Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.Formal statement
Dirichlet's principle states that, if the function is the solution to Poisson's equation
on a domain of with boundary condition
then u can be obtained as the minimizer of the Dirichlet energy
amongst all twice differentiable functions such that on . This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet.History
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 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 by the direct method in the calculus of variations.
