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In mathematics, Dirichlet’s principle in potential theory states that, if the function ${\displaystyle u(x)}$ is the solution to Poisson’s equation

${\displaystyle \Delta u+f=0}$

on a domain ${\displaystyle \Omega }$ of ${\displaystyle \mathbb {R} ^{n}}$ with boundary condition

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

then ${\displaystyle u}$ can be obtained as the minimizer of the Dirichlet’s energy

${\displaystyle E=\int _{\Omega }\left({\frac {1}{2}}|\nabla v|^{2}-vf\right)\mathrm {d} x}$

amongst all twice differentiable functions ${\displaystyle v}$ such that ${\displaystyle v=g}$ on ${\displaystyle \partial \Omega }$, provided only that there exists one such function making the Dirichlet’s integral finite. This concept is named after the German mathematician Lejeune Dirichlet.

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
• Green’s first identity

References

• Lawrence C. Evans (1998). Partial Differential Equations. American Mathematical Society. ISBN 978-0821807729.
• Weisstein, Eric W. "Dirichlet's Principle". MathWorld.

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