Neumann problemNeumannProblem2005-06-07 22:57:562006-06-24 03:18:49Definition% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thm}{Theorem}
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\newtheorem{cor}{Corollary}Suppose $\Omega$ is a region of $\sR^n$ and $\partial\Omega$ is the boundary of $\Omega$.
Further suppose $f$ is a function $f\colon\partial \Omega\to\sC$, 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
\emph{Neumann problem} is to find a function $\phi\colon \Omega\cup \partial \Omega \to\sC$
such that
\begin{eqnarray*}
\frac{\partial\phi}{\partial n} &=& f,\quad \text{on $\partial \Omega$}, \\
\nabla^2 \phi &=& 0,\quad \text{in $\Omega$}.
\end{eqnarray*}
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
\begin{align*}
\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
\end{align*}
and from the fact that $\nabla^2\phi=0$.