Geodesic
Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces
Analysis and Geometry in Metric Spaces, Tome 4 (2016) no. 1.

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This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem involving the functional, boundary values can be presented as a penalty term.
Mots-clés : calculus of variations, functionals of linear growth, relaxation, functions of bounded variation, analysis on metric measure spaces 
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     title = {Relaxation and {Integral} {Representation} for {Functionals} of {Linear} {Growth} on {Metric} {Measure} spaces},
     journal = {Analysis and Geometry in Metric Spaces},
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Hakkarainen, Heikki; Kinnunen, Juha; Lahti, Panu; Lehtelä, Pekka. Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces. Analysis and Geometry in Metric Spaces, Tome 4 (2016) no. 1. http://geodesic.mathdoc.fr/item/AGMS_2016_4_1_a12/