# Neumann problem

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

 $\displaystyle\frac{\partial\phi}{\partial n}$ $\displaystyle=$ $\displaystyle f,\quad\text{on \partial\Omega},$ $\displaystyle\nabla^{2}\phi$ $\displaystyle=$ $\displaystyle 0,\quad\text{in \Omega}.$

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

 $\displaystyle\int_{\partial\Omega}\frac{\partial\phi}{\partial n}d\sigma=\int_% {\Omega}\nabla\!\cdot\!(\nabla\phi)d\tau=\int_{\Omega}\nabla^{2}\phi\,d\tau$

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

Title Neumann problem NeumannProblem 2013-03-22 15:19:59 2013-03-22 15:19:59 dczammit (9747) dczammit (9747) 10 dczammit (9747) Definition msc 31B15 msc 31B05 msc 31A05 HarmonicFunction