction to the calculus of variations.

Lecture plan: (the plan is very preliminary and will be changed as I fall behind my schedule).

Lecture 1 6th September: Laplace equation part 1, pages 20-30 in Evans

(pp. 20-30 in 2nd edition)

Lecture 2 13th September: Laplace equation part 2, pages 30-44 in Evans

(pp. 30-44in 2nd edition)

Lecture 3 20th September: Heat equation, pages 44-65 in Evans

(pp. 44-65 in 2nd edition)

Lecture 4 27th September: Wave Equation, pages 65-85 in Evans

(pp. 65-84 in 2nd edition)

Lecture 5 4th October: CANCELED , pages 182-191 in Evans (pp. in 2nd edition)

Lecture 6 11th October: Sobolev Spaces I, pages 239-252 in Evans

(pp. 255-268 in 2nd edition)

Lecture 7 18th October: Sobolev spaces II: pages 261-269 in Evans

(pp. 277-286 in 2nd edition)

25th October: No Lecture due to exam period

Lecture 8 1st November: Sobolev spaces III: Pages 271-282 in Evans

(pp. 288-298 in 2nd edition)

Lecture 9 8th November: Elliptic PDE I: Pages 293-302 in Evans

(pp. 313-322 in 2nd edition)

Lecture 10 15th November: Elliptic PDE II: Pages 308-316 and 326-333 in Evans

(pp. 328-336 and 346-353 in 2nd edition) Note: I will not have time to cover the eigenvalueproblems from this lecture and they will not be on the final exam!

Lecture 11 22nd November: Elliptic PDE III: Pages 334-340 in Evans

(pp. 356-362 in 2nd edition)

Lecture 12 29th November: Parabolic Equations: Pages  349-377 we wil not focus on the regularity theory parts - they are similar to the elliptic case

(pp. 373-400 in 2nd edition)

Lecture 13 6th December: Hyperbolic Equations : Pages 377-394

(pp. 400-423 in 2nd edition)

Lecture 14 13th December: Calculus of variations I: Pages 431-453

(pp. 455-477 in 2nd edition)

Lecture 15 14th December at 8:15-10:00 in F11: Calculus of variations II: Pages 453-463

(pp. 477-490 in 2nd edition)

The course contains rather much material and we will therefore not be able to cover all pages during the lectures. I will try to give more detailed reading instructions during the term (which parts that can be skimmed over and which parts you must read carefully).

Examination: We will have approximately three homework assignments during term and the course will end with a written exam (probably in January).





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