Geodesic
Some concepts of regularity for parametric multiple-integral problems in the calculus of variations
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 741-758.

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We show that asserting the regularity (in the sense of Rund) of a first-order parametric multiple-integral variational problem is equivalent to asserting that the differential of the projection of its Hilbert-Carathéodory form is multisymplectic, and is also equivalent to asserting that Dedecker extremals of the latter $(m+1)$-form are holonomic.
Classification : 37Jxx, 49K10, 49N60, 53Cxx, 58E15, 70Gxx
Keywords: parametric variational problem; regularity; multisymplectic 
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     title = {Some concepts of regularity for parametric multiple-integral problems in the calculus of variations},
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Crampin, M.; Saunders, D. J. Some concepts of regularity for parametric multiple-integral problems in the calculus of variations. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 741-758. http://geodesic.mathdoc.fr/item/CMJ_2009__59_3_a13/