Let $M$ be a connected, oriented, smooth Riemannian n-manifold with non-empty boundary, and $\omega$ a smooth differential $p$-form on $M$ which is closed, co-closed, and vanishes on boundary of M. We will show that $\omega$ is identically zero. Since the result is local, we can take $M$ to be the upper half-space in $R^n$, with boundary $R^{n−1} $.
In the proof of Lemma2 (page 6) in (Sylvain Cappell, Dennis DeTurck, Herman Gluck Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary), the authors stated that extend the $p$-form $\omega$ to all of $R^n$, by making it odd with respect to reflection in $R^{n−1} $, so the extended $p$- form $\omega$ will be of class $C^1$ on all of $R^n$ and together with the vanishing of $\omega$ on $ R^{n−1} $, are enough to show that the first derivatives of the coefficients of $\omega$ vanish along $ R^{n−1} $, even when computed in the normal direction.. Moreover, they asserted that repeated differentiation of the equations which express the fact that $\omega$ is closed and co-closed, together with the vanishing of $\omega$ on $ R^{n−1} $, show that all higher partial derivatives of the coefficients of $\omega$ vanish on $ R^{n−1} $.
Could anyone please explain mathematically (with details) how the extended form will be smooth at the boundary (where it is just in class $C^1$)?. In other words the key conceptual step in the proof is to double the manifold across the boundary to get a closed manifold, and there is no reason to believe that the doubled $\omega$ is smooth at the boundary. Of course, the closed and co-closed $\omega$ , together with the vanishing of $\omega$ on $ R^{n−1} $ are the key point here but I do not understand how I can proof that.
More concretely, I need to understand their phrases with mathematical details “repeated differentiation of the equations which express the fact that $\omega$ is closed and co-closed, together with the vanishing of $\omega$ on $ R^{n−1} $, show that all higher partial derivatives of the coefficients of $\omega$ vanish on $ R^{n−1} $. ”