Journal of Convex Analysis, Vol. 6, No. 2, pp. 349-366 (1999)

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

Giovanni Bellettini and Guy Bouchitté and Ilaria Fragala

Dipartimento di Matematica, Universita di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy, and Département de Mathématiques (Laboratoire A.N.L.A.), Université de Toulon et du Var, BP 132, F-83957 La Garde, Cedex, France, and Dipartimento di Matematica "L. Tonelli", Universita di Pisa, Via Buonarroti, 2, 56127 Pisa, Italy,

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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