Dirichlet's principle, also known as Thomson's principle, states that there exists a function 
that minimizes the functional
![D[u]=int_Omega|del u|^2dV](/images/equations/DirichletsPrinciple/NumberedEquation1.svg) |
(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 ![D[u]](/images/equations/DirichletsPrinciple/Inline13.svg)
comes arbitrarily close without being forced to actually
reach it. Kneser, however, obtained a valid proof of Dirichlet's principle.
