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I am looking for an advanced book on partial differential equations that makes use of functional analysis as much as possible. All the books I have looked in so far either shy away from functional analysis and try to avoid even basic concepts, or present results from functional analysis I know anyway just to discuss some very basic applications to partial differential equations (say, semigroup theory applied to the heat equation).

The book I am looking for should

  • use functional analysis instead of hard analysis whenever possible (I am well aware of the fact that the theory of partial differential equations is not merely an application of functional analysis),
  • go into some advanced topics that are relevant for research, and
  • not spend too much space on covering the results of functional analysis itself - I have my references for that.

The background is that I am interested in operator equations that are not partial differential equations, yet methods from pde are often helpful. If it is relevant, I am mostly interested in elliptic and parabolic equations, although I don't want to limit the focus.

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    $\begingroup$ Really very, very broad. Maybe too much... Can you define "as much as possible"? I work in PDE theory, but the amount of functional analysis I need is just reasonable. Moreover, research papers in PDEs are often based on hard analysis. $\endgroup$
    – Siminore
    Oct 28, 2016 at 11:37
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    $\begingroup$ @Siminore: I know that hard analysis can't be avoided. I am just looking for a book that avoids hard analysis wherever it can be avoided. For example, whenever the spectral theorem can be applied, I prefer it over elementary proofs. $\endgroup$
    – MaoWao
    Oct 28, 2016 at 11:42
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    $\begingroup$ You might check out the 3 volume series Non-homogeneous Boundary Value Problems and Applications by J. L. Lions and E. Magenes. $\endgroup$
    – Disintegrating By Parts
    Oct 28, 2016 at 15:40
  • $\begingroup$ I think you are asking a very strange question from the point of view of pdes. The point is not to apply results from FA, but to solve equations and the tools for that depend on the equation at hand. There are some types of equations/problems that can be handled with quite a lot FA, say spectral problems and parabolic equtions and there are lots of books on these. $\endgroup$
    – mcd
    Oct 29, 2016 at 3:49
  • $\begingroup$ You might try "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Brezis. Very heavy on applications of functional analysis, but does spend time covering the basics, and may not be as advanced as you are looking for. $\endgroup$
    – Jeff
    Oct 30, 2016 at 1:11

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Here are some suggestions.

  1. Functional Analysis, Sobolev Spaces, and Partial Differential Equations by Haim Brezis. This violates your rule of not developing the functional analysis material, but is a very good book. You can skip the stuff you know and jump right to the PDE / operator bits.
  2. An Introduction to Partial Differential Equations by Michael Renardy and Robert Rogers. Here you want the last part of the book, say after chapter 8. There's a lot of nice stuff in Chapters 10-12 that uses lots of functional analysis to solve nonlinear elliptic problems, etc.
  3. Monotone Operators in Banach Space and Nonlinear PDE by Ralph Showalter. This is heavy functional / operator theoretic material used to solve some serious nonlinear problems.
  4. Nonlinear Differential Equations of Monotone Type in Banach Spaces by Biorel Barbu. This covers the same sort of material as the Showalter book.
  5. Applications of Functional Analysis and Operator Theory by Hutson and Pym. There's a lot more in here than applications in PDE, but you might find it interesting.
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