A New Notion of Conjugacy for Isoperimetric Problems

A New Notion of Conjugacy for Isoperimetric Problems For problems in the calculus of variations with isoperimetric side constraints, we provide in this paper a set of points whose emptiness, independently of nonsingularity assumptions, is equivalent to the nonnegativity of the second variation along admissible variations. The main objective of introducing a characterization of this condition should be, of course, to obtain a simpler way of verifying it. There are two other sets of points available in the literature, introduced by Loewen and Zheng (1994) and Zeidan (1996), for which this necessary condition implies their emptiness. However, we show that verifying membership of these sets may be more difficult than checking directly if that condition holds. Contrary to this behavior, we prove that the desired objective of characterizing that condition is achieved by means of the set introduced in this paper. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A New Notion of Conjugacy for Isoperimetric Problems

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Publisher
Springer-Verlag
Copyright
Copyright © 2004 by Springer
Subject
Mathematics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-004-0800-3
Publisher site
See Article on Publisher Site

Abstract

For problems in the calculus of variations with isoperimetric side constraints, we provide in this paper a set of points whose emptiness, independently of nonsingularity assumptions, is equivalent to the nonnegativity of the second variation along admissible variations. The main objective of introducing a characterization of this condition should be, of course, to obtain a simpler way of verifying it. There are two other sets of points available in the literature, introduced by Loewen and Zheng (1994) and Zeidan (1996), for which this necessary condition implies their emptiness. However, we show that verifying membership of these sets may be more difficult than checking directly if that condition holds. Contrary to this behavior, we prove that the desired objective of characterizing that condition is achieved by means of the set introduced in this paper.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2004

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