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Journal of Convex Analysis 06 (1999), No. 2, 349--366
Copyright Heldermann Verlag 1999

BV Functions with Respect to a Measure and Relaxation of Metric Integral Functionals

Giovanni Bellettini
Dip. di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy

Guy Bouchitté
Dép. de Mathématiques, Université de Toulon et du Var, 83957 La Garde, France

Ilaria Fragalà
Dip. di Matematica, Università di Pisa, Via Buonarotti 2, 56127 Pisa, Italy

We introduce and study the space of bounded variation functions with respect to a Radon measure μ on RN and to a metric integrand φ on the tangent bundle to μ. We show that it is equivalent to view such space as the class of μ-integrable functions for which a distributional notion of (μ, φ)-total variation is finite, or as the finiteness domain of a relaxed functional. We prove a quite, general coarea-type formula and then we focus our attention to the problem of finding an integral representation for the (μ, φ)-total variation.

Keywords: Bounded variation functions, Radon measures, relaxation, duality, integral representation.

MSC: 26A25; 49M20, 46N10

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