## APM 576: Theory of PDE I

## Fall 2017

Class Time | M,W 10:45 – 12:00 |

Class Location | Tempe – LL 264 |

Instructor | Sébastien Motsch |

Office | WXLRA 836 |

smotsch@asu[dot]edu | |

Office Hours | M,W 1:00pm – 3:00pm |

Class webpage | www.seb-motsch.com/graduate |

## Textbook

- L. Evans,
*Partial Differential Equations*(2nd edition) - Additional: H. Brézis,
*Functional Analysis, Sobolev Spaces and Partial Differential Equations*

### Course description

This course introduces rigorous methods to study partial differential equations such as existence theory and global behavior of solutions. The goal is to understand *intuitively* PDEs and then to develop analytic skills to * prove* results. This class is intended to be spread over two semesters, the first semester will be focused on **linear PDEs** (e.g. elliptic, parabolic equations) and the second semester on **non-linear PDEs** (e.g. conservation laws, Hamilton-Jacobi equations).

The course will be divided in four parts:

- a) Review PDEs
*(Chap. 1, Chap. 2.1-2.3)* - b) Functional analysis
*(Chap. 5.2-5.7)* - c) Elliptic PDEs
*(Chap. 6)* - d) Evolution equations
*(Chap. 7)*

### Class Schedule

Aug 21 – 23 | Intro. (chap. 1), Transport eq. (chap. 2.1) |

Aug 28 – 30 | Heat eq. (chap. 2.3) |

Sep 4 – 6 | Labor day – Laplace eq. (chap. 2.2) |

Sep 11 – 13 | Laplace eq. (chap. 2.2) – Functional analysis (chap. 5.2) |

Sep 18 – 20 | Functional analysis (chap. 5.2)(supplement weak convergence) |

Sep 25 – 27 | Approximation – Trace (chap. 5.3, 5.5) |

Oct 2 – 4 | Compact embedding (chap. 5.7), Elliptic eq. (chap. 6.1) |

Oct 9 – 11 | Fall Break |

Oct 18 – 20 | Weak solution – Lax-Milgram (chap. 6.1-6.2) |

Oct 23 – 25 | Eigenvalues-regularity (chap. 6.3-6.5) |

Oct 30 – Nov 1 | Linear evolution eq. (chap. 7.1) |

Nov 13 – Nov 15 | Linear evolution eq. (chap. 7.1) |

### Class homework

Sep 6 | HW1 (solution, TeX) |

Sep 27 | HW2 (solution) |

Oct 23 | HW3 (solution) |

Nov 13 | HW4 |