Dirichlet Problem

The Dirichlet problem is named after Lejeune Dirichlet, who proposed a solution by a variational method which became known as Dirichlet's principle. The existence of a unique solution is very plausible by the 'physical argument': any charge distribution on the boundary should, by the laws of electrostatics, determine an electrical potential as solution.

However, Weierstrass found a flaw in Dirichlet's argument, and a rigorous proof of existence was found only in 1900 by Hilbert. It turns out that the existence of a solution depends delicately on the smoothness of the boundary and the prescribed data.