Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary and that $M\subset \tilde{M}$ with $(\tilde{M},g)$ also a compact Riemannian manifold with smooth boundary. Let us consider a one-form $\alpha \in L^2(M;T^*M)$ with the additional property that $\nabla \cdot \alpha \in L^2(M)$, that is the divergence of $\alpha$ weakly makes sense as an element of $L^2(M)$. Does $\alpha$ admit an extension to $\tilde{M}$ such that the same exact regularity properties hold in the larger domain?

Thanks,