[Math] Advanced book on partial differential equations

Question

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.

Answer 1

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.