Relaxation and integral representation for functionals of linear growth on metric measure spaces

TL;DR

This paper develops an integral representation for linear growth functionals on metric measure spaces, revealing new features in the absolutely continuous part and applying it to variational problems with boundary penalties.

Contribution

It introduces a novel integral representation for functionals of linear growth on metric measure spaces, including a new constant in the absolutely continuous part.

Findings

Integral representation includes a constant in the absolutely continuous part.

Singular part representation aligns with the variation measure.

Application to variational problems with boundary penalties.

Abstract

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincar\'e inequality. Such a functional is defined through 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 related to the functional, boundary values can be presented as a penalty term.