Relaxation of functionals with linear growth: interactions of emerging measures and free discontinuities

TL;DR

This paper investigates the relaxation of certain integral functionals with linear growth involving interactions between functions and measures, addressing complexities due to discontinuities and concentration effects in the context of functions with bounded variations.

Contribution

It provides a detailed calculation of the relaxed functional in BV and Radon measures spaces, highlighting the role of measure-discontinuity interactions and concentration effects.

Findings

Derived explicit relaxation formulas for the functionals.

Identified the impact of measure-discontinuity interactions on relaxation.

Highlighted the significance of concentration effects in the relaxation process.

Abstract

For an integral functional defined on functions $(u,v)\in W^{1,1}\times L^1$ featuring a prototypical strong interaction term between $u$ and $v$, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities bring various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.