|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||Picard-Lindelöf 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||Rankine-Hugoniot 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|
|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||C2 estimates|
|Th 11/01||Dirichlet energy. Sobolev spaces. Weak and strong derivatives|
|Mo 11/05||Meyers-Serrin theorem. Weak solutions. Boundary behaviour. Interior L2 regularity|
|Th 11/08||Regularity up to the boundary. Hilbert space method|
|Fr 11/09||(at 11:30 in 1214) Spectral theory for compact self-adjoint 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 Hilbert-Schmidt 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) Faber-Krahn inequality. Pleijel's nodal domain theorem. Coercivity. Spectral resolution of heat equation.|
|Mo 11/26||Heat equation: semigroup property, smoothing, and decay. Well-posedness. Backward heat equation|
|Th 11/29||Log-energy 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|
|Fr 9/28||Lebesgue integration primer||
|Fr 10/5||Lie group methods||
|Fr 10/19||Sobolev embedding theorem||
|Fr 10/26||Lp spaces||
|Fr 11/02||Approximation and extensions||
|Fr 11/16||Rellich's theorem and trace inequalities||
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. Cauchy-Kowalevskaya 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 take-home midterm exam 20%, and the course project 40%.