18.152 - Introduction To PDEs (Spring 2020)

Lecture 1 : Higher order estimates for solution to Poisson equation and applications

  • Lecture 1 (A) Intro to the Kellogg's theorem
  • Lecture 1 (B) Example of existence results I
  • Lecture 1 (C) Example of existence results II

  • Lecture 2 : Schauder estimates I

  • Lecture 2 (A) Global Schauder estimates
  • Lecture 2 (B) Fundamental Schauder estimates

  • Lecture 3 : Schauder estimates II

  • Lecture 3 (A) Interior Schauder estimates I
  • Lecture 3 (B) Interior Schauder estimates II

  • Lecture 4 : Banach space

  • Lecture 4 (A) Interior Schauder estimates III
  • Lecture 4 (B) C^0 estimates
  • Lecture 4 (C) Banach space
  • Lecture 4 (C) Erratum Bounded C^k space is not sequentially compact

  • Lecture 5 : Introduction to the wave equation

  • Lecture 5 (A) Existence of solutions to linear elliptic PDEs
  • Lecture 5 (B) Introduction to the wave equation
  • Lecture 5 (C) Separation of variables

  • Lecture 6 : 1D Cauchy problem

  • Lecture 6 (A) d'Alembert's formula
  • Lecture 6 (B) Nonhomogeneous equation
  • Lecture 6 (C) Local energy

  • Lecture 7 : nD Cauchy problem

  • Lecture 7 (A) Euler-Poisson-Darboux equation
  • Lecture 7 (B) Kirchhoff formula and applications

  • Lecture 8 : Energy estimates

  • Lecture 8 (A) Intro to Sobolev Inequality
  • Lecture 8 (B) Energy estimates
  • Lecture 8 (C) Sobolev Inequality proof I

  • Lecture 9 : Eigenfunctions and Eigenvalues

  • Lecture 9 (A) Sobolev Inequality proof II
  • Lecture 9 (B) Intro to eigenfunctions
  • Lecture 9 (C) Fredholm alternative
  • Lecture 9 (D) The first eigenfunction and calculus of variations

  • Lecture 10 : Hilbert space and Existence of eigenfunctions

  • Lecture 10 (A) Construction of orthogonal eigenfunctions
  • Lecture 10 (B) Hilbert space
  • Lecture 10 (C) Existence of H^1_0 eigenfunctions

  • Lecture 11 : Weak convergence

  • Lecture 11 (A) Weak solutions and Regularity of eigenfunctions
  • Lecture 11 (B) Subsequential weak convergence of bounded sequences
  • Lecture 11 (C) Weak convergence is weak

  • Lecture 12 : Eigenfunctions span L^2 space

  • Lecture 12 (A) Rellich–Kondrachov theorem I
  • Lecture 12 (B) Rellich–Kondrachov theorem II and Divergence of digenvalues
  • Lecture 12 (C) Eigenfunctions span L^2 space
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