dg.Differential geometry – uniformity of harmonic shapes on manifolds with boundaries

To let $ M $ Be a smooth, limited, oriented Riemannian manifold with border. To let $ alpha $ be a harmonic differential $ p $Form $ M $subject to the constraint $ alpha wedge nu ^ sharp | partial M = 0 $ or $ iota_ nu alpha | partial M = 0 $, Here $ nu $ is the normal vector field along $ partial M $, $ nu ^ sharp $ is his dual $ 1 $Form and $ iota $ is the inner multiplication. Accept that $ alpha in W ^ {1,2} $,

The question is: Can we finish this? $ alpha in C ^ infty $?