PlanetMath: Neumann problem

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Neumann problem

(Definition)

Suppose $\Omega$ is a region of $\mathbb{R}^n$ and $\partial\Omega$ is the boundary of $\Omega$ . Further suppose 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

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

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 $\nabla^2\phi=0$ .

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"Neumann problem" is owned by dczammit. [ full author list (4) ]

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See Also: harmonic function

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Cross-references: identic equation, vanish, entire, integral, solution, harmonic function, operator, Laplacian, represents, point, normal, derivative, function, boundary, region
There is 1 reference to this entry.

This is version 7 of Neumann problem, born on 2005-06-07, modified 2006-06-24.
Object id is 7147, canonical name is NeumannProblem.
Accessed 3390 times total.

Classification:

AMS MSC:	31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
	31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
	31B15 (Potential theory :: Higher-dimensional theory :: Potentials and capacities, extremal length)

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