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.

There is no textbook for this course, and we will follow the class notes posted below. Some additional references are suggested in the course syllabus.

Classes meet on TR, 2:45-4pm at Stevenson Center 1312.

Contact Information and Office Hours
Instructor's office: A1017, 17^th & Horton (Sony bld).
Instructor's email: marcelo.disconzi@vanderbilt.edu.
Instructor's office hours: Tuesdays, 4:10-6:10pm, Thursdays, 12-1pm, or by appointment.
Instructor's office phone: (615) 322-7147.

Class notes
Click here for the class notes. If you find some inconsistency or something that seems wrong in the class notes, it is likely a typo. In that case, please let the instructor know.

Schedule
Below is schedule for the course which will be updated as the course progresses (for the academic calendar, click here). Sections refer to the class notes. Materials (including HW) will be posted below or on Brightspace.

Date	Material covered	Remarks
Aug 21	Introduction. Examples. Sections 1-4.	Homework 1.
Aug 26	Laplace's and Poisson's equations. Fundamental solution. Existence of solutions to Poisson's equations. Properties of harmonic functions (regularity, mean value, maximum principle, etc.) Section 5.	Homework 2.
Aug 28	Comments on the heat equation. The wave equation in 1, 2, 3, and arbitrary dimensions: existence and uniqueness, finite propagation speed, Duhamel's formula. Sections 7-8.	Homework 3, Homework 4.
Sept 2	Weak derivatives, basic properties. Section 9.1.	Homework 5.
Sept 4	Sobolev spaces. Basic properties. Approximation by smooth functions. Sections 9.2-9.3.
Sept 9	No class. Make-up class to be scheduled.
Sept 11	No class. Make-up class to be scheduled.
Sept 16	The segment condition. Approximation by smooth functions up to the boundary. Change of coordinates. Section 9.3.
Sept 18	The strong local Lipschitz, uniform C^k, the cone, and the uniform cone conditions. Extensions. Section 9.4.	Homework 6.
Sept 23	The Sobolev embedding theorem. Section 9.5.
Sept 25	Sobolev inequality. Compact embeddings. Sections 9.6-9.7.
Sept 30	Duality. Sobolev spaces of negative order. Comments on traces, Sobolev spaces defined via Fourier transform, and miscellaneous inequalities. Sections 9.8-9.11.	Homework 7.
Oct 2	Necessary and sufficient conditions for existence of solutions to linear PDEs. Egorov's example of a PDE with no solution. Sections 10-10.1.	Homework 8.
Oct 7	Linear elliptic equations. Existence of weak solutions and interior regularity. Sections 11-11.2.
Oct 9	Fall break, no class.
Oct 14	No class. Make-up class to be scheduled.
Oct 16	No class. Make-up class to be scheduled.
Oct 21	Boundary regularity for linear elliptic equations. Maximum principles. Sections 11.2-11.3.	Homework 9.
Oct 23	Nonlinear elliptic equations via examples: sub- and super-solutions, implicit function theorem, and continuity methods. Sections 12-12.3.
Oct 28	Linear first-order symmetric hyperbolic systems. Energy estimates. Sections 13-13.1.
Oct 30	Linear first-order symmetric hyperbolic systems. Existence and uniqueness. Section 13.1.	Homework 10.
Nov 4	Linear wave equations. Section 13.2.
Nov 6	Energy estimates for quasilinear wave equations. Section 14.
Nov 11	Local existence and uniqueness for quasilinear wave equations. Section 14.	Homework 11.
Nov 13	Local existence and uniqueness for quasilinear wave equations. Continuation criterion. Sections 14-14.1.
Nov 18	The role of the characteristics. General hyperbolic equations. Section 15.
Nov 20	More about characteristics. Section 15.
Nov 25	Thanksgiving holidays, no class.
Nov 27	Thanksgiving holidays, no class.
Dec 2
Dec 4

Anonymous feedback
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