**EDIT:** Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \times [0,1]$.


Thanks to David's comments that help me a lot, the problem I really concern is just the following.


**Question:** 


1. Now that we know if $M$ is non-compact then $C^\infty(M,N)$ is
modeled on spaces $C^\infty\_c(M,N)$. Since we know
$W^{k,p}\_0(\Omega)$ is the closure of $C^\infty\_c(\Omega) $ in
$W^{k,p}(\Omega)$ for $\Omega\subset \mathbb R$, in analogy, can we
say $W^{k,p}(M,N)$ is modeled on spaces $W^{k,p}\_0(M,f^\*TN)$? Any reference?
2. Can we consider $W^{k,p}\_0(M,f^\*TN)$ as a completion (in some
sense?) of $C^\infty\_c(M,f^\*TN)$, the spaces of smooth sections with compact support of pullback bundles along $f:M\to N$?




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Previous post:


**(1)** Can we define $W^{1,p}\_0(M,N)$ 
 in a similar manner that $W^{1,p}\_0(\mathbb R)$ is defined by the completions of $C^\infty\_0$ or $C^\infty\_c$?


* For example, Floer in [his paper](http://onlinelibrary.wiley.com/store/10.1002/cpa.3160410603/asset/3160410603_ftp.pdf?v=1&t=j5095eur&s=9385885258b1b9f84fc0142d956b95fa460d1a2e) (see Definition 2.1) actually
discusses (roughly) $\mathcal W:=W\_{loc}^{k,p}(\mathbb R \times
 [0,1], N)$ with a topology given by open sets as follows: $ \mathcal
 O\_{u,\rho,\epsilon} =\{ v\in \mathcal W \mid v= \exp\_u \xi
 ~\text{on}~ [-\rho,\rho]\times[0,1], ~\text{and}~ ||\xi||\_{W^{k,p}}
 <\epsilon \} $ On the other hand, Audin and Damian in [their
book](http://www.springer.com/us/book/9781447154952) (see Definition 8.2.2) consider Banach manifolds $\widetilde
 {\mathcal W} =W^{k,p}(\mathbb R \times [0,1], N)$ (actually $[0,1]$
should be replaced by $S^1$) in a quite different manner. Here open
sets are the space of maps of the form $v=\exp\_u \xi$ where $\xi \in
 W^{k,p}(\mathbb R \times [0,1], N)\equiv W\_0^{k,p}(\mathbb R \times
 [0,1], N)$ and where $u$ is smooth and converges in some decay at the
infinity.


Heuristically, $\mathcal W$ is like the completion of $C\_c^\infty$
while $\widetilde {\mathcal W}$ is like that of $C\_0^\infty$.


**(2)** Is $\mathcal W$ the same as $\widetilde{\mathcal W}$? Which one could be a better candidate for the definition of $W^{k,p}\_0(M,N)$? Are they both Banach manifolds as in the case $M$ is compact?


* Recently I notice that when considering (infinite-dimensional)
manifolds of mapping, say $C^\infty(M,N)$, we usually require $M$ to
be compact. (See Section 4.2 of this paper: [The inverse function
theorem of Nash and Moser](https://projecteuclid.org/euclid.bams/1183549049)). Notice that as long as the domain $M$
is compact, the questions (1), (2) and (3) become trivial.


**(3)** In general, if $M$ is non-compact, then is $C^\infty(M,N)$ still a Frechet manifold as in the case $M$ is compact?




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Manifold of mappings between $M$ and $N$, with non-compact source $M$
Let $M$ and $N$ be Riemannian manifolds.
In general, the space of Sobolev mappings $W^{k,p}(M,N)$ should not be defined as a completion of smooth mappings even if $k=1$ and manifolds are compact. The common definition (at least if $N$ is compact) is as follows. Take an isometric embedding of $N$ into a Euclidean space $\mathbb{R^\nu}$ and then define:
$$
W^{k,p}(M,N)=\{ f\in W^{m,p}(M,\mathbb{R}^\nu):\, f(x)\in N \text{ a.e.} \}.
$$
This space is equipped with the metric inherited from the Sobolev norm and in general smooth mappings are not dense [1]. There are some problems when $N$ is not compact. You can find more papers about higher order Sobolev mappings between manifolds, including the case of non-compact target, at the homepage of 
[Van Schaftingen](https://perso.uclouvain.be/jean.vanschaftingen/).


[1] P. Bousquet, A. C. Ponce, J. Van Schaftingen, Strong density for higher order Sobolev spaces into compact manifolds. *J. Eur. Math. Soc. (JEMS)* 17 (2015), no. 4, 763–817.


