Dirichlet's principle, also known as Thomson's principle, states that 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.