Lower semicontinuity for an integral functional in BV

We prove a lower semicontinuity result for a functional of linear growth initially defined by $\int_ΩF\left(\frac{dDu}{dμ}\right)\,dμ$ for $u\in BV(Ω;\mathbb{R}^N)$ with $Du\ll μ$. The positive Radon measure $μ$ is only assumed to satisfy $\mathcal L^n\ll μ$.

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