|W 1/16||Cauchy-Kovalevskaya theorem|
|F 1/18||Characteristic surfaces|
|W 1/23||Holmgren's theorem|
|F 1/25||Subspaces of distributions. Compactly supported distributions|
|W 1/30||Distributions as derivatives of functions. Point-supported distributions|
|F 2/1||Convolution of distribution-function pairs|
|W 2/6||Convolution of distributions|
|F 2/8||Fundamental solutions. Schwartz's theorem on hypoellipticity|
|T 2/12||Fourier transform|
|W 2/13||Paley-Wiener theorem. Tempered distributions|
|F 2/15||Malgrange-Ehrenpreis theorem. A necessary condition for hypoellipticity|
|W 2/20||Hörmander's characterization of hypoellipticity|
|F 2/22||Hypoelliptic polynomials. Petrowsky's theorem|
|W 2/27||Wave front set. A microlocal regularity theorem|
|F 3/1||Gårding hyperbolicity|
|W 3/13||The Cauchy problem. Petrowsky well-posedness|
|F 3/15||Strong hyperbolicity and Petrowsky parabolicity|
|W 3/20||Inhomogeneous equations. Duhamel's principle|
|F 3/22||Semilinear evolution equations|
|W 3/27||Multiplication in Sobolev spaces. The Navier-Stokes equations|
|T 4/2||Weak solutions of the Navier-Stokes equations|
|W 4/3||Bochner spaces|
|F 4/5||Boundary value problems in half space. Lopatinsky-Shapiro conditions|
|W 4/10||Overview of elliptic theory. Coercivity and strong ellipticity|
|F 4/12||Gårding inequality. Interior regularity|
|T 4/16||Regularity up to the boundary|
|T 2/5||Baire's theorem and its consequences||
|T 2/12||Make-up lecture||
|T 2/19||Banach-Alaoglu theorem||
|T 2/26||Interpolation of Banach spaces||
|T 3/12||Fredholm operators||
|T 3/19||Fredholm operators||
|T 3/26||Hille-Yosida theorem||
|T 4/2||Make-up lecture||
|T 4/9||Probabilistic approach to the Dirichlet problem||
|F 4/19||The Coulomb Hamiltonian||
|F 4/19||Scattering theory||
|F 4/19||Mean field games||
Lectures: WF Burnside Hall 1205 & 1234 (+ lecture on Apr 16)
Instructor: Dr. Gantumur Tsogtgerel
Prerequisite: MATH 580 (PDE1), MATH 355 (Honours Analysis 4) or equivalent
Note: If you plan to take this course without taking MATH 580, please consult with the instructor.
Topics: The main focus will be on general constant coefficient operators, semilinear evolution equations, and strongly elliptic systems.
The approach will be from the viewpoint of a general theory, rather than as apparently ad hoc treatments of specific examples.
If time allows, the next topics in line are the regularity theory for variational problems, problems with critical exponents, and quasilinear hyperbolic systems.
As such, distributions, Fourier transform, Sobolev spaces, and functional analytic methods will be heavily used.
More precisely, the planned topics are
Calendar description: 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.
There is no required textbook.
The followings are recommended.
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.
Course project: The course project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and giving a lecture.
Grading: The final grade will be the weighted average of homework 40%, the take-home midterm exam 20%, and the course project 40%.