PlanetMath: Dirichlet problem

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Dirichlet problem

(Definition)

Suppose $\Omega$ is a domain of $\mathbb{R}^n$ and $\partial\Omega$ is the boundary of $\Omega$ . Further suppose is a function $f\colon\partial \Omega\to\mathbb{C}$ . Then the Dirichlet problem is to find a function $\phi\colon \Omega\cup \partial \Omega \to\mathbb{C}$ such that

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

"Dirichlet problem" is owned by matte. [ full author list (2) ]

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See Also: harmonic function

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Cross-references: function, boundary, domain
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This is version 4 of Dirichlet problem, born on 2005-01-16, modified 2005-06-07.
Object id is 6646, canonical name is DirichletProblem.
Accessed 7394 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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