Geodesic
An analogue of Pontryagin's maximum principle in problems of minimization of multiple integrals
Izvestiya. Mathematics , Tome 81 (2017) no. 5, pp. 973-984.

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We prove a theorem on necessary conditions of Pontryagin's maximum principle type for an optimum of functionals given by multiple integrals. In contrast to the case of one-dimensional integrals, the maximum of the Pontryagin function is taken only over matrices of rank 1, not over all matrices. We give some examples.
Keywords: Pontryagin's maximum principle, multiple integrals, transversality conditions, necessary and sufficient conditions for strong or weak minima, semicontinuous extensions of variational problems, fields of extremals. 
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M. I. Zelikin. An analogue of Pontryagin's maximum principle in problems of minimization of multiple integrals. Izvestiya. Mathematics , Tome 81 (2017) no. 5, pp. 973-984. http://geodesic.mathdoc.fr/item/IM2_2017_81_5_a2/

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