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In this course, we study elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation
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English
English [CC]
FREE
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Description
Then we study Fourier and harmonic analysis, emphasizing applications of Fourier analysis. We will see some applications in combinatorics / number theory, like the Gauss circle problem, but mostly focus on applications in PDE, like the Calderon-Zygmund inequality for the Laplacian, and the Strichartz inequality for the Schrodinger equation. In the last part of the course, we study solutions to the linear and the non-linear Schrodinger equation. All through the course, we work on the craft of proving estimates
Course content
- Review of Harmonic Functions and the Perspective We Take on Elliptic PDE Unlimited
- Finding Other Second Derivatives from the Laplacian Unlimited
- Korn’s Inequality I Unlimited
- Korn’s Inequality II Unlimited
- Schauder’s Inequality Unlimited
- Using Functional Analysis to Solve Elliptic PDE Unlimited
- Sobolev Inequality I Unlimited
- Sobolev Inequality II Unlimited
- De Giorgi-Nash-Moser Inequality Unlimited
- Nonlinear Elliptic PDE I Unlimited
- Nonlinear Elliptic PDE II Unlimited
- Barriers Unlimited
- Minimal Graphs Unlimited
- Gauss Circle Problem I Unlimited
- Gauss Circle Problem II Unlimited
- Fourier Analysis in PDE and Interpolation Unlimited
- Applications of Interpolation Unlimited
- Calderon-Zygmund Inequality I Unlimited
- Calderon-Zygmund Inequality II Unlimited
- Littlewood-Paley Theory Unlimited
- Strichartz Inequality I Unlimited
- Strichartz Inequality II Unlimited
- The Nonlinear Schrödinger Equation Unlimited
Instructor
Massachusetts Institute of Technology
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