TITLE: PDEs on manifold: what changes from Euclidean case?
QUESTION [30 upvotes]: I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds.
For example, do things like Poincare's inequality for $H^1_0$ hold, or like the norms $|\Delta \cdot |_{L^2}$ being equivalent to $|\cdot|_{H^2}$ is still true?
My friend told me that nothing changes since on manifolds we just work locally in $\mathbb{R}^n$ but I think he is not entirely correct.
Also, I want to know whether the PDE theory I know (eg. Evans) I can apply it to surfaces on $\mathbb{R}^n$ like a sphere. Or if not a sphere some open surface. Or do I need to use the PDE on manifold theory.
Thanks for any replies

REPLY [7 votes]: As a first approximation, you are already doing PDEs on a manifold if you are studying variable coefficient PDEs. Once you know what you are doing, analytic arguments on closed manifolds tend to be technically simpler and cleaner because there is no boundary. Obviously, you have to be careful when you want to import global considerations from flat domains to manifolds. But this does not mean that the manifold case is more difficult; boundaries can cause a lot of trouble in $\mathbb{R}^n$.