Lectures WF 1:05pm–2:25pm, Seminars M 1:05pm–2:25pm, Burnside Hall 1205

Class schedule

Note: Click on the date to get Ibrahim's class notes. He has also written some supplementray notes:
  • Matrix exponentials

    Wed 1/11 Idea of distributions. Topological vector spaces.
    Fri 1/13 Topological vector spaces. Hausdorff property.
    Wed 1/18 Locally convex spaces. Seminorms. Fréchet spaces.
    Fri 1/20 LF spaces.
    Wed 1/25 Distributions. Radon measures.
    Fri 1/27 Subspaces of distributions. Basic operations on distributions.
    Wed 2/1 Sheaf structure of distributions.
    Fri 2/3 Local structure of distributions.
    Wed 2/8 Convolution.
    Fri 2/10 Constant coefficient operators. Fundamental solutions. Hypoellipticity.
    Wed 2/15 Schwartz theorem. Laurent expansion. Analytic hypoellipticity.
    Fri 2/17 Fourier transform. Liouville's theorem.
    2/20–2/24 Study break
    Mon 2/27 Hörmander's characterization of hypoelliptic polynomials.
    Fri 3/2 Problems in half-space. Cauchy problem. Petrowsky well-posedness.
    Wed 3/7 Boundary value problems. Lopatinsky-Shapiro condition.
    Fri 3/9 Strongly hyperbolic and p-parabolic systems.
    Wed 3/14 Strong hyperbolicity. Inhomogeneous Cauchy problem.
    Fri 3/16 Well-posedness of a general class of Cauchy problems. Parabolicity.
    Wed 3/21 Semilinear evolution equations.
    Fri 3/23 Multiplication in Sobolev spaces. Derivative nonlinearities.
    Wed 3/28 Elliptic boundary value problems. Gårding inequality.
    Fri 3/30 Gårding inequality proof. Dirichlet problem. Lax-Milgram lemma.
    Wed 4/4 Friedrichs inequality. Rellich-Kondrashov compactness.
    Fri 4/6 Good Friday
    Wed 4/11 L2-regularity theory.
    Fri 4/13 Spectral theory. Semigroups.


  • Assignment 1 due Wednesday January 25
  • Assignment 2 due Friday February 3
  • Assignment 3 due Friday February 17
  • Assignment 4 [tex] due Friday March 2
  • Assignment 5 [tex] due Friday March 16
  • Assignment 6 [tex] due Friday March 30
  • Assignment 7 [tex] due Monday April 16

    Final project

    4/16 Morgane Henry Wave maps
    4/16 Ibrahim Al Balushi Nonlinear diffusion
    4/16 Olga Yakovlenko Introduction to the Ricci flow
    4/30 Yang Guo Einstein equations
    4/30 Sebastien Picard Introduction to the Yang-Mills equations
    4/30 Spencer Frei DeGiorgi-Nash-Moser regularity theory
    4/30 Mario Palasciano A nonlocal aggregation model
    4/30 Olivier Mercier Mean curvature flow
    4/30 Joshua Lackman Pseudodifferential operators
    4/30 Andrew MacDougall Some topics in semiclassical analysis

    The final project consists of the student studying an advanced topic, typing up expository notes, and presenting it in class. Here are some ideas for the project:

  • Maxwell-Klein-Gordon equations
  • Harmonic maps
  • Harmonic map heat flow
  • Wave maps
  • Yamabe problem
  • Positive mass theorem
  • Einstein's constraint equations
  • Yang-Mills equations
  • Mean curvature flow
  • Ricci flow
  • Einstein equations
  • Plateau's problem
  • Monge-Ampere equations
  • Nirenberg's solution of Weyl and Minkowski problems
  • Comparative study of the regularity works of De Giorgi, Nash, and Moser
  • Krylov-Safanov estimates
  • Nash embedding theorem
  • Nash-Moser iteration (implicit function theorem)
  • Partial regularity for the Navier-Stokes equations
  • Can one hear the shape of a drum?
  • Pseudodifferential operators

    Weekly seminars

    PDE questions from previous qualifying exams for download.

    Suggested reading
    1/16 Baire's theorem and consequences Rudin Ch2, Tao Gantumur
    1/23 Hahn-Banach theorem Rudin §3.1-3.7, Tao Spencer
    1/30, 2/6 Closed range theorem Rudin §4.1-4.15 Ibrahim
    2/13, 2/29 Fredholm operators Rudin §4.16-4.25, McLean 2.14-2.17, Tao Mario
    3/5 Banach-Alaoglu theorem Rudin §3.8-3.18, Tao Andrew
    3/12, 3/19 Spectral theorem Rudin Ch13, Jaksic Sébastien
    3/26 Hille-Yosida theorem Rudin §13.34-13.37 Yang
    4/2 Navier-Stokes equations Tao, Clay, Lei-Lin Gantumur


  • Walter Rudin, Functional analysis. McGraw-Hill.
  • François Trèves, Topological vector spaces, distributions and kernels. Dover 1995.
  • Gregory Eskin, Lectures on linear partial differential equations. AMS 2011.
  • Izrail Moiseevich Gelfand and Georgi Evgenevich Shilov, Generalized functions III. Academic Press 1967.
  • François Trèves, Basic linear partial differential equations. Dover 2006.
  • Heinz-Otto Kreiss and Jens Lorenz, Initial-boundary value problems and the Navier-Stokes equations. SIAM 2004.
  • Gerald Budge Folland, Introduction to partial differential equations. Princeton 1995.
  • Joseph Theodor Wloka, Partial differential equations. Cambridge 1987.

    Topics to be covered

  • Distributions and constant coefficient operators: Introduction to distributions, fundamental solutions, parametrices, hypoellipticity and ellipticity, introduction to Fourier transform, hyperbolicity and parabolicity;
  • Linear elliptic operators: Dirichlet principle, Sobolev spaces, Poincaré inequality, variable coefficients, stationary Stokes problem and linear elasticity, Harnack estimates, regularity theory, Sobolev embedding, Hodge-type decompositions, introduction to spectral and Fredholm theories;
  • Nonlinear elliptic equations: Lagrange multipliers, semilinear problems with subcritical, critical, and supercritical exponents, direct method of calculus of variations, De Giorgi-Nash-Moser regularity theory;
  • Evolution equations: Heat, wave, Schrödinger, and Stokes propagators, Navier-Stokes, Euler and magnetohydrodynamics equations, turbulence models, nonlinear heat and wave examples, weak and strong solutions, regularity theory, viscosity solutions, linear scattering theory (if time permits)


    Dr. Gantumur Tsogtgerel
    Office: Burnside Hall 1123. Phone: (514) 398-2510.
    Email: gantumur -at- math.mcgill.ca.
    Office hours: Just drop by or make an appointment

    Online resources

    PDE Lecture notes by Bruce Driver (UCSD)

    Xinwei Yu's page (Check the Intermediate PDE Math 527 pages)

    John Hunter's teaching page at UC Davis (218B is PDE)

    Textbook by Ralph Showalter on Hilbert space methods

    Lecture notes by Georg Prokert on elliptic equations


    MATH 355 or equivalent, MATH 580.

    Catalog 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.


    Homework 40%, take-home midterm exam 20%, final project 40%.
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