Appl Math Optim 50:209–228 (2004)
2004 Springer Science+Business Media, Inc.
A New Notion of Conjugacy for Isoperimetric Problems
Javier F. Rosenblueth
IIMAS–UNAM, Apartado Postal 20-726,
M´exico DF 01000, M´exico
Abstract. For problems in the calculus of variations with isoperimetric side con-
straints, 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.
Key Words. Isoperimetric problems, Calculus of variations, Conjugate points,
AMS Classiﬁcation. 49K15.
In order to illustrate the main objective of this paper, we start by brieﬂy considering the
following example. Suppose that we are interested in minimizing
I (x) =
(t) − 4x
This work was done during a sabbatical stay at The Weizmann Institute of Science, Rehovot, Israel. The
support given by Consejo Nacional de Ciencia y Tecnolog´ıa and Direcci´on General de Asuntos del Personal
Acad´emico, UNAM, is gratefully appreciated.