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I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf).

Let

  1. $(M,g)$ be a Riemannian manifold with boundary;
  2. $E\to M$ be an hermitian fiber bundle;
  3. $\Delta$ be a Laplace-Beltrami operator defined from a connection over $E$ compatible with the fiber metric;
  4. $\Gamma(E)$ be the space of smooth sections of $E$;
  5. $H^s(E)$ be the $s$-sobolev space over $E$.

If $\psi_1,\psi_2\in \Gamma(E)$ I can compute $\langle \Delta \psi_1,\psi_2\rangle-\langle \psi_1,\Delta\psi_2\rangle$ integrating by parts and I get $$ \langle \Delta \psi_1,\psi_2\rangle-\langle \psi_1,\Delta\psi_2\rangle=\int_{\partial M} [(\dot\varphi_1, \varphi_2)-(\varphi_1,\dot\varphi_2)]\,\mathrm d\sigma $$ where $\varphi_a=\psi_a|_{\partial M}$ and $\dot \varphi_a$ is the outer normal derivative along $\partial M$.

The article says that this formula is valid even if $\psi_1,\psi_2\in H^2(E)$. In this case $(\varphi_a,\dot \varphi_a)$ is defined as the image of $\psi_a$ under the map $b\colon H^2(E)\to H^{3/2}(\partial E)\oplus H^{1/2}(\partial E)$, which is the unique continuous extension of the boundary map $\psi\in \Gamma(E)\mapsto (\varphi,\dot \varphi)\in \Gamma(\partial E)\oplus \Gamma(\partial E)$ (that's the Lions-Magenes theorem).

I don't get how to obtain this formula when $\psi_1,\psi_2\in H^2(E)$.

Can you help me? Thanks in advance.

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  • $\begingroup$ Are you happy with the statement in $\mathbb R^d$ and you don't see why it still works on a manifold, or the flat case also worries you? $\endgroup$
    – username
    May 8, 2022 at 19:06
  • $\begingroup$ I'm happy only with the smooth case. Even if $M=\mathbb R^d$ and the metric is flat, I have problem integrating by parts with sobolev function (I'm a beginner with sobolev space) $\endgroup$ May 8, 2022 at 19:49
  • $\begingroup$ So the way to do it is to use density, with a smooth approximating sequence, and notice that the convergence is strong e.g. in L^2 of the boundary. $\endgroup$
    – username
    May 8, 2022 at 20:48
  • $\begingroup$ I know that $H^2(E)$ is the closure of $\Gamma(E)$ with respect to the sobolev norm $||u||=\sqrt{||u||_2+||\Delta u||_2}$. Do you reccomend this approximation? $\endgroup$ May 9, 2022 at 10:59
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    $\begingroup$ Thanks a lot, now I see! $\endgroup$ May 9, 2022 at 20:55

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