Marcelo Mendes Disconzi
Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu
office: Stevenson Center 1515
phone: (615) 322 7147   fax: (615) 343 0215
mail to: 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240

Vanderbilt












Marcelo Mendes Disconzi
MATH 294 - Fall 13

General Information.
For a description of the course, including the grading policy, consult the course syllabus. Students are responsible for reading the syllabus and being aware of all the course and university's policies.

Textbook: Partial Differential Equations: Second Edition, by Lawrence C. Evans.

Time and Location: MWF 9:10-10:00am, SC 1310.

Contact Information and Office hours.
Lecturer's office: Stevenson Center 1515.
Lecturer's email: marcelo.disconzi@vanderbilt.edu.
Lecture's office hours: MWF 10:10am–11:10am, or by appointment.
Lecturer's office phone: (615) 615-7147.
General Math Office: Stevenson Center 1326, phone: (615) 322-6672.

Resources and extra material.
Multivariable caulculus is an important pre-requisite for this course. If you need to brush up your Calculus skills, there is a nice set of lecture notes by Gregory Naber, containing several examples, at:

http://www.gregnaber.com/lectures/calculus/

If typically you can learn things only by seeing tons of examples, you should have a look at the textbook "Schaum's outline of Partial Differential Equations" (or one of their variants), which contains more than 1000 problems solved. It can be found in the library or bought at a reasonable price on Amazon.

Exams.

Test 1
Wed, Oct 9, in class
Final Exam
Take-home


Announcements.
No announcements.

Schedule.
Below is a schedule for the course. Unless stated otherwise, all problems and sections are from the textbook.

Date
Material Covered
Suggested HW problems
Assignments Remarks
Aug 21 Introduction.


Aug 23 Review of multivariable calculus, chapter 1, appendix C. HW1

Aug 26 More review.
Intro to PDEs. Basic definitions. General form of a PDE. Linear PDEs.


If possible, read pages 20 to 22 of the textbook prior to Wed class.
Aug 28 Existence theorem for Poisson's equaion.


Aug 30 Existence theorem for Poisson's equaion.


Sept 2 Mean value formulas and maximum principle. Problems 2, 3, 4, 5, 6, 10 and 11 of section 2.5.
Prove theorems 14 on p. 37, and 15 on p. 41.
Problems 2, 3, 4, and 6 of section 2.5.
Due: Sept 11.

Sept 4 Mollifiers, regularization.
Regularity of harmonic functions.

Prove theorem 7 on p. 29 and theorem 10 on p. 31.
Due: Sept 18th.

Sept 6 Green's function for the Laplacian, analyticity of harmonic functions, Harnack inequality.


Sept 9 Fundamental solution for the heat equation. Initial value problem.


Sept 11 Student presentation: distributions.
Do these problems.
Due: Sept 18th.

Sept 13 Wave equation in 1d; D'Alembert's formula.


Sept 16 Wave equation in even and odd dimensions.
Problems 13, 14, 15, 18, 19, 21, 24 of section 2.5. Do also these problems.
Due Sept 25th.

Sept 18 Finite propagation speed for the wave equation; uniqueness. Review of measure theory.


Sept 20 Review of functional analysis.
Extra credit.
Due: Sept 30th.

Sept 23 Weak derivatives.


Sept 25. Sobolev spaces.
Problems 1, 3, 4, 9, 10, 11, 14, 15 of chapter 5. Also these problems.
Due: Oct 4th.
Extra credit: 2, 5, 6 of chapter 5.
Due: Oct 4th.

Sept 27 Sobolev embedding.


Sept 30 Sobolev embedding.


Oct 2 Sobolev embedding
and other properties of Sobolev functions.



Oct 4 Sobolev embedding
and other properties of Sobolev functions.



Oct 7 Extension and difference quotients for Sobolev functions.


Oct 9 Test.


Oct 11 Fall Break.


Oct 14 Test corrections
Classification of equations in elliptic, parabolic and hyperbolic.



Oct 16
Definition and generalities about weak solutions of second order linear elliptic equations.



Oct 18
Existence of weak solutions for second order linear elliptic equations.



Oct 21
Fredholm alternative. Statement and applications.

Problems 1, 2, and 3 of section 6.6.
Due: Oct 28th.

Oct 23
Fredholm alternative.



Oct 25
Fredholm alternative.



Oct 28
Elliptic regularity.



Nov 1 Elliptic regularity.
Weak maximum principle.

HW.
Due Nov 11.

Nov 6 Boundary maximum principle


Nov 8
Strong maximum principle



Nov 11
Negative norm Sobolev spaces


Notes.
Nov 13
Applications of negative norm Sobolev spaces: necessary and sufficient condition for existence of linear PDEs.



Nov 15
Seminar


The PDE seminar (4:10pm, SC 1307) will replace the class.
Dec 2nd
Local solvability. Egorov's counter-example.


Notes.
Dec 4
Discussion of the take-home final exam.






Anonymous feedback.
Students are encouraged to bring suggestions and to discuss with the course instructor any concerns they may have, including something they think is not being properly handled in the course. But if you do not feel comfortable about doing that, here you have the opportunity to send some anonymous feedback.