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

TL;DR

This paper establishes that BV maps into projective space can be lifted to BV maps into the sphere with optimal estimates, with applications to liquid crystals and an explicit formula for intrinsic BV-energy of manifold-valued maps.

Contribution

It proves the existence of BV liftings for projective space maps with optimal BV-estimates and introduces an explicit intrinsic BV-energy formula for manifold-valued maps.

Findings

BV maps into $ p^{d-1}$ can be lifted to $S^{d-1}$ with optimal BV-estimates.

Application to uniaxial Q-tensors in liquid crystal models.

Explicit formula for intrinsic BV-energy of manifold-valued maps.

Abstract

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 to liquid crystals, this result is also stated for $BV$ maps with values into the set of uniaxial $Q$-tensors. In order to quantify $BV$ liftings, we prove an explicit formula for an intrinsic $BV$-energy of maps with values into any compact smooth manifold.