Course projects

  • T. Salvador - Mean field games
  • B. Landon - Introductory scattering theory
  • A. Kumar - Kato's theorem on the Coulomb Hamiltonian
  • M.-A. Mandich - Probabilistic approach to the Dirichlet problem


  • Assignment 1 [tex] due Wednesday January 30
  • Assignment 2 [tex] due Friday February 15
  • Assignment 3 [tex] due Friday March 1
  • Assignment 4 [tex] due Friday March 15
  • Assignment 5 [tex] due Tuesday April 2
  • Assignment 6 [tex] due Tuesday April 16

    Class schedule

  • Wednesdays 13:05–14:25, Burnside Hall 1205
  • Fridays 10:05–11:25, Burnside Hall 1234
  • Tue Apr 16 will follow a Wednesday schedule
  • No lectures in the first week, we'll arrange make-up lectures later

    Date Topics
    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

    Student seminar

  • Tuesdays 11:35–12:55, Burnside Hall 1214

    Date Topics Speaker
    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

    Course outline

    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

  • Cauchy-Kovalevskaya theorem, Holmgren's theorem
  • Distributions, convolution, Fourier transform
  • Constant coefficient operators, fundamental solution, hypoellipticity
  • Malgrange-Ehrenpreis theorem
  • Hörmander's theorem on hypoelliptic polynomials
  • Problems in half-space, shades of hyperbolicity, parabolicity, and ellipticity
  • Fourier analytic treatment of Sobolev spaces
  • Semilinear evolution equations, Duhamel's principle
  • The Navier-Stokes equations
  • Overview of elliptic theory, Lopatinsky-Shapiro condition
  • Strongly elliptic systems, Gårding inequality
  • Elliptic regularity
  • Variational minimization problems, gradient flows
  • Semilinear elliptic problems with critical exponents (if time permits)
  • Quasilinear hyperbolic systems (if time permits)

    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.

    Books: There is no required textbook. The followings are recommended.

  • Gerald Budge Folland, Introduction to partial differential equations. Princeton 1995.
  • Gregory Il'ich Eskin, Lectures on linear partial differential equations. AMS 2011.
  • Joseph Theodor Wloka, Partial differential equations. Cambridge 1987.

    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%.

    Online resources

  • Stuff from last year, from last semester
  • Lecture notes by Bruce Driver (UCSD)
  • Teaching page of John Hunter (UC Davis)
  • Textbook on Hilbert space methods by Ralph Showalter (Texas State)
  • Notes on variational methods by Georg Prokert (Eindhoven)
  • Qing Han's page (Notre Dame)
  • Review article on hyperbolic initial-boundary value problems by Olivier Sarbach and Manuel Tiglio