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..)