Vol. 104, No. 2, 1983

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Basic calculus of variations

Edward Silverman

Vol. 104 (1983), No. 2, 471–482

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.

Mathematical Subject Classification
Primary: 49A50, 49A50
Received: 12 December 1980
Published: 1 February 1983
Edward Silverman