# 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? $$alpha in C ^ infty$$?