Vol. 104, No. 2, 1983

Recent Issues
Vol. 331: 1
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
Basic calculus of variations

Edward Silverman

Vol. 104 (1983), No. 2, 471–482
Abstract

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
Milestones
Received: 12 December 1980
Published: 1 February 1983
Authors
Edward Silverman