758Tsikkou
Spring 2016
Course Information
Description: Theory of Partial Differential Equations 2
Instructor: Charis Tsikkou. Email
Class Schedule: Tuesdays, Thursdays 02:30-03:45 PM in Armstrong Hall 203
Office Hours: Tuesdays 04:00-05:00 PM, Thursdays 01:00-02:00 PM, and by appointment, in Armstrong Hall 405
Info Sheet containing more or less the stuff on this webpage.
Textbook
Textbook:
Lawrence C. Evans, Partial Differential Equations: Second Edition (Graduate Studies in Mathematics)
Errata for “Partial Differential Equations”, AMS Press, Second Edition by Lawrence C. Evans
Alternative Textbook:
Emmanuele DiBenedetto, Partial Differential Equations: Second Edition (Cornerstones)
Gerald B. Folland, Introduction to Partial Differential Equations: Second Edition
Fritz John, Partial Differential Equations (Applied Mathematical Sciences)
Robert C. McOwen, Partial Differential Equations: Methods and Applications (2nd Edition)
Jeffrey Rauch, Partial Differential Equations (Graduate Texts in Mathematics)
Walter Strauss, Partial Differential Equations: An Introduction
Evaluation
Grading scheme
- 35% Final exam
- 25% Midterm exam
- 40% Homework Assignments
- Letter grades will be assigned according to the scheme
- A 90-100% | B 80-90% | C 70-80% | D 60-70% | F 0-60%
Homework
- Homework will be assigned once every three weeks, and due two weeks later (please see the course schedule below for exact dates).
- Homework will always be collected at the beginning of the class.
- Late turn-ins will not be accepted.
Midterm Exam
- There will be one take-home exam, posted: Thursday, March 10, due: Tuesday, March 15.
- No make-up exam will be given.
Final Exam
- Due: Thursday, May 5.
- Final is cumulative.
Course Schedule (subject to changes)
Date | Topic | Resources | HW |
---|---|---|---|
Tuesday, Jan 12 | Introduction to Conservation Laws; Shocks, Entropy Condition | 3.4.1 | |
Thursday, Jan 14 | Examples; Shock and Rarefaction Waves; Entropy Solution | 3.4.1 - 3.4.3 | |
Tuesday, Jan 19 | Riemann's Problem; Separation of Variables; Examples | 3.4.4, 4.1.1 | |
Thursday, Jan 21 | Similarity Solutions; Plane and Traveling Waves; Solitons | 4.2.1 | HW 1 posted, solutions |
Tuesday, Jan 26 | Examples | 4.2.1 | |
Thursday, Jan 28 | Similarity Under Scaling; Fourier Transform | 4.2.2, 4.3.1 | |
Tuesday, Feb 2 | Plancherel's Theorem; Properties | 4.3.1 | |
Thursday, Feb 4 | Examples | 4.3.1 | |
Tuesday, Feb 9 | Laplace Transform; Examples; Cole-Hopf Transformation | 4.3.3, 4.4.1 | |
Thursday, Feb 11 | Examples; Holder Spaces; Weak Derivatives | 4.4.1, 5.1, 5.2.1 | HW 2 posted, solutions |
Tuesday, Feb 16 | Sobolev Spaces; Definitions ; Examples | 5.2.2 | |
Thursday, Feb 18 | Elementary Properties | 5.2.3 | |
Tuesday, Feb 23 | Sobolev Spaces as Function Spaces | 5.2.3 | |
Thursday, Feb 25 | Local Approximation by Smooth Functions | 5.3.1 | |
Tuesday, Mar 1 | Global Approximation by Smooth Functions | 5.3.2 | |
Thursday, Mar 3 | Global Approximation by Functions Smooth up to the Boundary | 5.3.3 | HW 3 posted, solutions |
Tuesday, Mar 8 | Extensions | 5.4 | |
Thursday, Mar 10 | Extension Theorem | 5.4 | Midterm Exam posted, solutions |
Tuesday, Mar 15 | Problem Solving | ||
Thursday, Mar 17 | Traces | 5.5 | |
Tuesday, Mar 22 | No Class (Spring Recess) | ||
Thursday, Mar 24 | No Class (Spring Recess) | HW 4 posted, solutions | |
Tuesday, Mar 29 | Traces; Sobolev Inequalities | 5.5, 5.6.1 | |
Thursday, Mar 31 | Gagliardo-Nirenberg-Sobolev Inequality | 5.6.1 | |
Tuesday, Apr 5 | Estimates for W^{ 1,p} and W_{0}^{ 1,p}, p < n | 5.6.1 | |
Thursday, Apr 7 | Morrey's Inequality | 5.6.2 | |
Tuesday, Apr 12 | Estimates for W^{ 1,p}, n < p | 5.6.2 | |
Thursday, Apr 14 | General Sobolev Inequalities | 5.6.3 | HW 5 posted, solutions |
Tuesday, Apr 19 | Rellich-Kondrachov Compactness Theorem | 5.7 | |
Thursday, Apr 21 | Hardy's Inequality | 5.8.4 | |
Tuesday, Apr 26 | The Dual space H^{ -1}; Second-Order Elliptic Equations | 5.9.1, 6.1.1 | |
Thursday, Apr 28 | Weak Solutions; Lax-Milgram Theorem; Energy Estimates | 6.1.2 - 6.2.2 | |
Thursday, May 5 | Final Exam | Final Exam posted, solutions |
Inclusivity statement
The West Virginia University community is committed to creating and fostering a positive learning and working environment based on open communication, mutual respect, and inclusion. If you are a person with a disability and anticipate needing any type of accommodation in order to participate in this class, please advise me and make appropriate arrangements with the Office of Accessibility Services (293-6700). For more information on West Virginia University's Diversity, Equity, and Inclusion initiatives, please see http://diversity.wvu.edu.
Accessibility Needs
If you are a person with a disability and anticipate needing any type of accommodation in order to participate in this class, please advise me and make appropriate arrangements with the Office of Disability Services (304-293-6700).