# Math 647: Theory of Partial Differential Equations 1

## Contents

## Catalog Information

### Title

Theory of Partial Differential Equations 1.

### Credit Hours

3

### Prerequisite

Math 541, 547. It is proposed that Math 547 be dropped as a prerequisite, as these courses have always operated independently of each other.

### Description

Proposed: Classical theory of canonical linear PDEs. Introduction to Sobolev spaces.

## Desired Learning Outcomes

### Prerequisites

Students should understand analysis at the first-year graduate level.

### Minimal learning outcomes

Outlined below are topics that all successful Math 647 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

- Classical theory for canonical linear PDEs
- Transport equation
- Laplace's equation
- Fundamental solution
- Mean-value and maximum principles
- Energy methods

- Heat equation
- Fundamental solution
- Mean-value and maximum principles
- Energy methods

- Wave equation
- Spherical means
- Energy methods

- Method of characteristics
- Sobolev spaces
- Traces
- Sobolev inequalities
- Compactness

### Textbooks

Possible textbooks for this course include (but are not limited to):

- Lawrence C. Evans,
*Partial Differential Equations (Second Edition)*, American Mathematical Society, 2010.

### Additional topics

If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.