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.

The name "Dirichlet's principle" is due to Riemann, who applied it in the study of complex analytic functions.[1]

Riemann (and others such as Gauss and Dirichlet) knew that Dirichlet's integral is bounded below, which establishes the existence of an infimum; however, he took for granted the existence of a function that attains the minimum. Weierstrass published the first criticism of this assumption in 1870, giving an example of a functional that has a greatest lower bound which is not a minimum value. Weierstrass's example was the functional

In 1900, Hilbert later justified Riemann's use of Dirichlet's principle by developing the direct method in the calculus of variations.[4]