Source: https://arxiv.org/pdf/1705.00069.pdf
On page 4, it says that the surface Laplacian of a function $u$ (I will use different letters here) defined on a neighborhood of the boundary $\partial M$ is $$\Delta_M u= \Delta u - 2H \frac{\partial u}{\partial n}-\frac{\partial^2u}{\partial n^2}$$ where $H$ denotes the mean curvature of $S$ and $\frac{\partial u}{\partial n}=n \cdot\nabla u$.
My question(s):
- What is $\frac{\partial^2u}{\partial n^2}?$
- Multiplying the right side with a test function $v$ , then integrating and using Greens first identity I should get $$\int_M \nabla u \nabla v \, dx-2H \int_{\partial M}\frac{\partial u}{\partial n}v \, ds - \int_{\partial M} ??? \, ds$$
What should the third integrand be?
We have that $$\frac{\partial^2u}{\partial\eta^2}={\rm Hess }u\left(\frac{\partial}{\partial\eta},\frac{\partial}{\partial\eta}\right).$$
Or, just $$\frac{\partial^2u}{\partial\eta^2}=\eta\cdot\nabla(\eta\cdot\nabla u)$$