Partial Differential Equations. I

- Lectures : MWF 12:00-12:50, Math B0431.

- Instructor : Pierre-Emmanuel Jabin
- Contact information, Phone : 301 405 5122, Office : 2307 in Math. Bldg and 4117 in Cscamm, e-mail : pjabin@cscamm.umd.edu
- Office hours: Monday-Friday 10:30-12:00

- Textbook: Lawrence C. Evans, Partial Differential Equations (Graduate Studies in Mathematics, V. 19), AMS
- Material: A more complete description of the material covered week by week and assignmnents can be found at the address http://www2.cscamm.umd.edu/~jabin/Schedule_673_14.html
- Homework: There will be several homework assignments and/or projects..
- Examination: 1 midterm and a final exam.
- Grades: 40% for the homework, 30% for the midterm and 30% for the final exam. Letter grades will be based on the accumulated points at the end of the semester, according to the following scheme: 90%-A; 80%-B; 70%-C; 50%-D; less than 50%-F.
- Disabilities : Please inform me as soon as possible if you need accommodations because of a disability.
- Communication : Informations about the course will be given during class and through this web page. In specific cases (unexpected class cancellations), students will be contacted by e-mail.
- Academic integrity : The University of Maryland, College Park has a nationally recognized Code of Academic Integrity, administered by the Student Honor Council. This Code sets standards for academic integrity at Maryland for all undergraduate and graduate students. As a student you are responsible for upholding these standards for this course. It is very important for you to be aware of the consequences of cheating, fabrication, facilitation, and plagiarism. For more information on the Code of Academic Integrity or the Student Honor Council, please visit www.shc.umd.edu. To further exhibit your commitment to academic integrity, remember to sign the Honor Pledge on all examinations and assignments: "I pledge on my honor that I have not given or received any unauthorized assistance on this examination (assignment)."
- Some examples about academic integrity : It is all right for students to discuss between them about the homework assignments. It is similarly permitted to look for tips on the internet, from other students or other resources. It is wrong to simply copy the solution to a homework exercise from any student, a book, a web page or any other source.

Course Description (Preliminary)

This class will give an overview of the main types of Partial Differential Equations and some of the classical methods to analyse them. The corresponding PDE's are motivated by important applications to Physics and Biology. The material covered roughly corresponds to the first 4 chapters of Evans' book. We will start by the main linear models :

- Transport equations

- Laplace's equation (analysis of boundary value problems)
- Heat and Wave equations (initial value problems)

Some classical numerical methods will also be presented.

We will then proceed to Nonlinear first-order PDE and to some representation formulas.

From the Mathematical point of view, the class will deal with

- Basic results of L^p spaces (no proofs just statement of Holder, Cauchy-Schwartz, completeness and approximation by convolution)
- Notions of weak solutions, distributions, Schwartz class.
- Fundamental solution of Laplace equations in the whole space
- Weak solutions to Laplace equations, using the fundamental solution
- Mean value theorem and maximum principle for Laplace
- Maximum principle for classical solutions to 2nd order linear elliptic equations with smooth coefficients
- Fundamental solution for the heat equation in the whole space
- Classical maximal principle for the heat equation
- Weak solutions for the heat equation (using the fundamental solution)
- Fundamental solution for the 1d wave equation in the whole space
- Fourier transform on L^2, Schwartz class
- Solution to the multi-d wave equation in L^2 by Fourier transform.
- Energy method for the multi-d wave equation
- Characteristics for transport eqs in the whole space with non constant coefficients.
- Weak solutions for transport eqs