MATH 581, Partial Differential Equations II, Winter 2018

Course web page: http://www.math.mcgill.ca/jakobson/courses/math581.html

Lectures:
  • Organizational meeting: Thursday, January 11, 12noon, Burnside 1234.
  • Schedule: Thursday, 9:30-11:30, Burnside 1120. Friday, 9:00-10:00, Burnside 1120.
    • Instructor: D. Jakobson
    Office: Burnside Hall 1220, Office Hours: TBA
    Tel: 398-3828
    E-mail: dmitry.jakobson AT mcgill.ca
    Web Page: www.math.mcgill.ca/jakobson
    Prerequisite: Math 355 or equivalent, Math 580
    Texts
  • Required: Lawrence C. Evans. Partial Differential Equations. Second edition. Graduate Studies in Mathematics, Vol. 19, AMS, Providence, RI 2010.
  • Recommended: David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Springer, 2001.
  • Recommended: Haim Brezis. Functional Analysis, Sobolev Spaces and Partial Diffrerential Equations. Springer, Universitext, 2010.
  • Recommended (Spectral geometry): Y. Canzani, Analysis on Manifolds via the Laplacian
    • Syllabus:
  • Continuation of 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.
    We shall cover remaining material from Chapters 6 in Evans (sections 6.4 and 6.5); we shall then cover selected material from Chapters 7, 8, 9 and possibly 10. Other topics (e.g. Geometric PDE, spectral theory of elliptic operators on compact manifolds etc) will be discussed as time permits.
    • Tests
  • There will be a midterm that will count 30% of the grade. You can write a take-home midterm, in-class midterm or both. Your grade will be the maximum of the 2 scores.
  • Further details to be announced
    • 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. Lecture should be approximately 30 minutes long. The topics are chosen by students in consultation with the instructor. The project is worth 40% of the final mark.
    List of possible topics (work in progress):
  • Operators with singular continuous spectrum
  • Heat invariants
  • Shape optimization
  • Poincare's inequalities
  • Isoperimetric inequalities
  • Eigenvalues of non-symmetric elliptic operators (Evans, section 6.5.2)
  • Non-variational techniques (Evans, chapter 9), several possible topics there
  • Minimal surfaces
  • Yamabe problem
  • Nirenberg problem (curvature prescription)
  • Einstein equations in GR
    • Grading:
  • Your final mark will be the following average: 30% Assignments + 30% Midterm + 40% course project
  • In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.
    • MyCourses: Your scores on assignments, midterm, course project, and your final mark will be posted on MyCourses
    Supplemental: There will be a supplemental exam, counting for 100% of the supplemental grade. No additional work will be accepted for D, F or J.
    Homework will be assigned in class and will be due by 5pm by the specified deadline.

    Assignments

  • The assignments will be worth 30% of the mark.
  • Assignment 1. Due Friday, February 16: Assignment 1
  • Assignment 1, Part 2. Due Friday, February 16: Assignment 1, Part 2
  • Extra credit problem, Cantor staircase function, due date TBA: EC 1
  • Assignment 2. Due date to be announced. Assignment 2

    • Lecture notes on related topics

  • D. Jakobson, N. Nadirashvili and J. Toth. Geometric Properties of Eigenfunctions (ps)
  • D. Jakobson. Spectra, dynamical systems, and geometry. Lecture at the SMS in Geometric and Computational Spectral Theory CRM, Montreal, June 2015.
  • A. Hatcher Notes on Introductory Point-Set Topology
  • V. Jaksic. Topics in Spectral theory (including complex measures, Fourier transform of measures, Poisson transform, spectral theory of self-adjoint operators, spectral theory of rank one perturbations)
  • Steve Zelditch. Eigenfunctions of the Laplacian of Riemannian manifolds
  • Peter Gilkey. The spectral geometry of operators of Dirac and Laplace type Handbook of Global Analysis, 2007.
  • A 1990 paper by E. Lieb, containing the proof of Hausdorff-Young inequality with sharp constants and characterizing the minimizers.
  • K. Datchev and H. Hezari. Inverse problems in Spectral geometry, arxiv:1108.5755
    • Course material from previous related courses at McGill:
  • Math 581: G. Tsogtgerel, W2014, G. Tsogtgerel, W2013, G. Tsogtgerel, W2012.
  • Math 580: G. Tsogtgerel, F2014, G. Tsogtgerel, F2013, G. Tsogtgerel, F2012.
  • L. Chen and D. Jakobson. MATH 599: Topics in Geometry and Topology (Fall 2016)
  • Old Math 564, Fall 2008 and Math 564, Fall 2009 web pages, D. Jakobson. Old Math 564, Fall 2010 web page, V. Jaksic. Math 564, Fall 2016 web page, Jeff Galkowski.
  • Old Math 565, Winter 2009, and Math 565, Winter 2010, Math 565, Winter 2012 and Math 565, Winter 2016 web pages, D. Jakobson. Old Math 565, Winter 2011 web page, V. Jaksic.
  • Old Math 354 and Math 355 web pages, D. Jakobson
  • Sam Drury's lecture notes for MATH 354 and MATH 355
  • Old Math 366 web page, D. Jakobson, Fall 2007
  • Vojkan Jaksic's Lecture Notes in Spectral Theory, ps and pdf.
    • Web links in Analysis
  • Metric space, Topology glossary, Functional analysis in Wikipedia
  • Norm, Holder's inequality, Minkowski inequality, Lp space, Hilbert space, Banach space, Cantor set, p-adic numbers in Wikipedia
  • Companion notes to Rudin's (undergraduate!) book
  • Harmonic Analysis page by Terry Tao (there is a lot of advanced stuff there)
    • HELPDESK and their email: helpdesk@math.mcgill.ca
    NOTICE: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for more information).
    NOTICE: In accord with McGill University's Charter of Student Rights, students in this course have the right to submit in English or in French any work that is to be graded.
    NOTICE: In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change