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 thefunctions 
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 beingforced to actually reach it. Kneser, however, obtained a valid proof of Dirichlet's principle.See also Dirichlet's Box Principle, Dirichlet Integrals- Togliatti Surface
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