For the classical one-dimensional
problem in the calculus of variations, a necessary condition that the integral be
lower semicontinuous is that the integrand be convex as a function of the
derivative. We shall see that, if the problem is properly posed, then this condition
is also necessary for the k-dimensional problem. For the one-dimensional
problem this condition is also sufficient. For the k-dimensional problem this
condition is shown to be sufficient subject to an additional hypothesis. For the
one-dimensional problem there is an existence theorem if the integrand grows
sufficiently rapidly with respect to the derivative, and this result also holds for the
k-dimensional problem, subject to an additional hypothesis. Some of these
additional hypotheses are automatically satisfied for the one-dimensional
problem.
