I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf).
Let
- $(M,g)$ be a Riemannian manifold with boundary;
- $E\to M$ be an hermitian fiber bundle;
- $\Delta$ be a Laplace-Beltrami operator defined from a connection over $E$ compatible with the fiber metric;
- $\Gamma(E)$ be the space of smooth sections of $E$;
- $H^s(E)$ 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.