Marcelo Mendes Disconzi

Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu

office: Stevenson Center 1515

phone: (615) 322 7147 fax: (615) 343 0215

mail to: 1326 Stevenson Center Ln, Vanderbilt University, Nashville TN 37240

Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu

office: Stevenson Center 1515

phone: (615) 322 7147 fax: (615) 343 0215

mail to: 1326 Stevenson Center Ln, Vanderbilt University, Nashville TN 37240

MATH 8110 - Graduate partial differential equations

General
Information

For a description of the course, including the grading policy, consult the course syllabus. Students are responsible for reading the syllabus and being aware of all the course and university policies.

For a description of the course, including the grading policy, consult the course syllabus. Students are responsible for reading the syllabus and being aware of all the course and university policies.

Textbook: No textbook will be required. See the course syllabus for suggested references.

Classes meet on TR, 11:10-12:25pm at Stevenson Center 1210.

We will be following the COVID-19-related guidelines stipulated by Vanderbilt.

Contact
Information and Office Hours

Instructor's office: Stevenson Center 1515 (5th floor of the Mathematics Building).

Instructor's email: marcelo.disconzi@vanderbilt.edu.

Instructor's office hours: Tuesdays, 1-3pm, Thursdays, 10-11am, or by appointment.

Instructor's office phone: (615) 322-7147.

Instructor's office: Stevenson Center 1515 (5th floor of the Mathematics Building).

Instructor's email: marcelo.disconzi@vanderbilt.edu.

Instructor's office hours: Tuesdays, 1-3pm, Thursdays, 10-11am, or by appointment.

Instructor's office phone: (615) 322-7147.

Schedule

Below is an ongoing schedule for the course (for the academic calendar,
click here).
Click here for the class notes, and
here for the class notes in handwritten form.
The class notes will be updated and typeset as the course progress. Since typesetting
them might take some time, typically the most up-to-date version of the notes will be
in the handwritten form. Non-math majors taking the course can consult
these notes
on preliminary notions for some background on mathematical concepts, conventions,
and notation that will be used in the course.

Date | Material covered | Remarks |
---|---|---|

Aug 27 | Introduction. Examples of PDEs. | HW1 posted on Brightspace. |

Aug 31 | Fundamental solution to the Laplacian. Solutions to Poisson's equation. Properties of harmonic functions. | HW2 posted on Brightspace. |

Sept 2 | Maximum principle. Green's function. | |

Sept 7 | The initial-value problem for the heat equation in R^n. | HW3 posted on Brightspace. |

Sept 9 | The Cauchy problem for the 1d wave equation; D'Alembert's formula. Finite propagation speed for the wave equation in n dimensions. Domains of dependence and influence. | HW4 posted on Brightspace. |

Sept 14 | Solutions to the wave equation in two and three dimensions. The inhomogeneous wave equation. | |

Sept 16 | Weak derivatives. | HW5 posted on Brightspace. |

Sept 21 | Sobolev spaces. Basic properties. Approximation by smooth functions. | |

Sept 23 | The segment condition. Approximation by smooth functions up to the boundary. | HW6 posted on Brightspace. |

Sept 28 | The strong local Lipschitz condition and the uniform C^k condition. Extensions. | |

Sept 30 | The cone condition. The Sobolev embedding theorem. | |

Oct 5 | More embeddings. Compact embeddings. | |

Oct 7 | Traces. L^2-based Sobolev spaces of negative and fractional order in R^n. | HW7 posted on Brightspace. |

Oct 12 | Duality. Negative-order Sobolev spaces on domains. | |

Oct 14 | Fall break, no class. | |

Oct 19 | Necessary and sufficient conditions for the existence of weak solutions to linear boundary-value problems. Egorov's example of a linear PDE that is not locally solvable at the origin. | HW8 posted on Brightspace. |

Oct 21 | Linear elliptic equations. Existence of weak solutions. | |

Oct 26 | The Fredholm alternative. Elliptic regularity. | |

Oct 28 | Elliptic regularity up to the boundary. Maximum principles. | HW9 posted on Brightspace. |

Nov 2 | Nonlinear elliptic equations. | |

Nov 4 | Linear hyperbolic equations. Energy estimates for linear first-order symmetric hyperbolic systems. | |

Nov 9 | Existence and uniqueness for linear first-order symmetric hyperbolic systems. | HW10 posted on Brightspace. |

Nov 11 | Existence and uniqueness for linear wave equations. Quasilinear wave equations: preliminaries. | |

Nov 16 | Class canceled. A make-up class will be announced at a later time. | |

Nov 18 | Class canceled. A make-up class will be announced at a later time. | |

Nov 23 | Thanksgiving break, no class. | |

Nov 25 | Thanksgiving break, no class. | |

Nov 30 | Quasilinear wave equations: energy estimates and local uniquencess. | |

Dec 2 | Quasilinear wave equations: local existence. | HW11 posted on Brightspace. |

Dec 7 | Einstein's equations: preliminaries, set-up, and the constraint equations. | |

Dec 9 | Einstein's equations: the Cauchy problem. | |

Dec 14 | General hyperbolic equations; the role of the characteristics. | Make-up class. |

Dec 16 | Informal discussion. | Make-up class. |

Anonymous
feedback

Students are encouraged to bring suggestions and discuss with the course instructor any concerns they may have, including something they think is not being properly handled in the course. But if you do not feel comfortable doing that, here you have the opportunity to send anonymous feedback.

Students are encouraged to bring suggestions and discuss with the course instructor any concerns they may have, including something they think is not being properly handled in the course. But if you do not feel comfortable doing that, here you have the opportunity to send anonymous feedback.