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.
@article{bwmeta1.element.doi-10_1515_agms-2016-0013,
author = {Heikki Hakkarainen and Juha Kinnunen and Panu Lahti and Pekka Lehtel\"a},
title = {Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces},
journal = {Analysis and Geometry in Metric Spaces},
volume = {4},
year = {2016},
zbl = {1354.49027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_agms-2016-0013}
}
Heikki Hakkarainen; Juha Kinnunen; Panu Lahti; Pekka Lehtelä. Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces. Analysis and Geometry in Metric Spaces, Tome 4 (2016) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_agms-2016-0013/