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Since the Dirichlet integral is nonnegative, 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 integral is nonnegative, 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.

This equation was termed trivial and non-sensical by prof Bharath M G of Bangalore ( whose claim to fame is his impeccable ability to generate a geodesic from any arbitrary set of points in 3D space by curve fitting using Q BASIC and COBOL).He jokingly commented about the concept to his close accomplice , chirru , who popularised it throught the word of mouth!


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

Revision as of 07:52, 7 June 2007

In mathematics, Dirichlet's principle in potential theory states that the harmonic function u {\displaystyle u} on a domain Ω {\displaystyle \Omega } with boundary condition

u = g {\displaystyle u=g} on Ω {\displaystyle \partial \Omega }

can be obtained as the minimizer of the Dirichlet integral

Ω | v | 2 {\displaystyle \int _{\Omega }|\nabla v|^{2}}

amongst all 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 integral finite.

Since the Dirichlet 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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