Hi!
Let $M$ be a compact manifold possibly with boundary with $\dim(M)=m$, let $N$ be a non compact manifold with $\dim(N)=n$. Let me recall the definition of the sobolev space $W^{1,p}(M,N)$. Embedding isometrically $N$ in some $\mathbb{R}^K$ we can define $$W^{1,p}(M,N):=\lbrace f\in W^{1,p}(M,\mathbb{R}^K)\textrm{ s.t. } f(p)\in N\textrm{ for a.e. }p\in M \rbrace$$ what can be said about the density of $C^{\infty}(M,N)$ in $W^{1,p}(M,N)$?
I'm particularly interested in the case $M=\overline{\Delta}$ ($\overline{\Delta}$ is the 2-dimensional closed unit disk) and $p=2$
Thank you in advance.
