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