I've been looking for reference about solving PDE with Fourier Series. I have a lot of references about Harmonic Analysis like "Fourier Analysis" by Javier Duoandikoetxea and Classical Fourier Analysis by Loukas Grafakos. But all them dedicate their books built in the background of Fourier Analysis and don't apply it to PDE, just a little bit as an example of applications. Other books like "Elementary Applied PDE" by Richard Haberman, solve PDEs using Fourier Series, but do it in a less modern approach, just for smooth functions. I need a book that uses Fourier Series using modern approach, for function in $L^p$ spaces and Sobolev spaces.
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4$\begingroup$ Taylor, vol I, chapters 3,4. $\endgroup$peek-a-boo– peek-a-boo2025-07-11 15:05:27 +00:00Commented Jul 11 at 15:05
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My recommendations:
- Mitrea, D. Distributions, Partial Differential Equations, and Harmonic Analysis.
- Brezis, H. Functional analysis, Sobolev spaces and partial differential equations.
- Paul Sacks. Techniques of Functional Analysis for Differential and Integral Equations.
I use Sacks' book as a textbook and Brezis's as reference for a PDE-oriented Functional Analysis class. I think any book that covers spectral theory for operators (elliptic PDE for example) will have Fourier analysis-flavored material here and there.
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$\begingroup$ I didn't know Paul Sacks. An excellent book! $\endgroup$Reginaldo Demarque da Rocha– Reginaldo Demarque da Rocha2025-07-18 18:01:35 +00:00Commented Jul 18 at 18:01