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IMA Journal of Mathematical Control and Information Advance Access published online on March 10, 2007
IMA Journal of Mathematical Control and Information, doi:10.1093/imamci/dnm005
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Weakening the strengthened condition of Weierstrass for the isoperimetric problem in the calculus of variations

Gerardo Sánchez Licea{dagger}

Universidad Autónoma de la Ciudad de México, Prolongación San Isidro No. 151, San Lorenzo Tezonco, Delegación Iztapalapa, C.P. 09790, México, D.F

{dagger} Email: gslicea{at}apolo.acatlan.unam.mx

Received on 21 March 2006; Accepted on 3 January 2007


  Abstract

An alternate sufficiency proof for the fixed end-point isoperimetric problem in the calculus of variations is presented. This technique not only shows how the problem need not be transformed into a problem of Lagrange but also shows how we can weaken the classical strengthened condition of Weierstrass. The usefulness of this sufficiency result is illustrated with an example which cannot be transformed into a problem of Lagrange and for which it is possible to apply the alternate sufficiency theorem in order to conclude that a given extremal affords a strict strong minimum. On the other hand, we show that the classical sufficiency theorem does not respond for this case.

Keywords: calculus of variations; isoperimetric problems; the strengthened condition of Weierstrass; sufficiency for local minima.


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