| Date | Topics |
| W 9/5 | Examples of harmonic functions. Green's identities. Fundamental solutions. |
| F 9/7 | Green's representation formula. Mean value property. |
| W 9/12 | Harnack inequality. Liouville's theorem. Maximum principles. Green's function. |
| F 9/14 | Poisson kernel. Poisson's formula. |
| W 9/19 | Removable singularities. Converse to the mean value property. Derivative estimates. |
| F 9/21 | Analyticity. Harnack's convergence theorem. |
| W 9/26 | Harnack's principle. Perron's method for the Dirichlet problem. |
| F 9/28 | Boundary regularity. |
| W 10/3 | Minimization of the Dirichlet energy. |
| F 10/5 | Sobolev spaces. Strong derivatives. Weak solutions. |
| W 10/10 | A density theorem. Friedrichs inequality. |
| F 10/12 | Mollifiers. Weyl's lemma. |
| W 10/17 | Weak derivatives. Meyers-Serrin theorem. |
| F 10/19 | Trace map. Boundary behavior of weak solutions. |
| W 10/24 | Poisson problems. Moduli of continuity. |
| F 10/26 | Nikolsky spaces. Interior regularity. |
| W 10/31 | Higher regularity. Sobolev's lemma. |
| F 11/2 | Analyticity. Regularity in half space. |
| W 11/7 | Boundary flattening. |
| F 11/9 | Localization. Classical regularity up to the boundary. |
| W 11/14 | Resolvent of the Laplacian. |
| F 11/16 | Rellich's compactness lemma. |
| W 11/21 | Hilbert-Schmidt theory. |
| F 11/23 | Laplace eigenproblems. Rayleigh quotient. Explicit examples. |
| W 11/28 | Minimax principle. Monotonicity properties. Courant's nodal domain theorem. |
| F 11/30 | Weyl's law. |
| W 12/12 | Final exam (Burnside 1205, 6:30pm) |
Instructor: Dr. Gantumur Tsogtgerel
Prerequisite: MATH 475 (Honours PDE) or equivalent
Topics: The main focus of the course is going to be on linear first and second order equations, and Sobolev spaces.
Rather than trying to build everything in full generality, we will study prototypical examples in detail to establish good intuition.
Roughly speaking, most of the topics from the calendar description of Math 580 and some from that of Math 581 will be covered.
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-Kovalevskaya theorem, power 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 2 weeks.
Exams: A final exam.
Reading seminars: We will organize weekly student seminars on reading assignments, problem solving, and other stuff related to the course. Attendance is optional.
Grading: Assignments 60%, Final exam 40%.