Syllabus and Course Details
MATH 7540 - Spring 2025
This page constitutes the official syllabus and course policy.
Course Information
Title: Partial Differential Equations II
Code:: Math 7540 - Spring 2025
Lecture: MWF 9:00am - 9:50am - Gibson Room 400D
Instructor Information
Instructor: Sam Punshon-Smith
Office: Gibson 423
Email: spunshonsmith@tulane.edu
Office Hours:
- Monday: 10:00 AM - 11:00 AM
- Wednesday: 10:00 AM - 11:00 AM
- Friday: 10:00 AM - 11:00 AM
Description
This course delves into the advanced theory of Partial Differential Equations (PDEs) with a strong emphasis on functional analysis techniques. We will begin by establishing essential concepts in functional analysis, including Banach and Hilbert spaces, operators, and duality. We will then transition into the theory of distributions and Sobolev spaces, which are crucial tools for the modern analysis of PDEs. We will study the regularity theory of elliptic equations, and introduce linear evolution equations and semigroup theory. Time permitting, we will explore advanced topics such as DeGiorgi-Nash-Moser theory or Hormander's theory of hypoellipticity. Throughout the course, we will emphasize rigorous mathematical proofs and a deep understanding of the underlying analytical structures.
For a detailed outline of the course topics by week, see the course schedule page.
Course Goals
This course aims to cultivate a deep understanding of the modern analytical foundations of Partial Differential Equations. By the end of the course, students will:
- Develop an appreciation for how functional analytic tools, particularly distributions and Sobolev spaces, provide a rigorous framework for studying PDEs.
- Gain proficiency in the use of distributions, Fourier analysis, and Sobolev spaces, and understand their power in analyzing the existence, uniqueness, and regularity of solutions to PDEs.
- Extend and deepen your analytical toolkit for studying PDEs, including advanced techniques for analyzing the existence, uniqueness, regularity, and long-term behavior of solutions.
- Develop the ability to read and understand advanced mathematical texts and research articles in PDE theory, and to formulate and explore their own research questions.
- Recognize the broad applicability of PDE theory in modeling real-world phenomena and appreciate the elegance and depth of the mathematical structures involved.
Learning Objectives
Upon successful completion of this course, students will be able to:
- Apply concepts from functional analysis, including Banach and Hilbert spaces, duality, and weak convergence, to analyze the well-posedness of PDEs.
- Utilize the theory of distributions to make sense of weak solutions and generalized functions, and understand their role in PDE theory.
- Apply Sobolev spaces to study the existence, uniqueness, and regularity of solutions to elliptic and evolution equations.
- Understand and apply advanced techniques, such as the Lax-Milgram theorem, maximum principles, and semigroup theory to analyze the existence, uniqueness, regularity, and long-term behavior of solutions to linear elliptic and evolution equations.
- (Depending on time and student interest) Explore advanced topics in PDE theory, such as the DeGiorgi-Nash-Moser theory, and gain exposure to current research frontiers.
- Articulate complex mathematical ideas clearly and concisely, both orally and in writing, through problem sets, exams, and potential project presentations.
Prerequisites
Must have a solid background in real analysis (measure theory, Lebesgue integration). Some prior exposure to functional analysis is helpful but not strictly required.
Course Materials
The only material required for the course is the following textbook by L.C Evans:
- Required Text Evans, Lawrence C. Partial Differential Equations (Second Edition). United States, American Mathematical Society, 2010. Amazon Link
This book is a classic text for learning graduate PDE and a highly recommended addition to your library. You can find copies online if you look (not very hard). Make sure you are using the second edition!
The material for the course will closely follow this text and problems will be assigned out of it. The following books will be used a supplementary texts:
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer 2011.
- E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
Additional notes on distributions, Fourier Analysis and regularity will be provided by the instructor.
Communication and Help
I will be available during regular business hours on weekdays via email and during my office hours listed above. Please give me at least 24 hrs to respond to emails. If you cannot attend my office hours, you may request to schedule a one-on-one appointment with me over email.
Assignments and Exams
Problem Sets
There will be weekly problem sets on most weeks. They will be assigned on the course schedule one week or more in advance. Problems must be written clearly and legibly with sufficient exposition and explanation. Illegible solutions or solutions without sufficient explanation will be penalized. It is highly recommended (though not required) for you to write your assignments in LaTeX. All problems sets are due in lecture at the beginning (due dates are listed on the schedule) in paper form. Assignments are listed in the course schedule.
Late Assignments
Late work will be deducted 10% per twenty-four hour period that elapses after the due date (this includes weekends). You may submit electronic versions of your assignment of your assignment over the weekend to avoid further penalty, but must submit an identical physical copy later. If foreseen or unforeseen circumstances prevent you from completing an assignment on time, you may request an extension. Extensions must be requested in advance of the due date. If the situation warrants an extension, we will determine a new due date for the assignment based on your individual circumstances.
Exams
There will also be two exams, one midterm exam and one final exam. The exams will be take home exams due on the dates listed on the course schedule. Topics covered are listed on the course schedule. The dates will be fixed at least two weeks before the exam. The final will be cumulative.
Grading
The breakdown and percentages for all graded materials are described in the following table:
| Category | Percentage |
|---|---|
| Problem Sets | 60% |
| Midterm | 20% |
| Final Exam | 20% |
| Total | 100% |
Grades will tentatively be assigned using the following cutoffs. Grade modifiers (+-) may be added to grades within ~1% of the boundary points on a case by case basis. Eg: 89% = B+
| Percentage | Grade |
|---|---|
| 90%-100% | A |
| 85-89% | A- |
| 80%-84% | B+ |
| 75%-79% | B |
| 70-74% | B- |
| 65%-69% | C+ |
| 60-64% | C |
| 55%-59% | C- |
| 50%-54% | D |
| < 50% | F |
Code of Academic Conduct
The Code of Academic Conduct applies to all undergraduate students, full-time and part-time, in Tulane University. Tulane University expects and requires behavior compatible with its high standards of scholarship. By accepting admission to the university, a student accepts its regulations (i.e., Code of Academic Conduct and Code of Student Conduct) and acknowledges the right of the university to take disciplinary action, including suspension or expulsion, for conduct judged unsatisfactory or disruptive.
ADA/Accessibility
Tulane University is committed to offering classes that are accessible. If you anticipate or encounter disability-related barriers in a course, please contact the Goldman Center for Student Accessibility to establish reasonable accommodations. If approved by Goldman, make arrangements with me as soon as possible to discuss your for medical documentation from you to support potential accommodation needs. Goldman Center contact information: Email: goldman@tulane.edu; Phone (504) 862-8433; Website: accessibility.tulane.edu
Diversity
This course is designed to support an inclusive learning environment where diverse perspectives are recognized, respected and seen as a source of strength. It is our intent to provide materials and activities that are respectful of various levels of diversity: mathematical background, gender, sexuality, disability, age, socioeconomic status, ethnicity, race, and culture.
- 2025