Course projects

  • Mashbat Suzuki - Hodge theory
  • Patrick Munroe - Uniformization
  • Selim Tawfik - Yamabe problem
  • Ilia Polotskii - Maximum principles
  • Alexandru Verzea - Minimal surfaces
  • Siyuan Lu - Nash embedding theorem
  • Aditya Kumar - Thomas-Fermi model
  • Tiago Salvador - Monge-Ampere equations
  • Michael Snarski - Calderon-Zygmund estimate
  • Dustin Connery-Grigg - Control theory and PDE
  • Ben Landon - Spectral theory of Schrödinger operators
  • Marc-Adrien Mandich - Morrey and Campanato spaces
  • Christophe Morris - Exact solutions of the Einstein equations
  • Sisi Shen - Harnack inequalities on complete Riemannian manifolds
  • Simon Szatmari - Similarity solutions in shallow water-type equations


  • Assignment 1 [tex] due Monday September 24
  • Assignment 2 [tex] due Thursday October 11
  • Assignment 3 [tex] due Thursday October 25
  • Assignment 4 [tex] due Thursday November 8
  • Assignment 5 [tex] due Thursday November 22
  • Assignment 6 [tex] due Wednesday December 5

    Suggested references

  • Distributions: my notes, Duistermaat-Kolk, Hörmander, Rudin
  • First order equations: my notes, Doelman-Duistermaat, Yu, Evans
  • Harmonic functions: my notes, Axler-Bourdon-Ramey, Han, Evans
  • Dirichlet problem: Doelman-Duistermaat, Jost, Evans, Han
  • Laplace eigenproblem: Doelman-Duistermaat, Jost, Evans
  • Heat and wave equations: Doelman-Duistermaat, Han, Evans

    Class schedule

  • M 10:05–11:25, R 8:35–9:55, Burnside Hall 1205 (+ lecture on Dec 5)

    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 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
    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 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

    Student seminar

  • F 11:35–12:55, Burnside Hall 1214

    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 Lp spaces
    Fr 11/02 Approximation and extensions
    Fr 11/09 Lecture
    Fr 11/16 Rellich's theorem and trace inequalities
    Fr 11/23 Lecture


    Some ideas for the project:
  • Pseudoholomorphic functions
  • Integral equation method
  • Moving plane method
  • Reaction-diffusion equations
  • Conservation laws
  • Heat equation on closed manifolds
  • Li-Yau inequalities
  • Schauder theory
  • Special solutions of the Navier-Stokes equations

    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.
  • Jeffrey Rauch, Partial differential equations. Springer 1999.

    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)
  • Lecture notes by Bruce Driver (UCSD)
  • Harmonic function theory by Sheldon Axler, Paul Bourdon, and Wade Ramey
  • Textbook on Hilbert space methods by Ralph Showalter (Texas State)
  • Qing Han's page (Notre Dame)
  • Alberto Bressan's page (Penn State)

    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 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

  • Introduction to distributions, seminormed spaces
  • First order equations, method of characteristics
  • Conservation laws, shocks, entropy solutions
  • Classification of second order equations, maximum principles
  • Green's identities, harmonic functions, Harnack inequality
  • Fundamental solution, Poisson's formula, potential estimates
  • Perron's method, barriers, boundary regularity
  • Introduction to Sobolev spaces, Dirichlet principle
  • Introduction to variational methods, boundary conditions
  • Spectral theory of compact self-adjoint operators
  • Spectral resolution of heat and wave equations
  • Cauchy problem for evolution equations
  • Duhamel's principle, Huygens principle
  • Symmetric hyperbolic systems (if time allows)
  • Nonlinear problems (if time allows)

    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%.