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

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

can be obtained as the minimizer of the Dirichlet integral

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

amongst all functions

${\displaystyle v}$ such that ${\displaystyle v=g}$ on ${\displaystyle \partial \Omega }$,

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

• Plateau's problem

References

• Weisstein, Eric W. "Dirichlet's Principle". MathWorld.

• Mathematical analysis
• Calculus of variations
• Partial differential equations
• Harmonic functions