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Existence, semicontinuity and representation for the integrals of the calculus of variations. The BV case. (English) Zbl 0613.49030

The authors establish existence, semicontinuity and representation theorems for the Burkill-Cesari-Weierstrass (BCW) integration in a rather general setting.
To be specific the set functions considered here are of the form \(\Phi (I)=F(X(I),\phi (I))\), where \(F\) satisfies the usual regularity assumptions employed for these problems and \(X(I)\) is a general set function which replaces the usual choice function \(x(\omega (I))\). Thanks to the great generality of the set function \(X(I)\) the authors are able to apply their results to the one-dimensional BCW integral over BV curves and to the one-dimensional weighted variation. This latter application contains and extends the existence, semicontinuity and representation theorems given by M. Boni [Atti Sem. Mat. Fis. Univ. Modena 25 (1976), 195–210 (1977; Zbl 0407.26003)].
Reviewer: P. Pucci

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
26B15 Integration of real functions of several variables: length, area, volume
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
49J45 Methods involving semicontinuity and convergence; relaxation

Citations:

Zbl 0407.26003
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