Syllabus for MATH 612, SECTION 600, Spring 2020

Partial Differential Equations

Office hours: TR 1:00-2:00; W 3:00-4:00;  Also, by appointment.

Class times and place: MWF 11:30-12:20, Blocker 121.

Section web page: www.math.tamu.edu/~phoward/M612.html

Textbook: Partial Differential Equations, 2nd Edition, by Lawrence C. Evans, Graduate Studies in Mathematics vol. 19 (2010), American Mathematical Society.

Prerequisites: The main prerequisite is M611, the first semester of the PDE sequence.  Students will need to be familiar with some graduate analysis (M607), but not more than we covered in M611.   

Catalog Description: Theory of linear partial differential equations. Sobolev spaces. Elliptic equations (including boundary value problems and spectral theory). Linear evolution equations of parabolic and hyperbolic types (including initial and boundary value problems). As time permits, additional topics might be included. Prerequisite: MATH 611 and MATH 607 or MATH 641 or approval of instructor.

Homework Assignments: A homework assignment will be made on most Fridays, due the following Friday. Homework assignments will typically consist of four problems, worth ten points each. Work will be accepted up to a week late, though five points will be deducted for each class period by which the assignment is late.  

Exams: There will be two exams, a midterm and a final. The midterm exam will be in the evening, 7:00-9:00 p.m., Wednesday March 4. The final exam for this class will be on Tuesday, May 5, 10:30 a.m.-12:30 p.m. Please make a note of these dates.

Grades: Final grades will be determined in the following manner: Homework assignments: 50%; Exams: 25% each. Grade ranges will be graduate standard: 89.50-100, A; 79.50-89.49, B; 69.50-79.49, C, 59.50-69.49, D; below 59.50, F.

Learning outcomes: Students will be able to: distinguish between Frechet and Gateaux derivatives, and compute Gateaux derivatives; find exact solutions to special cases of the Hamilton-Jacobi equation via the Hopf-Lax formula; find entropy solutions of scalar conservation laws using characteristics; find entropy solutions of scalar conservation laws using the Lax-Oleinik formula; compute Fourier transforms of L² functions, and use Fourier transforms to solve PDE; work comfortably with functions in Holder and Sobolev spaces, including facility with Sobolev embeddings; work comfortably with various aspects of functional analysis, including duality and the geometry of Hilbert spaces; develop weak formulations of linear elliptic PDE and prove existence, uniqueness and regularity for such solutions; work comfortably with time-dependent function spaces; develop weak formulations of linear parabolic PDE and prove existence, uniqueness and regularity of such solutions.

Make-up policy: Make-ups for exams will only be given if the student can provide a documented University-approved excuse (see University Regulations). According to University Student Rules students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise the student forfeits his or her right to a make-up.

Scholastic Dishonesty: Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. "An Aggie does not lie, cheat, or steal or tolerate those who do." Please refer to the Honor Council Rules and Procedures, available at the Office of the Aggie Honor System.

Copyright policy: All printed materials disseminated in class or on the web are protected by copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Students with Disabilities: The following statement was provided by the Department of Disability Services: The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Resources, currently located in the Disability Resources building at the Student Services at White Creek complex on west campus or call 979-845-1637. For additional information visit http://disability.tamu.edu.

Class Schedule: Roughly speaking, we should cover the following material on the following schedule (for brevity I've omitted several short topics that I'll comment on as we go):

Week of Monday	Material Covered
January 13	Fourier transforms in L2. (Fri. Jan. 17 is the last day for drop/add.)
January 20	Hölder and Sobolev spaces: motivation; basic definitions. (No class Mon., Jan. 20.)
January 27	Hölder and Sobolev spaces: approximation.
February 3	Elements of functional analysis.
February 10	Sobolev spaces: extensions; trace operators.
February 17	Sobolev spaces: embedding theorems.
February 24	Second-order elliptic PDE: existence of weak solutions.
March 2	Second-order elliptic PDE: existence of weak solutions (continued). (Midterm exam, Wed. Mar. 4, 7-9 pm.)
March 9	Spring Break.
March 16	Classes canceled by the University.
March 23	Second-order elliptic PDE: interior regularity.
March 30	Second-order elliptic PDE: boundary regularity.
April 6	Function spaces involving time. (No class Friday, Apr. 10.)
April 13	Second-order parabolic PDE: existence/uniqueness of weak solutions. (Tues. Apr. 14 is last day for Q-drop.)
April 20	Second-order parabolic PDE: regularity.
April 27	Second-order parabolic PDE: higher regularity. (Tuesday, Apr. 28 is the last day of classes, redefined as Friday.)