The Importance of Being Normal, Regular and Proper in the Calculus of Variations

Abstract

Some classical references in the literature include second-order necessary conditions for the isoperimetric problem of Lagrange in the calculus of variations involving inequality and equality constraints. Those conditions take into account, in the sets of critical directions, the sign of the Lagrange multipliers of the extremal under consideration. However, they hold under a strong assumption of normality relative to a set defined only by equality constraints for active indices. In this paper, we show how the same conditions can be preserved under a weaker assumption, thus providing a wider range of applicability.