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Revision as of 09:30, 24 September 2008 editThumperward (talk | contribs)Administrators122,782 editsm moved Dirichlet’s principle to Dirichlet's principle: per MoS← Previous edit Revision as of 06:29, 25 September 2008 edit undoMichael Hardy (talk | contribs)Administrators210,264 editsmNo edit summaryNext edit →
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In ], '''Dirichlet's principle''' in ] states that, if the function <math>u(x)</math> is the solution to Poisson's equation In ], '''Dirichlet's principle''' in ] states that, if the function <math>u(x)</math> is the solution to Poisson's equation


:<math>\Delta u + f = 0</math> :<math>\Delta u + f = 0\,</math>


on a ] <math>\Omega</math> of <math>\mathbb{R}^n</math> with ] on a ] <math>\Omega</math> of <math>\mathbb{R}^n</math> with ]


:<math>u=g</math> on <math>\partial\Omega</math>, :<math>u=g\text{ on }\partial\Omega,\,</math>


then <math>u</math> can be obtained as the minimizer of the ] then ''u'' can be obtained as the minimizer of the ]


:<math>E = \int_\Omega \left(\frac{1}{2}|\nabla v|^2 - vf\right)\mathrm{d}x</math> :<math>E = \int_\Omega \left(\frac{1}{2}|\nabla v|^2 - vf\right)\,\mathrm{d}x</math>


amongst all twice differentiable functions <math>v</math> such that <math>v=g</math> on <math>\partial\Omega</math>, provided only that there exists one such function making the Dirichlet's integral finite. This concept is named after the German mathematician ]. amongst all twice differentiable functions <math>v</math> such that <math>v=g</math> on <math>\partial\Omega</math>, provided only that there exists one such function making the Dirichlet's integral finite. This concept is named after the German mathematician ].


Since the Dirichlet's integral is nonnegative{{dubious}}, the existence of an ] is guaranteed. That this infimum is attained was taken for granted by ] (who coined the term ''Dirichlet's principle'') and others until ] gave an example of a functional that does not Since the Dirichlet's integral is nonnegative{{dubious}}, the existence of an ] is guaranteed. That this infimum is attained was taken for granted by ] (who coined the term ''Dirichlet's principle'') and others until ] gave an example of a functional that does not attain its minimum. ] later justified Riemann's use of Dirichlet's principle.
attain its minimum. ] later justified Riemann's use of Dirichlet's principle.


==See also== ==See also==

Revision as of 06:29, 25 September 2008

Not to be confused with Pigeonhole principle.

In mathematics, Dirichlet's principle in potential theory states that, if the function u ( x ) {\displaystyle u(x)} is the solution to Poisson's equation

Δ u + f = 0 {\displaystyle \Delta u+f=0\,}

on a domain Ω {\displaystyle \Omega } of R n {\displaystyle \mathbb {R} ^{n}} with boundary condition

u = g  on  ∂ Ω , {\displaystyle u=g{\text{ on }}\partial \Omega ,\,}

then u can be obtained as the minimizer of the Dirichlet's energy

E [ v ( x ) ] = ∫ Ω ( 1 2 | ∇ v | 2 − v f ) d x {\displaystyle E=\int _{\Omega }\left({\frac {1}{2}}|\nabla v|^{2}-vf\right)\,\mathrm {d} x}

amongst all twice differentiable functions v {\displaystyle v} such that v = g {\displaystyle v=g} on ∂ Ω {\displaystyle \partial \Omega } , provided only that there exists one such function making the Dirichlet's integral finite. This concept is named after the German mathematician Lejeune Dirichlet.

Since the Dirichlet's integral is nonnegative, the existence of an infimum is guaranteed. That this infimum is attained was taken for granted by Riemann (who coined the term Dirichlet's principle) and others until Weierstraß gave an example of a functional that does not attain its minimum. Hilbert later justified Riemann's use of Dirichlet's principle.

See also

References

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