## Dirichlet's Principle

Also known as Thomson's Principle. There exists a function that minimizes the functional

(called the Dirichlet Integral) for or among all the functions which take on given values on the boundary of , and that function satisfies in , , . Weierstraß showed that Dirichlet's argument contained a subtle fallacy. As a result, it can be claimed only that there exists a lower bound to which comes arbitrarily close without being forced to actually reach it. Kneser, however, obtained a valid proof of Dirichlet's principle.