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.
See also Dirichlet's Box Principle, Dirichlet Integrals
© 1996-9 Eric W. Weisstein
1999-05-24