Notes

Sobolev Space Reading Course

Together with Nathan Willis, I am co-organising a reading course about Sobolev space in Fall 2018. This is a graduate-student led reading course with guidance from Fernando Guevara Vasquez, Christel Hohenegger, and Akil Narayan. The textbook is L. C. Evans "Partial Differential Equations". The plan is to read Chapter 5 (Sobolev spaces) and if time permits, Chapter 6 (second-order elliptic PDEs). We meet Thursday from 1-3PM in JWB 108. The lectures are intended to be approximately 1.5 hours a week, follow by 30 minutes for questions and discussion.

August 30: Chee Han, Section 5.2 [Notes]
  • Motivation, weak derivatives, definition of Sobolev spaces ($W^{k,p}(\Omega)$, $H^{k}(\Omega)$, $W_0^{k,p}(\Omega)$), examples of Sobolev functions, properties of weak derivatives, completeness of Sobolev spaces

  • September 6 - Chee Han, Section 5.3, Appendix C.4 [Notes]
  • Mollifiers, mollification of $L_\textrm{loc}^1(\Omega)$ functions and its properties, local (interior) approximation of $W^{k,p}(\Omega)$ functions by mollifiers, global approximation of $W^{k,p}(\Omega)$ functions by $C^{\infty}(\Omega)$ functions
  • Read ourselves: Global approximation of $W^{k,p}(\Omega)$ functions by $C^{\infty}(\overline\Omega)$ functions

  • September 13 - Huy, Section 5.4, Appendix C.1 [Notes]
  • $C^k$ boundary, extension of $W^{k,p}(\Omega)$ to $W^{k,p}(\mathbb{R}^n)$

  • September 20 - Zexin, Section 5.5
  • Trace operators, characterisation of $W_0^{k,p}(\Omega)$ functions

  • September 27 - Dihan, Section 5.6.1 [Notes]
  • Sobolev conjugate, Gagliardo-Nirenberg-Sobolev inequality
  • Sobolev embedding of $W^{k,p}(\Omega)$ for $p\in[1,n)$
  • Poincaré inequality for $W_0^{1,p}(\Omega)$ functions

  • October 4 - China, Section 5.6.2
  • Hölder space, Morrey's inequality, Sobolev embedding for $W^{k,p}(\Omega)$ functions with $k\gt 1$

  • October 18 - Rebecca, Section 5.7
  • Rellich-Kondrachov theorem

  • October 25 - Nathan, Section 5.8.1 and 5.8.2 [Notes]
  • Poincaré inequality for $W^{1,p}(\Omega)$ functions
  • Difference quotients

  • November 1 - Nathan, Section 5.8.3 and 5.8.4 [Notes]
  • Characterisation of $W^{1,\infty}(\Omega)$ functions
  • Differentiability almost everywhere
  • Characterisation of $H^k(\mathbb{R}^n)$ by Fourier transform

  • November 8 - Huy, Section 5.9
  • Definition of $H^{-1}(\Omega)$ and its characterisation

  • November 15 - Zexin, Section 6.2.2
  • Divergence form elliptic operator and its bilinear form
  • Energy estimates for bilinear form
  • Existence of weak solutions for linearly perturbed elliptic operator
  • Definition of adjoint operator

  • November 29 - China, Section 6.2.3
  • Fredholm alternative

  • December 6 - Rebecca

  • Math 6640: Introduction to Optimisation

    This course is currently taught by Braxton Osting in Fall 2018. The textbook is A. Beck "IntroductIon to Nonlinear Optimization:Theory, Algorithms, and Applications with MATLAB". The lecture notes would be written jointly with Rebecca Hardenbrook and Nathan Willis.
  • Notes (ongoing)

  • Math 6710: Applied Linear Operator and Spectral Methods

    This course was taught by Fernando Guevara Vasquez in Fall 2015. The textbook is E. Kreyszig "Introductory Functional Analysis with Application".
  • Hahn-Banach theorem
  • Solution to Chapter 1 (Metric Spaces)
  • Solution to Chapter 2 (Normed Spaces & Banach Spaces)

  • Math 6410: Ordinary Differential Equations

    This course was taught by Paul Bressloff in Fall 2015. The main textbook is C. Chicone "Ordinary Differential Equations with Applications".
  • Notes (incomplete)

  • Math 6610: Analysis of Numerical Method I

    This course was taught by Yekaterina Epshteyn in Fall 2016. The main textbook is L. N. Trefethen "Numerical Linear Algebra”.
  • Notes (Chapter 3 and 4 are incomplete)

  • Math 6620: Analysis of Numerical Method II

    This course was taught by Yekaterina Epshteyn in Spring 2017. The main textbook is K. Atkinson “An Introduction to Numerical Analysis”.
  • Notes

  • Math 6730: Asymptotic and Perturbation Methods.

    This course was taught by Paul Bressloff in Fall 2017. The textbook is M. H. Holmes "Introduction to Perturbation Methods". The note was written together with Hyunjoong Kim.
  • Notes

  • Math 6880: Fluid Dynamics

    This course was taught by Aaron Fogelson and Christel Hohenegger in Fall 2017 and Spring 2018. The course’s webpage for the first half can be found here.
  • Notes (in progress)

  • Miscellaneous

    Some other useful notes:
  • Asymptotic Expansions.



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