Classification of area-strict limits of planar BV homeomorphisms

Abstract

We present a classification of area-strict limits of planar $BV$ homeomorphisms. This class of mappings allows for cavitations and fractures but fulfil a suitable generalisation of the INV condition of Müller 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 $W^{1,p}$ closure of planar homeomorphisms. In the current paper, we show that a suitable version of this concept is equivalent with a map, $f$, 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 $|D_1{f}_k|\left(\Omega \right)+|D_2{f}_k|\left(\Omega \right)\to |D_{1}f|\left(\Omega \right)+|D_{2}f|\left(\Omega \right)$). Further, we show that the no-crossing BV condition is equivalent with a seemingly stronger version of the same condition.