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.