Marcelo Mendes Disconzi
Department of Mathematics, Vanderbilt University
email: marcelo.disconzi at vanderbilt.edu
office: A1017, 17th & Horton (Sony bld)
phone: (615) 322 7147
mail to: 1326 Stevenson Center Ln, Vanderbilt University, Nashville, TN, 37240
Vanderbilt










Marcelo Mendes Disconzi
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.


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.

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.


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