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We prove that a $BV$ map with values into the projective space$\mathbb{RP}^{d-1}$ has a $BV$ lifting with values into the unit sphere$\mathbb S^{d-1}$ that satisfies an optimal $BV$-estimate. As an application toliquid crystals, this result is also stated for $BV$ maps with values into theset of uniaxial $Q$-tensors. In order to quantify $BV$ liftings, we prove anexplicit formula for an intrinsic $BV$-energy of maps with values into anycompact smooth manifold.

Lifting of $\mathbb{RP}^{d-1}$-valued maps in $BV$ and applications to  uniaxial $Q$-tensors. With an appendix on an intrinsic $BV$-energy for  manifold-valued maps