Final exam

The final exam is scheduled on Thursday December 11, at 9am, in Burnside 1205.

It is a closed book exam. Please make sure you review the following topics before the exam.

  • Semilinear first order equations
  • Cauchy-Kovalevskaya theorem
  • Explicit formulas for wave equation, energy method
  • Heat kernel, Tychonov's uniqueness theorem
  • Basic properties of harmonic functions
  • Perron's method

    There will be 5 problems, 3-4 of which are heavily inspired by the homework problems and these practice problems.

    Lecture notes

  • First order equations (Update on Sep 19: Examples 24 and 25 added)
  • The Cauchy-Kovalevskaya theorem
  • Harmonic functions
  • Perron's method

    Assignments

  • Assignment 1 [tex] due Monday September 22
  • Reading assignment: Section 7 of the First order equations notes
  • Assignment 2 [tex] due Wednesday October 8
  • Assignment 3 [tex] due Monday October 27
  • Assignment 4 [tex] due Monday November 17
  • Assignment 5 [tex] due Monday December 1
  • Reading assignment: Harmonic functions

    Class schedule

  • MW 10:05–11:25, Burnside Hall 920 (+ lecture on Dec 4)

    Date Topics
    W 9/3 Ordinary differential equations. Local existence and uniqueness.
    M 9/8 Maximal solutions. Blow-up criterion.
    W 9/10 Flow maps. Continuity of flow maps.
    M 9/15 Differentiability of flow maps.
    W 9/17 Homogeneous linear first order equations. Characteristic curves.
    M 9/22 Noncharacteristic surfaces. Semilinear and quasilinear equations.
    W 9/24 Real analytic functions.
    M 9/29 Analytic solutions to ODE's.
    W 10/1 The Cauchy-Kovalevskaya theorem for first order linear systems.
    M 10/6 Linear equations. Characteristic surfaces.
    W 10/8 Kovalevskaya's example. Well posedness.
    M 10/13 Thanksgiving
    W 10/15 Basic classifications. The wave equation in 1D. Finite speed of propagation.
    M 10/20 Kirchhoff's formula for the wave equation.
    W 10/22 Hadamard's method of descent. Justification of Kirchhoff's formula.
    M 10/27 Asymptotic solutions of the wave equation.
    W 10/29 Geometric optics. The Euclidean heat kernel.
    M 11/3 Maximum principles. Tychonov's uniqueness theorem.
    W 11/5 Nonuniqueness in the heat equation. Mean value property for harmonic functions.
    M 11/10 Harnack inequality. Liouville's theorem. Maximum principles.
    W 11/12 Poisson's formula. Koebe's theorem.
    M 11/17 Removable singularities. Derivative estimates and analyticity.
    W 11/19 Harnack's first theorem. Normal families. Subharmonic functions.
    M 11/24 Perron's method. Boundary regularity.
    W 11/26 Poincaré's and Osgood's criteria. Lebesgue's spine.
    M 12/1 The Riemann mapping theorem (cancelled).
    W 12/3 Hamilton-Jacobi equations (by prof. Oberman).
    R 12/4 Hamilton-Jacobi equations (by prof. Oberman).
    R 12/11 Final exam (Burnside 1205, 9am).

    Tutorial sessions

    Instructor: Ibrahim Al Balushi

  • Th 14:35–15:55, Burnside Hall 1234

    Reference books

  • Lawrence Craig Evans, Partial differential equations. AMS 1998.
  • Qing Han, A basic course in partial differential equations. AMS 2011.
  • Fritz John, Partial differential equations. Springer 1982.
  • Jürgen Jost, Partial differential equations. Springer 2007.

    Online resources

  • Stuff from last year: Math 580, Math 581
  • Intermediate PDE Math 527 pages by Xinwei Yu (Alberta)
  • Lecture notes by Arjen Doelman (Leiden)
  • Harmonic function theory by Sheldon Axler, Paul Bourdon, and Wade Ramey
  • Qing Han's page (Notre Dame)

    Course outline

    Instructor: Dr. Gantumur Tsogtgerel

    Prerequisite: MATH 375 (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

  • First order equations, method of characteristics
  • Cauchy problem for heat and wave equations
  • Duhamel's, Huygens, and maximum principles
  • Green's identities, harmonic functions, Harnack inequality
  • Fundamental solution, Green's function, Poisson's formula
  • Dirichlet problem: Perron's method, barriers, boundary regularity
  • Poisson equations: potential estimates, Kellog's theorem
  • Sobolev spaces, weak and strong derivatives, Dirichlet principle
  • Variational formulation, boundary conditions
  • Elliptic regularity, Sobolev embedding
  • Laplace eigenvalues and eigenfunctions (if time permits)

    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 (counting the take-home midterms as homework).

    Exams: Two take-home midterm exams, and a final exam.

    Grading: Homework + Midterms 60%, Final exam 40%.