Abstract

We introduce and study the space of bounded variation functions with respect to a Radon measure μ on RN\mathbb{R}^N 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.

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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 Buonarroti 2, 56127 Pisa, Italy

Keywords

  • Bounded variation functions
  • Radon measures
  • relaxation
  • duality
  • integral representation

Mathematics Subject Classification

26A25; 49M20, 46N10

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Suggested citation

G. Bellettini, G. Bouchitté, I. Fragalà. “BV Functions with Respect to a Measure and Relaxation of Metric Integral Functionals.” Journal of Convex Analysis 6 (1999), No. 2, 349–366. https://doi.org/10.68381/jca06020

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Published by Heldermann Verlag, 1999. Rights now held by Banach Press.