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.
Read more about Dirichlet Problem: General Solution, Example: The Unit Disk in Two Dimensions, Methods of Solution, Generalizations
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