We know that the Laplace-Beltrami Operator is widly used in many areas, whose definition can be found here. I have learned basic knowledge about PDE and Sobolev spaces, but I know little about the elliptic lifting theorem of Laplace-Beltrami Operator, which some authors directly used without any detail in their article. So I wonder if any one know something about elliptic lifting theorem of Laplace-Beltrami Operator. I would also be grateful if anyone can provide some books or articles about this theorem.
The part of the original article reads as follows:
Suppose $\Omega$ is a bounded domain with sufficiently smooth boundary and $\psi$ satisfies $$ \left\{\begin{array}{l} \Delta \psi=0, \quad \text { in } \Omega, \\ \frac{\partial \psi}{\partial \boldsymbol{n}}=\boldsymbol{E} \cdot \boldsymbol{n}, \quad \text { on } \partial \Omega, \\ \int_{\Omega} \psi d x=0 . \end{array}\right. $$ Hence, applying the elliptic lifting theorem for the Laplace-Beltrami operator on smooth surfaces, we have $$ \begin{aligned} &\|\psi\|_{H^{\frac{3}{2}}(\partial \Omega)} \\ &\leq C \left\|\operatorname{div}_{T} \nabla_{T} \psi\right\|_{H^{-\frac{1}{2}}(\partial \Omega)}+C \left|\int_{\partial \Omega} \psi d S\right|. \end{aligned} $$
(Here $\operatorname{div}_{T} \nabla_{T}$ is the Laplace-Beltrami Operator.)