Necessity for isoperimetric inequality constraints

Jorge A. Becerril; Javier F. Rosenblueth

doi:10.3934/dcds.2017047

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2017, Volume 37, Issue 3: 1129-1158. Doi: 10.3934/dcds.2017047

This issue Previous Article Next Article Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers

Necessity for isoperimetric inequality constraints

IIMAS, Universidad Nacional Autóonoma de México, Apartado Postal 20-126, México DF 01000

Received: May 2016

Revised: September 2016

Published: May 2017

The second author was supported by the PASPA-DGAPA program from Universidad Nacional Autónoma de México.

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Abstract

In this paper we deal with second order necessary conditions for the problem of Lagrange in the calculus of variations posed over piecewise smooth trajectories and involving inequality and equality isoperimetric constraints. We provide a review of different approaches to derive second order necessary conditions for this problem and prove that, surprisingly, though the solution set to the problem where the conditions hold may vary, all approaches impose the same strong assumption of normality relative to the set defined by equality constraints for active indices. Based on these approaches, we also give some applications to certain optimization problems with mixed constraints.

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Mathematics Subject Classification: Primary:49K15, 49K21;Secondary:34H05.

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