I have the following basic question about Sobolev-spaces which take their values in a Riemannian manifold $(M,g)$, i.e. functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that for every chart $\varphi : M \to \mathbb{R}^d$, the composition $\varphi\circ u$ is in the usual $H^1$ space.

Question: 1. Is this notion well-defined (this is only easy to show for higher order smoothnes or $n =1$)? 2. Does the manifold-valued Sobolev space admit the structure of a Hilbert manifold? 3. Do there exist good references for manifold-valued Sobolev spaces (and their relations with Hilbert manifolds)? 4. Is there a better way to define manifold-valued Sobolev spaces?

Many thanks for your answers!