Differential analysis ii: partial differential equations and fourier analysis

Lectures: 3 sessions / week, 1 hour / session

18.155 Differential Analysis

In this course, we study elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation. 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. For a much more detailed description of the class, please see the course summary.

There is no required textbook. Lecture notes are available.

If you want to brush up on Fourier analysis, this undergraduate book by Stein and Shakarchi is very readable, and it covers the background in the subject that we use.

Stein, Elias M., Rami Shakarchi. Fourier Analysis: An Introduction (Princeton Lectures in Analysis). Princeton University Press, 2003. ISBN: 9780691113845.

Fourier analysis quick review (PDF) is a super-quick summary of some fundamental facts about Fourier analysis that you can review before we start the Fourier analysis unit.

There are 6 problem sets plus optional further problems for study and review.

The grade was based on an average of the six problem sets.

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