Geodesic
On the question of extension of multidimensional variational problems
Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 1-19

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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. 
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     title = {On the question of extension of multidimensional variational problems},
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F. V. Guseinov. On the question of extension of multidimensional variational problems. Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 1-19. http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a0/