I know that on a connected, compact, oriented Riemannian manifold without boundary the Hodge Laplacian $\Delta_k=(d+\delta)^2$ (acting on $k$-forms) admits an orthogonal eigenspace decomomposition of $L^2\Lambda^k(M)$, consisting of smooth eigenforms, where all eigenvalues have finite multiplicity, are non-negative and only accumulate at infinity. A reference for this is for instance:

The Laplacian on a Riemannian manifold: An Introduction to Analysis on Manifolds (1997), Cambridge University Press. By Steven Rosenberg [Theorem $1.30$]

I was wondering whether the same holds true for compact manifolds with boundary, provided we impose some appropriate boundary conditions. Rosenberg deals first with the case of $0$-forms before proving the general result. He mentions after the $\Lambda^0(M)$ case, that we may allow boundary if we impose Dirichlet or Neumann boundary conditions. I also found references for this case. However he does not say anything about a similar result for general $k$-forms on manifolds with boundary.

So my question is, do we also have an orthogonal eigenspace decomposition of $L^2\Lambda^k(M)$ in the case of compact manifolds with boundary (under some appropriate boundary condtions, e.g. vanishing normal component)?

So far I was not able to find any reference for such a result. If anyone is aware of a paper or book dealing with this problem I would highly appretiate it.

Thanks in advance


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