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, 2:20–3:35pm 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, 4–6pm, Thursdays, 5–6pm, 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	The role of the characteristics. Hyperbolicity.
Dec 3	Einstein's equations: the basic set-up.
Dec 7 (make-up class)	The Einstein constraint equations. The meaning of the Cauchy problem for Einstein's equations. Wave coordinates.
Dec 8 (make-up class)	Local existence for the Cauchy problem for Einstein's equations. Uniqueness: the maximal globally hyperbolic development of the data.
Dec 9 (make-up class)	The characteristic geometry of quasilinear wave equations. Motivation: global existence and decay of solutions.
Dec 10 (make-up class)	Null frames and eikonal functions.
Dec 11 (make-up class)	The null-structure equations.

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.