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Dirichlet's Principle


Dirichlet's principle, also known as Thomson's principle, states that there exists a function u that minimizes the functional

 D[u]=int_Omega|del u|^2dV

(called the Dirichlet integral) for Omega subset R^2 or R^3 among all the functions u in C^((1))(Omega) intersection C^((0))(Omega^_) which take on given values f on the boundary partialOmega of Omega, and that function u satisfies del ^2=0 in Omega, u|_(partialOmega)=f, u in C^((2))(Omega) intersection C^((0))(Omega^_). Weierstrass showed that Dirichlet's argument contained a subtle fallacy. As a result, it can be claimed only that there exists a lower bound to which D[u] comes arbitrarily close without being forced to actually reach it. Kneser, however, obtained a valid proof of Dirichlet's principle.


See also

Dirichlet's Box Principle, Dirichlet Integrals

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References

Monna, A. F. Dirichlet's Principle: A Mathematical Comedy of Errors and Its Influence on the Development of Analysis. Utrecht, Netherlands: Osothoek, Scheltema, and Holkema, 1975.

Referenced on Wolfram|Alpha

Dirichlet's Principle

Cite this as:

Weisstein, Eric W. "Dirichlet's Principle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletsPrinciple.html

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