Consider $\Omega\subset\mathbb{R}^N$ and $\Gamma$ its boundary. If a function $v$ in $\Omega$ such that $v=0$ and $\frac{\partial v}{\partial n}=h$ on $\Gamma$ with $n$ being normal vector, then how can we prove that $\frac{\partial^2 v}{\partial n^2}=-h\sum_j \frac{\partial n}{\partial s_j}.\mathcal{T}_j$ where $(\mathcal{T}_1(x),...,\mathcal{T}_{N-1}(x))$ is the basis of tangent plan to $\Gamma$ in $x$ and $s_1,...,s_{N-1}$ are local coordinates. Moreover, how we can show that $\Delta v= -\sum_j \frac{\partial n}{\partial s_j}.\mathcal{T}_j\frac{\partial v}{\partial n}+\frac{\partial^2 v}{\partial n^2}$ on $\Gamma$? Thank you in advance.
Peter,