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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
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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