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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 5, Pages 92–104 (Mi izv8622)  

An analogue of Pontryagin's maximum principle in problems of minimization of multiple integrals

M. I. Zelikinab

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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.

DOI: https://doi.org/10.4213/im8622

Full text: PDF file (493 kB)
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English version:
Izvestiya: Mathematics, 2017, 81:5, 973–984

Bibliographic databases:

UDC: 517.977
MSC: 49K20, 49Q20, 58E15, 70H05
Received: 26.10.2016
Revised: 27.01.2017

Citation: M. I. Zelikin, “An analogue of Pontryagin's maximum principle in problems of minimization of multiple integrals”, Izv. RAN. Ser. Mat., 81:5 (2017), 92–104; Izv. Math., 81:5 (2017), 973–984

Citation in format AMSBIB
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  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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