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

Vanderbilt












Marcelo Mendes Disconzi
MATH 8110 - Theory of 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.

This course will be taught through a hybrid online/in-person model. See the course syllabus for details.

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

Classes meet on TR, 2:203:35pm at Stevenson Center 1308 (ground floor of the Mathematics Building).

Due to the COVID-19 outbreak, we will be following all guidelines stipulated by Vanderbilt University. See the return to campus website, the Vanderbilt COVID-19 response website, and the Vanderbilt COVID-19 dashboard for more information.

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, 46pm, Thursdays, 56pm, or by appointment. Office hours will be held virtually. If you want to meet in person, please schedule an 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 (these will be updated on a regular basis). Starting Monday, November 30th, classes will be online only.

Date Material covered Remarks
Aug 25
Introduction, notation, differential operators.

Aug 27
Laplace's and Poisson's equations. Fundamental solution. Properties of harmonic functions. Green's function.

Sep 1
The heat equation. Fundamental solution. Smoothing and infinite propagation speed. Duhamel's principle.

Sep 3
The wave equation. D'Alembert's, Kirchhoff's, and Poisson's formulas. Solutions for n even and odd. Domains of dependence and influence.

Sep 8
Weak derivatives and Sobolev spaces: basic properties.

Sep 10
Approximation by smooth functions: interior and up-to-the boundary approximation; the segment condition. Extensions.

Sep 15
The cone condition. The Sobolev embedding theorem. Sobolev's inequality.
Sep 17
More on Sobolev embedding: embedding into Holder spaces.

Sep 22
Canceled. To be rescheduled at a later time.

Sep 24
Canceled. To be rescheduled at a later time.
Sep 29
Compact embedding. Traces. Fractional and negative order Sobolev spaces.

Oct 1
Dualities. More on negative Sobolev spaces.

Oct 6
On necessary and sufficient conditions for existence of solutions to linear PDEs. A PDE that is not locally solvable at the origin (Egorov's counter-example).

Oct 8
Canceled. To be rescheduled at a later time.
Oct 13
Linear elliptic equations; existence and uniqueness of weak solutions.

Oct 15 Elliptic regularity.

Oct 20 Maximum principles. Semi-linear elliptic equations: the method of sub- and super-solutions. Applications: the uniformization theorem for Riemann surfaces.

Oct 22 Nonlinear elliptic equations. The continuity method and implicit function theorem methods.

Oct 27 Canceled. To be rescheduled at a later time.
Oct 29 Linear hyperbolic equations. Gronwall's lemma. Energy estimates for first-order symmetric hyperbolic systems.

Nov 3 Existence and uniqueness for linear first-order symmetric hyperbolic systems.

Nov 5 Existence and uniqueness for linear second order hyperbolic systems (linear wave equations).
Nov 10 Local existence and uniqueness for quasilinear wave equations. Preliminaries. Uniqueness.

Nov 12 Local existence and uniqueness for quasilinear wave equations. Energy estimates.
Nov 17 Local existence and uniqueness for quasilinear wave equations. Existence.
Nov 19 Canceled. To be rescheduled at a later time.
Nov 23 - Nov 27 Thanksgiving holidays, no classes.

Dec 1

Dec 3

Dec 7 (make-up class)

Dec 8 (make-up class)

Dec 9 (make-up class)

Dec 10 (make-up class)

Dec 11 (make-up class)


Anonymous feedback
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