TITLE: What should everyone know about PDEs?
QUESTION [6 upvotes]: Suppose I'm going into a field where it's unlikely that I will ever see a PDE.
Assuming I have a graduate level background in analysis, what should I know about PDEs to have a general understanding of the topic? The book by Evans, for example, seems more comprehensive than necessary. I'd like to reduce it to a list of around say, five, chapters or topics I should know well.

REPLY [3 votes]: Here is my attempt:

Harmonic functions
The Dirichlet problem on bounded domains 

Perron's method
$H^1$ method: Existence, regularity (Weyl's lemma)

Poisson equations: Regularity in $H^k$ scale, Schauder estimates
Spectral properties of the Laplacian on bounded domains

Hilbert-Schmidt theory, variational characterization of the eigenvalues
Eigenvalue comparison theorems, Weyl's asymptotic law
Courant's nodal domain theorem, Pleijel's theorem

The heat equation: Heat kernel, Tychonov's uniqueness theorem, maximum principles, backward uniqueness
The wave equation: Explicit formulas, finite speed of propagation, Huygens principle
First order equations: Method of characteristics, Hamilton-Jacobi equations, conservation laws