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This course will focus on the elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation. Then we also cover Fourier and harmonic analysis, emphasizing applications of Fourier analysis, number theory, Gauss circle problem, applications in PDE, and the Strichartz inequality for the Schrodinger equation. At the last part, we emphasize solutions to linear and the nonlinear Schrodinger equation
Assessment
This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.
Certification
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Course Credit: MIT
Course Curriculum
Module: 01 | |||
Lec1 Review of Harmonic Functions and the Perspective We Take on Elliptic PDE | 00:15:00 | ||
Lec2 Finding Other Second Derivatives from the Laplacian | 00:10:00 | ||
lec3 Korn’s Inequality I | 00:05:00 | ||
lec4 Korn’s Inequality II | 00:15:00 | ||
lec5 Schauder’s Inequality | 00:15:00 | ||
lec6 Using Functional Analysis to Solve Elliptic PDE | 00:10:00 | ||
lec7 Sobolev Inequality I | 00:10:00 | ||
lec8 Sobolev Inequality II | 00:15:00 | ||
Module: 02 | |||
lec9 De Giorgi-Nash-Moser Inequality | 00:20:00 | ||
lec10 Nonlinear Elliptic PDE I | 00:15:00 | ||
lec11 Nonlinear Elliptic PDE II | 00:10:00 | ||
lec12 Barriers | 00:10:00 | ||
lec13 Minimal Graphs | 00:15:00 | ||
lec14 Gauss Circle Problem I | 00:15:00 | ||
lec15 Gauss Circle Problem II | 00:05:00 | ||
lec16 Fourier Analysis in PDE and Interpolation | 00:15:00 | ||
Module: 03 | |||
lec17 Applications of Interpolation | 00:10:00 | ||
lec18 Calderon-Zygmund Inequality I | 00:10:00 | ||
lec19 Calderon-Zygmund Inequality II | 00:10:00 | ||
lec20 Littlewood-Paley Theory | 00:10:00 | ||
lec21 Strichartz Inequality I | 00:15:00 | ||
lec22 Strichartz Inequality II | 00:10:00 | ||
lec23 The Nonlinear Schrödinger Equation | 00:30:00 | ||
Assessment | |||
Submit Your Assignment | 00:00:00 | ||
Certification | 00:00:00 |
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