Relaxation and Integral Representation forFunctionals of Linear Growth on MetricMeasure spaces Article Swipe

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Heikki Hakkarainen , Juha Kinnunen , Panu Lahti , Pekka Lehtelä ·

YOU? · · 2016 · Open Access · · DOI: https://doi.org/10.1515/agms-2016-0013 · OA: W2492393789

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.