MATH 472 - Analysis III

Time
Location
Instructor
Office Hours
E-mail
TR 09:40-10:55 Hylan 1101 Dan-Andrei Geba TR 11:00-12:00 or by appointment, Hylan 806 dangeba@math.rochester.edu

Syllabus: first-order partial differential equations, principles for higher-order equations, the wave equation, the Laplace equation, the heat equation, complex interpolation, Marcinkiewicz interpolation, Littlewood-Paley theory.

Prerequisites: MATH 471 (Analysis I) and MATH 467 (Analysis II).

Textbook: Lawrence C. Evans, Partial Differential Equations (2nd edition), AMS, Graduate Studies in Mathematics, Volume 19.

In addition to the textbook, one could also consult

All these books will be on reserve at the Carlson Library reserve collection.

Course philosophy: This is a first graduate class on the topic of partial differential equations and the analytic methods used in analyzing them. It is very challenging and it requires a lot of time commitment. It moves fast, with reading assignments being assigned on a weekly basis. Proficiency will be achieved only by hard work and massive problem solving. Please take full advantage of my office hours.

Grading: Homework 30%, Midterm 30%, Final 40%.

Homework

Six homework sets will be assigned throughout the semester, each being based on the material covered in two weeks worth of lectures. Hard copies of the homework are due in class two weeks from the day they were assigned. Out of the 6 homework sets, only the best 5 will count towards your course grade. There will be no make-up homework.

There will be about 10 problems per homework from which 3, arbitrarily chosen, will be graded. Each of these problems is worth 3 points and, for a maximum grade of 10, 1 point will be awarded for substantial efforts on all of the homework problems.

Exams

The Midterm is an in-class exam, scheduled for Tuesday, 10/18. It is based on material covered from the start of the semester up to and including the 10/6 lecture.

The Final is scheduled for 12/21, from 16:00 to 19:00, and it will be a comprehensive exam.

Course policies

1. The course average is not based on a curve, nor on previously fixed scales. It will reflect how well the class is doing, and it will be high if everyone is working hard on the homework problems and is performing well on exams.

2. Incomplete "I" grades are almost never given. The only justification is a documented serious medical problem or a genuine personal/family emergency. Falling behind in this course or problems with workload on other courses are not acceptable reasons.

3. If you miss the Midterm with a valid excuse (e.g., illness or emergency), you must notify the instructor and provide supporting documentation verifying your excuse as soon as possible. For a valid excuse with supporting documentation, the Final will count as your make-up test (i.e., the Final will count towards 70% of your grade). If you miss the Final, you are in trouble. No make-up exams will be given for any reason. If you miss an exam without a valid excuse (and supporting documentation), you will receive a score of 0 on that test.

4. You are responsible for knowing and abiding by the University of Rochester's academic honesty policy. Any violation of academic honesty will be pursued according to the specified procedures. Furthermore, the following Mathematics Department policy also applies to this class:

Any usage whatsoever of online solution sets or paid online resources (chegg.com or similar) is considered an academic honesty violation and will be reported to the Board on Academic Honesty. In particular, any assignment found to contain content which originated from such sources is subject to a minimum penalty of zero on the assignment and a full letter grade reduction at the end of the semester (e.g. a B would be reduced to a C). Depending on the circumstances, this may apply even if the unauthorized content was obtained through indirect means (through a friend for instance) and/or the student is seemingly unaware that the content originated from such sources. If you have any questions about whether resources are acceptable, please check with your instructor.

5. This course follows the College credit hour policy for four-credit courses. This course meets 3 academic hours per week. Students may also be expected to deepen their understanding of the course material through close examination/evaluation of the readings assigned in the course.

Tentative weekly schedule:


Week of Topic Reading assignment Homework problems
8/29 Introduction (sections 1.1 - 1.3), transport equation (section 2.1), first order PDE (subsection 3.2.1) Characteristics (subsections 3.2.2-3.2.5)
9/5 Worked-out examples of first order PDE, boundary conditions and local solutions for the characteristic ODE (subsections 3.2.2-3.2.4) Homework 1 (due 9/22)
9/12 Local solutions for the characteristic ODE, applications (subsections 3.2.4-3.2.5), conservation laws, shocks (subsection 3.4.1) Power series (section 4.6)
9/19 Power series (section 4.6), Lewy's example, classification of semilinear 2nd order PDE on R^2 Laplace's equation (pg. 20-28) Homework 2 (due 10/6)
9/26 Classification and canonical forms for semilinear 2nd order PDE on R^2, Laplace's equation (pg. 20-27) Laplace's equation (pg. 29-36, 39-43)
10/3 Laplace's equation (pg. 29-36, 39-41) Homework 3 (due 10/27)
10/10 Fall break (no lecture on 10/11), Laplace's equation (pg. 42-43), heat equation (pg. 44, 54-56)
10/17 Midterm (10/18), heat equation (pg. 45-48, 57, 62-65)
10/24 Heat equation (pg. 49-51, 59-61), wave equation (pg. 65-68, 70-71, 81-82) Wave equation (pg. 71-74, 82-84) Homework 4 (due 11/10)
10/31 Wave equation (pg. 71-74, 82-84)
11/7 Fourier transform (definition, properties, Parseval's and Plancherel's identities, solution formulas to PDE, Heisenberg's uncertainty principle, localization) Homework 5 (due 11/29)
11/14 Short review of Lebesgue spaces (definition, dense sets, duality, basic inequalities), three-lines lemma, the Riesz-Thorin interpolation theorem, applications to Young and Hausdorff-Young inequalities
11/21 Weak Lebesgue spaces, Marcinkiewicz interpolation, the uncentered maximal function, Thanksgiving break (no lecture on 11/24)
11/28 Hardy-Littlewood-Sobolev inequality and applications, temperate distributions, Sobolev spaces Homework 6 (due 12/13)
12/5
12/12