Let $\Omega \subset \mathbb R^n$ be a bounded domain with smooth boundary and $N$ be a closed Riemannian manifold of dimension $n$ isometrically embedded into some $\mathbb R^{\ell}.$ For $1 \leq p < \infty,$ define,
$$ W^{1,p}(\Omega,N) = \{ u \in W^{1,p}(\Omega,\mathbb R^{\ell}) \mid u(x) \in N\, \text{ for a.e. } x \in \Omega \}.$$
This is viewed as a closed subspace of $W^{1,p}(\Omega,\mathbb R^{\ell}) \simeq \oplus_{i=1}^{\ell} W^{1,p}(\Omega),$ from which it inherits a Banach space structure.
Question: Does there exist $V \supset \overline{\Omega}$ and a bounded linear operator, $$ E : W^{1,p}(\Omega,N) \rightarrow W^{1,p}(V,N), $$ such that for all $u \in W^{1,p}(\Omega,N)$ we have $Eu = u$ almost everywhere in $\Omega$?
This is a generalisation of the classical extension theorem for $W^{1,p}(\Omega).$ For the scalar case we can prove it (following Evans 5.4) by,
- By picking $x \in \partial \Omega$ and flattening the boundary, consider the case of a half-plane.
- Use a reflection technique to give a local extension of $u$ near x.
- Patch together local extensions using a partition of unity.
This argument breaks down in this general case, as in step 2 the reflection may no longer take values in $N$ and similarly the global extension using partitions of unity need not take values in $N$ either.
Motivation: I am reading about approximation of elements in $W^{1,p}(\Omega,N)$ using smooth functions. A result of this form would be useful for proving density in the case $p=n$ (when it is known to be true), as convolving an element $u$ requires an extension into some open neighbourhood (which is the approach this paper by Hajłasz seems to suggest). I would also be interested more generally in references which establish these more basic properties about these manifold-valued Sobolev mappings, as most papers I consult seem to gloss over these points.