7
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Craioveanu, Puta and Rassias define the Sobolev space $H_k(M)$ (pg. 106) on a compact Riemannian manifold $M$ as the completion of $C^\infty(M)$ with respect to the norm $$ \|f\|_{H_k(M)} = \|f \|_{H_{k-1}(M)} + \int_M \langle \nabla^k f, \nabla^k f \rangle_M\ d\mu_g $$ where $\nabla^k$ is the iterated covariant derivative with respect to the Levi-Civita connection on $M$, $d\mu_g$ is the Borel measure induced by the volume form, and $H_0(M)$ is defined as $L_2(M)$. Let $\Delta_g$ be the Laplace-Beltrami operator on $M$ with eigenvalues $0 = \lambda_0 < \lambda_1 \leq \lambda_2 \ldots$ and associated eigenfunctions $\{\varphi_j\}$ assuming Dirichlet boundary conditions. These eigenfunctions form an orthonormal basis for $L_2(M)$. The book then proves (pg. 134) that \begin{equation} H_1(M) = \left\{ f = \sum_{j=0}^\infty \alpha_j \varphi_j\ \middle|\ \sum_{j=0}^\infty \lambda_j \alpha_j^2 < \infty \right\} \tag{1}\end{equation} They state without proof that a similar result holds for $H_2(M)$, that is \begin{equation} H_2(M) = \left\{ f = \sum_{j=0}^\infty \alpha_j \varphi_j\ \middle|\ \sum_{j=0}^\infty \lambda_j^2 \alpha_j^2 < \infty \right\}\tag{2}\end{equation} and use this fact to extend the domain and range of $\Delta_g$ from $C^\infty(M)$ to $H_2(M)$ and $L_2(M)$ respectively.

Questions:

  1. How does one prove (2)?
  2. Can it be generalised to $H_k(M)$?
  3. The proof of the first statement uses Green's formula and the fact the covariant derivative for functions coincides with the gradient. Is there an analogue of Green's formula for iterated covariant derivatives i.e. something like $$ \int_M \langle \nabla^k f, \nabla^k g \rangle_M = \int_M \langle \Delta^k f , g \rangle_M $$ when $f$ and $g$ vanish on the boundary of $M$?
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  • $\begingroup$ Answer to 1 is obvious if I have understood the fact that the $\lambda_j$'s denote the eigenvalues of the Laplacian. Answer to 2: Yes. Answer to 3: Not that I am aware of. $\endgroup$ – AmorFati Nov 23 '17 at 11:04
  • $\begingroup$ To get a higher integration by parts formula you need to assume not only that $f,g$ vanish on the boundary, but also some of their derivatives. The correct formula probably has a bunch of curvature terms. $\endgroup$ – Anthony Carapetis Nov 23 '17 at 11:16
  • $\begingroup$ @KyleBroder Could you elaborate on 1? It isn't obvious to me, perhaps I'm missing something simple. I updated the question to make it more clear - I understand how $\Delta$ is extended to $H_2(M)$ once we have the second equality, but I don't know how to obtain the second equality itself. $\endgroup$ – Abhinav V Natarajan Nov 23 '17 at 13:01

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