Why the Sobolev space $W^{1,2}(M,N)$ weak-sequencially closed in $W^{1,2}(\mathbb R^K)$?

Question

Given two Riemannian manifolds $M,N$ and an imbedding $N\subset \mathbb R^K$, we define \begin{array} \;W^{1,2}(M,N):=\{v\in W^{1,2}(M,\mathbb R^K):v(x)\in N \;{\rm a.e.}\;x\in M\} \end{array}

My question is how to prove that $W^{1,2}(M,N)$ is weak-sequentially closed in $W^{1,2}(M,\mathbb R^K)$.

Is it true for general $W^{p,q}$?

Thank you.

Answer 1

For a sequence $u_n\in W^{1,2}(M,N)\subset W^{1,2}(M,\mathbb R^K)$, which converges weakly to $u\in W^{1,2}(M,\mathbb R^K)$. There exists a subsequence such that, $u_k\to u$ a.e. Then $u(m)\in N$ a.e.