**
Czechoslovak Mathematical Journal, Vol. 59, No. 3, pp. 741-758, 2009
**

#
Some concepts of regularity for parametric multiple-integral problems in

the calculus of variations

##
M. Crampin, D. J. Saunders

* M. Crampin*, Address for correspondence: 65 Mount Pleasant, Aspley Guise, Beds MK17 8JX, UK, e-mail: ` Crampin@btinternet.com`; Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium; * D. J. Saunders*, Department of Algebra and Geometry, Palacky University, 779 00 Olomouc, Czech Republic

**Abstract:** 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-Caratheodory form is multisymplectic, and is also equivalent to asserting that Dedecker extremals of the latter $(m+1)$-form are holonomic.

**Keywords:** parametric variational problem, regularity, multisymplectic

**Classification (MSC 2000):** 58E15, 49N60

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