Date  Topics 
Th 9/6  Idea of distributions. Seminorms 
Tu 9/11  Test functions and distributions 
Th 9/13  Operations on distributions 
Mo 9/17  Vector valued functions. ODE. Contraction mapping principle 
Th 9/20  PicardLindelöf theorem 
Mo 9/24  First order semilinear equations. Method of characteristics 
Th 9/27  First order quasilinear equations. Conservation laws. Wave breaking 
Mo 10/1  RankineHugoniot conditions. Green's identities. Fundamental solutions of the Laplacian 
Th 10/4  Mean value property. Harnack inequalities. Koebe's theorem. Derivative estimates. Liouville's theorem 
Mo 10/8  Thanksgiving 
Th 10/11  Analyticity. Maximum principles. Green's function approach 
Mo 10/15  Poisson's formula. Removable singularity. Harnack's convergence theorems 
Th 10/17  Dirichlet problem. Perron's method 
Mo 10/22  Perron's method. Barriers. Boundary regularity 
Th 10/24  Poisson equations. Newtonian potential 
Mo 10/29  C^{2} estimates 
Th 11/01  Dirichlet energy. Sobolev spaces. Weak and strong derivatives 
Mo 11/05  MeyersSerrin theorem. Weak solutions. Boundary behaviour. Interior L^{2} regularity 
Th 11/08  Regularity up to the boundary. Hilbert space method 
Fr 11/09  (at 11:30 in 1214) Spectral theory for compact selfadjoint positive operators 
Mo 11/05  No lecture (moved to Fr 11/09) 
Th 10/08  No lecture (moved to Fr 11/23) 
Mo 11/19  Application of the HilbertSchmidt theory to the Dirichlet and Neumann Laplacians on bounded domains 
Th 11/22  Courant's minimax principle. Weyl's law. Courant's nodal domain theorem 
Fr 11/23  (at 11:30 in 1214) FaberKrahn inequality. Pleijel's nodal domain theorem. Coercivity. Spectral resolution of heat equation. 
Mo 11/26  Heat equation: semigroup property, smoothing, and decay. Wellposedness. Backward heat equation 
Th 11/29  Logenergy convexity. Backward uniqueness. Cauchy problem for the heat equation. Heat kernel. Parabolic maximum principles. 
Mo 12/03  Tychonov's example. Spectral resolution of the wave equation. Kirchhoff's formula 
We 12/05  (at 10:05 in 1205) Method of descent. Energy estimates for waves. Spacetime energy flux. Basic classifications of PDEs. 
Tu 12/11  Deadline for preliminary version of the report 
Th 12/13  Student presentations 
Fr 12/14  Student presentations 
Date  Topics  Speaker 
Fr 9/28  Lebesgue integration primer 

Fr 10/5  Lie group methods 

Fr 10/12  cancelled 

Fr 10/19  Sobolev embedding theorem 

Fr 10/26  L^{p} spaces 

Fr 11/02  Approximation and extensions 

Fr 11/09  Lecture 

Fr 11/16  Rellich's theorem and trace inequalities 

Fr 11/23  Lecture 

Instructor: Dr. Gantumur Tsogtgerel
Prerequisite: MATH 375 (Honours PDE) or equivalent
Topics: The main focus of the course is going to be on linear second order and quasilinear first order equations.
If time allows, we will discuss symmetric hyperbolic systems and some nonlinear problems.
Rather than trying to build everything in full generality,
we will study prototypical examples in detail to establish good intuition.
Distributions, Sobolev spaces, and functional analytic methods will be introduced.
This essentially means that we will end up covering the topics from the calendar description of Math 580, some from that of Math 581, and some additional topics.
More precisely, the planned topics are
Calendar description: Classification and wellposedness of linear and nonlinear partial differential equations; energy methods; Dirichlet principle. Brief introduction to distributions; weak derivatives. Fundamental solutions and Green's functions for Poisson equation, regularity, harmonic functions, maximum principle. Representation formulae for solutions of heat and wave equations, Duhamel's principle. Method of Characteristics, scalar conservation laws, shocks.
Calendar description of Math 581: Systems of conservation laws and Riemann invariants. CauchyKowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.
Homework: Assigned and graded roughly every other week.
Weakly seminars: We will organize weekly seminars on problem solving, standard results from analysis and geometry, and other stuff related to the course. Attendance is optional.
Course project: The course project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and presenting it in class as a short lecture.
Grading: The final grade will be the weighted average of homework 40%, the takehome midterm exam 20%, and the course project 40%.