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UA Course Catalog Prerequisites:
This course has two prerequisites.
1. Basic knowledge of PDE (Math 541 or 642).
2. Real Analysis (Math 580 and 681).
Course Description and Credit Hours
Advanced course in real analysis. Topics may include harmonic analysis (the Fourier transform, Hardy-Littlewood maximal operator, interpolation, singular integral operators, BMO and Hardy spaces, weighted norm inequalities) or analysis and PDEs (Sobolev spaces, weak solutions to PDEs, Lax-Milgram theory, the Fredholm alternative, existence and regularity for elliptic and parabolic equations).
MODERN THEORY OF PARTIAL DIFFERENTIAL EQUATIONS
This course concerns the modern theory of partial differential equations (PDE). The classical approach of PDE is the search for explicit formulas for the solutions. However, many times formulas do not provide as much useful information as we hope. What’s more, most PDEs do not possess a formula for their solutions. Hence it is essential to understand the properties of their solutions without a formula.
In this course, we will abandon the search for explicit solutions. Instead we will concentrate on modern techniques in the theoretical study of linear and nonlinear PDEs. This is a course that provides students the opportunity to learn PDE in topics beyond the classical theory covered in Math 541 and 642. And students will be prepared for research in PDE-related areas. In his now definitive textbook Partial Differential Equations, L. Craig Evans described six principles:
PDE theory is (mostly) not restricted to two independent variables.
Many interesting equations are nonlinear.
Understanding generalized solutions is fundamental.
PDE theory is not a branch of functional analysis.
Notation is a nightmare.
Good theory is (almost) as useful as exact formulas.
These principles are also the guidelines for our course.
Required Texts from UA Supply Store:
- EVANS / PARTIAL DIFFERENTIAL EQUATIONS (V19) (Required)
- EVANS (RENTAL) / (RENTAL) PARTIAL DIFFERENTIAL EQUATIONS (V19) (RENTAL)
Student Learning Outcomes
Students will analyze the properties of PDEs.
Other Course Materials
Weak Convergence Methods for Nonlinear Partial Differential Equations, (Regional Conference Seriess in Mathematics, No 74) CBMS/74, by Lawrence C. Evans, 1990
Elliptic Partial Differential Equations, Second Edition, by Qing Han and Fanghua Lin
Outline of Topics
Sobolev spaces; weak solutions (existence, uniqueness, and regularity) for linear and nonlinear PDEs; calculus of variations; non variational techniques for nonlinear PDEs.
Exams and Assignments
There will be no written homework or written exam. Rather, each student will give a presentation on some problems chosen from the textbook.
The grade will be determined by class participation, and the quality of each student's presentation.
Policy on Missed Exams and Coursework
Each student will give at least one presentation on topics from the textbook. If a student has to miss a scheduled presentation due to legitimate reasons or emergencies, he/she should inform the instructor as soon as possible.
Attendance is required.
Notification of Changes
The instructor will make every effort to follow the guidelines of this syllabus as listed; however, the instructor reserves the right to amend this document as the need arises. In such instances, the instructor will notify students in class and/or via email and will endeavor to provide reasonable time for students to adjust to any changes.
Statement on Academic Misconduct
Students are expected to be familiar with and adhere to the official Academic Misconduct Policy provided in the Online Catalog.
Statement On Disability Accommodations
Contact the Office of Disability Services (ODS) as detailed in the Online Catalog.
Severe Weather Protocol
Please see the latest Severe Weather Guidelines in the Online Catalog.
Pregnant Student Accommodations
Title IX protects against discrimination related to pregnancy or parental status. If you are pregnant and will need accommodations for this class, please review the University’s FAQs on the UAct website.
Under the Guidelines for Religious Holiday Observances, students should notify the instructor in writing or via email during the first two weeks of the semester of their intention to be absent from class for religious observance. The instructor will work to provide reasonable opportunity to complete academic responsibilities as long as that does not interfere with the academic integrity of the course. See full guidelines at Religious Holiday Observances Guidelines.
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