TITLE: PDE/Analysis graduate courses
QUESTION [9 upvotes]: I'm just starting my graduate studies in Analysis and PDE's and am a bit lost about what topics should I cover in order to do a good Phd program.
I`ve already done the usual undergrad courses, plus Real and complex analysis (graduate level), functional analysis and measure theory.
So, if you guys can recommend me which courses I should do, (I can get my university to open new courses as needed), and which books I should study, it'd make me really happy

REPLY [2 votes]: Fixed Point Theory is an important part of analysis to cover. And if you want to mix analysis with a little bit geometry, you MUST check the two brilliant books by I. Chavel: eigenvalues in riemannian geometry and isoperimetric inequalities. They do reveal beatiful applications of PDE's to geometric problems.