Misplaced Pages

Dirichlet's principle: Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editNext edit →Content deleted Content addedVisualWikitext
Revision as of 08:28, 22 July 2006 editDraicone (talk | contribs)2,734 edits Disambiguation link repair - You can help!← Previous edit Revision as of 16:48, 26 August 2006 edit undoLinas (talk | contribs)Autopatrolled25,539 edits →External links: see also Plateau's problemNext edit →
Line 16: Line 16:
attain its minimum. ] later justified Riemann's use of Dirichlet's principle. attain its minimum. ] later justified Riemann's use of Dirichlet's principle.


==External links== ==See also==
* ]

==References==
* {{MathWorld | urlname=DirichletsPrinciple | title=Dirichlet's Principle}} * {{MathWorld | urlname=DirichletsPrinciple | title=Dirichlet's Principle}}


Revision as of 16:48, 26 August 2006

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

Categories: