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 17:06, 5 October 2011 editGiftlite (talk | contribs)Autopatrolled, Extended confirmed users, Pending changes reviewers39,601 edits −wlink already included in the main article← Previous edit Revision as of 15:02, 1 November 2011 edit undoNick Number (talk | contribs)Autopatrolled, Extended confirmed users, File movers, Pending changes reviewers, Rollbackers197,952 editsm repaired link(s) to disambiguation page - (you can help) - HilbertNext edit →
Line 14: Line 14:
amongst all twice differentiable functions <math>v</math> such that <math>v=g</math> on <math>\partial\Omega</math> (provided that there exists at least one 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 that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician ].


Since the Dirichlet's integral is bounded from below, 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. Since the Dirichlet's integral is bounded from below, 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.


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

Revision as of 15:02, 1 November 2011

Not to be confused with Pigeonhole principle.

In mathematics, Dirichlet's principle in potential theory states that, if the function 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 that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Lejeune Dirichlet.

Since the Dirichlet's integral is bounded from below, 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: