BASIC CALCULUS OF VARIATIONS

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 ^-dimensional problem. For the one-dimensional problem this condition is also sufficient. For the A>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 ^-dimensional problem, subject to an additional hypothesis. Some of these additional hypotheses are automatically satisfied for the one-dimensional problem. Let G be a bounded domain in R*, A — G X R^, Z be the space of (N X A:)-matrices andFGφX Z). If y: G -> R* is smooth, let IF(y) = JGF(xy y(x), y\x)) dx where y\x) is the matrix of partial derivatives oiy.