Let
,
and
be continuous
on
with
convex,
for all
real
and
for all
where
. If
then
is a permissible integrand for the two-dimensional parametric variational
problem.
Let
be a simple
closed curve in
,
be the closed unit circle
in the plane,
be the
collection of functions
continuous on
into
for
which
and
. Suppose
that
is
not empty. It was shown in ‘A problem of least area’, [7], that the problem of minimizing
over
is equivalent
to minimizing
over
where
,
and both integrals
are taken over
. The
minimizing solution of
is known to have differentiability properties corresponding to
, and this solution
also minimizes
.
The function
is simple,
that is, for each
, each
supporting linear functional to
is simple. If
,
then, of course, each parametric integrand is simple. In this
paper we show that for each simple parametric integrand
there exists
, satisfying the conditions
imposed upon
,
such that
is
obtained from
as
was
obtained from
.
|