General Information
For a description of the course, including the grading policy, consult the course syllabus.

Textbook: No textbook will be adopted. Support references are given in the syllabus.

Classes meet on TR, 2:35–3:50pm, at Stevenson Center 1313 (3rd floor of the Mathematics Building).

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: Thursdays, 4:00–7:00pm, 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). This will be updated regularly and, therefore, students should check this webpage frequently. The due date for each assignment will be posted as the course progresses.

Date	Material covered	Remarks
Aug 24	Introduction
Aug 29	Notation and generalities. Harmonic and sub- super- harmonic functions. Mean value formulas, maximum principle, and Harnack's inequality.
Sept 5	Fundamental solution and Green's function, representation formulas, Poisson integral. C^0 sub-harmonic functions and its properties.
Sept 7	Solution to the Dirichlet problem for Laplace's equation via the method of sub-harmonic functions.
Sept 12	The wave equation. D'Alembert's formula, Kirchhoff's formula, and finite speed of propagation.
Sept 19	More on the wave equation. The heat equation.
Sept 21	Weak derivatives. Sobolev spaces.
Sept 26	Basic properties of Sobolev spaces.
Oct 3	More properties of Sobolev spaces. Density and trace theorems.
Oct 5	More properties of Sobolev spaces. Extensions. Holder spaces.
Oct 10	Sobolev inequalities.
Oct 17	More on Sobolev inequalities and compact embeddings.
Oct 27	Linear elliptic equations. Weak solutions. Lax-Milgram.
Oct 31	The Fredholm alternative in Banach spaces.
Nov 2	The Fredholm alternative in Hilbert spaces.
Nov 7	Existence and uniqueness of weak solutions to linear elliptic equations. The weak maximum principle.
Nov 9	Elliptic regularity.
Nov 14	Elliptic regularity up to the boundary.
Nov 16	Basic calculus in Frechet and topological vector spaces.
Nov 28	Inverse and implicit function theorems.
Nov 30	Picard iteration and Newton's method in Banach spaces.
Dec 5	The Nash implicit function theorem.
Dec 7	The Nash isometric embedding theorem.

Anonymous feedback
Students are encouraged to bring suggestions and to 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 some anonymous feedback.