Classification of area-strict limits of planar BV homeomorphisms

We present a classification of area-strict limits of planar BV$BV$ homeomorphisms. This class of mappings allows for cavitations and fractures but fulfil a suitable generalisation of the INV condition of M & uuml;ller and Spector (Arch. Rational Mech. Anal. 131 (1995), no. 1, 1-66). As pointed out by J. Ball, these features are expected in limit configurations of elastic deformations. De Philippis and Pratelli introduced the no-crossing condition which characterises the W1,p$W<^>{1,p}$ closure of planar homeomorphisms. In the current paper, we show that a suitable version of this concept is equivalent with a map, f$f$, being the area-strict limit of BV homeomorphisms. This extends our results from Campbell et al. (J. Funct. Anal. 285 (2023), no. 3, Paper No. 109953, 30), where we proved that the no-crossing BV condition for a BV map was equivalent with the map being the m-strict limit of homeomorphisms (i.e. and |D1fk|(Omega)+|D2fk|(Omega)->|D1f|(Omega)+|D2f|(Omega)$|{D}_{1}{f}_{k}|(\mathrm{\Omega})+|{D}_{2}{f}_{k}|(\mathrm{\Omega})\to |{D}_{1}f|(\mathrm{\Omega})+|{D}_{2}f|(\mathrm{\Omega})$). Further, we show that the no-crossing BV condition is equivalent with a seemingly stronger version of the same condition.