TITLE: What should a PDE/analysis enthusiast know?
QUESTION [11 upvotes]: What are the cool things someone who likes PDE and functional analysis should know and learn about? What do you think are the fundamentals and the next steps? I was thinking it would be good to know how to show existence or even to know where to start to show existence of any non-linear PDE I come across. 
For example, I only recently found about about how people can use the inverse theorem to prove existence of a non-linear PDE. This involved Frechet derivatives which I have never seen before. And I don't fully appreciate the link between normal derivative, Gateaux derivative and Frechet derivative. So I thought how many other things I have no idea about in PDEs.
And PDEs on surface are interesting (but I'm just learning differential geometry so a long wait till I look at that in detail) but it seems studied to death.
So anyway what do you think is interesting in this field? I am less interested in constructing solutions to PDEs and more into existence. PS: you can assume the basic knowledge (Lax-Milgram, linear elliptic and parabolic existence and uniqueness, etc..)

REPLY [4 votes]: One nice idea along these lines is the Leray-Schauder Fixed Point approach to non-linear elliptic existence problems.
Roughly speaking, for example, if I would like to solve the quasilinear problem
$a_{ij}(x,u,Du)D_{ij}u + b(x,u,Du) = 0$
$u = \varphi$ on $\partial\Omega$, 
say, I can set up a map which takes a function $v$ and sends it to the unique solution $u$ of the linear problem
$a_{ij}(x,v,Dv)D_{ij}u + b(x,v,Dv) = 0$
$u = \varphi$ on $\partial\Omega$.
Call this map $T$. Obviously I need to know existence for the linear problem. Then, by considering this as a map (not a linear map) between appropriate Banach spaces (for which I need some regularity for solutions of the linear problem) I can see that a solution to the quasilinear problem is precisely a fixed point of $T$.
Googling Leray-Schauder fixed point theorem seems to turn up a nice set of notes by Leon Simon on the subject. Also Chapter 11 of Gilbarg and Trudinger explains the method well.
The method is so-named because of the Leray-Schauder fixed point theorem - an abstract fixed point theorem (that is to say it is just about maps between Banach spaces and not specifically about PDE) which, under certain conditions gives a fixed point of $T$. The task of verifying the conditions under which the theorem is applicable is one of gaining apriori estimates for solutions of the quasilinear problem, so this approach is a good way to see the need for the apriori philosophy in (elliptic, at least) existence problems.