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Bandyopadhyay, Saugata; Dacorogna, Bernard; Sil, Swarnendu

Calculus of variations with differential forms. (English) Zbl 1317.49015

J. Eur. Math. Soc. (JEMS) 17, No. 4, 1009-1039 (2015).
In order to investigate variational problems involving differential forms, the authors introduce extensions of convexity and some of its multifarious generalizations, such as quasi-convexity, polyconvexity and so on. They provide a detailed analysis of the relationships between the new notions, to which the prefix “ext.” is added, and the classical ones. In the considered cases, “ext.” may stand either for the exterior product or the exterior derivative. The interesting new notions are characterized in several concrete cases, besides that the authors give an example of an ext. one convex function that is not ext. quasiconvex. For this they make use of a well-known result of V. Šverák in [Proc. R. Soc. Edinb., Sect. A 120, No. 1–2, 185–189 (1992; Zbl 0777.49015)]. The authors conclude their paper exhibiting a nice application of the ext. quasiconvexity notion to a minimization problem.
Reviewer: Antonio Roberto da Silva (Rio de Janeiro)

Cited in 3 Reviews
Cited in 10 Documents

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J40 Variational inequalities
58A10 Differential forms in global analysis

Keywords:

calculus of variations; differential forms; quasiconvexity; polyconvexity; ext. one convexity

Citations:

Zbl 0777.49015
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References:

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