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Journal of Convex Analysis, Vol. 6, No. 2, pp. 349-366 (1999)
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BV Functions with Respect to a Measure and Relaxation of Metric Integral Functionals

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Giovanni Bellettini and Guy Bouchitté and Ilaria Fragala

Dipartimento di Matematica, Universita di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy, belletti@axp.mat.uniroma2.it and Département de Mathématiques (Laboratoire A.N.L.A.), Université de Toulon et du Var, BP 132, F-83957 La Garde, Cedex, France, bouchitte@univ-tln.fr and Dipartimento di Matematica "L. Tonelli", Universita di Pisa, Via Buonarroti, 2, 56127 Pisa, Italy, fragala@dm.unipi.it

**Abstract:** We introduce and study the space of bounded variation functions with respect to a Radon measure $\mu$ on $\mathbb{R}^N$ and to a metric integrand $\varphi$ on the tangent bundle to $\mu$. We show that it is equivalent to view such space as the class of $\mu$-integrable functions for which a distributional notion of $(\mu, \varphi)$-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 $(\mu, \varphi)$-total variation.

**Keywords:** Bounded variation functions, Radon measures, Relaxation, Duality, Integral representation

**Classification (MSC2000):** 26A45, 49M20, 46N10

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