Transferring a result in PDE from open domain to a manifold

Question

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ to a compact manifold? This would use partitions of unity and patching things together in some way. It could be a density result for example.

Thanks

Answer 1

Both of the following books have good treatments of these ideas:

• Sobolev Spaces on Riemannian Manifolds by Emmanuel Hebey
• Some Nonlinear Problems in Riemannian Geometry by Thierry Aubin