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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 $ f$ 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 MSC31A05 (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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Dirichlet problem link to domain by emoldoveanu on 2005-09-06 14:03:30
I think the author does not mean domain as in the page of the link!

Emil
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