On the question of extension of multidimensional variational problems

The extension of multidimensional variational problems is investigated. In particular, a multidimensional analogue of the Bogolyubov theorem on extension of a simple problem of the calculus of variations is proved. The problem of the closure of the class of functions of several variables with derivatives in a given set is considered. This problem is closely connected with the extension of variational problems.
Bibliography: 16 titles.