Legendre conditions for a variational problem with one-sided phase constraints

This paper is concerned with a variational problem with inequality phase constraints. The authors present second-order necessary conditions (Legendre condition) for weak minimal solutions of the variational problem. Our optimality conditions give an information about the Hessian of the integrand at not only inactive points but also active points. Since the present constraints do not include cx, the present conditions differ from the Legendre-Clebsch condition.