Neumann problem

Suppose $\mathrm{\Omega }$ is a region of ${ℝ}^n$ and $\partial \mathrm{\Omega }$ is the boundary of $\mathrm{\Omega }$.Further suppose $f$ is a function $f:\partial \mathrm{\Omega }\to ℂ$, and suppose $\frac{\partial }{\partial n}$ corresponds to taking a derivative in a direction normal to the boundary $\partial \mathrm{\Omega }$ at any point. Then theNeumann problem is to find a function $\varphi :\mathrm{\Omega }\cup \partial \mathrm{\Omega }\to ℂ$such that

${\displaystyle \frac{\partial \varphi }{\partial n}}$$=$$f,\text{on }\partial \mathrm{\Omega },$

${\nabla }^2\varphi$$=$$0,\text{in }\mathrm{\Omega }.$

Here ${\nabla }^2$ represents the Laplacian operator and the second condition is that $\varphi$ be a harmonic function on $\mathrm{\Omega }$. The condition for the existence of a solution $\varphi$ of the Neumann problem is that integral of the normal derivative of the function $\varphi$, calculated over the entire boundary $\partial \mathrm{\Omega }$, vanish. This follows from the identic equation

${\displaystyle {\int }_{\partial \mathrm{\Omega }}}{\displaystyle \frac{\partial \varphi }{\partial n}}𝑑\sigma ={\displaystyle {\int }_{\mathrm{\Omega }}}\nabla \cdot (\nabla \varphi )𝑑\tau ={\displaystyle {\int }_{\mathrm{\Omega }}}{\nabla }^2\varphi d\tau$

and from the fact that ${\nabla }^2\varphi =0$.