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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


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.


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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
Submit Your Assignment 00:00:00
Certification 00:00:00

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