Classification of area-strict limits of planar BV homeomorphisms

TL;DR

This paper classifies area-strict limits of planar BV homeomorphisms, linking them to a no-crossing condition and extending previous results on BV limits and deformation features.

Contribution

It establishes the equivalence between area-strict limits of BV homeomorphisms and a no-crossing condition, extending prior work on BV limits and deformation properties.

Findings

Area-strict limits of BV homeomorphisms are characterized by a no-crossing condition.

The no-crossing BV condition is equivalent to a stronger version of itself.

Results extend previous work on BV limits and elastic deformation features.

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 generalization of the INV condition. As pointed out by J. Ball [4], these features are expected in limit configurations of elastic deformations. In [12], De Philippis and Pratelli introduced the \emph{no-crossing} condition which characterizes 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 [10], where we proved that the \emph{no-crossing BV} condition for a BV map was equivalent with the map being the m-strict limit of homeomorphisms (i.e. $f_k$ converges $w^*$ to $f$ and $|D_1f_k|(\Omega)+|D_2f_k|(\Omega) \to |D_1f|(\Omega)+|D_2f|(\Omega)$). Further we show that the \emph{no-crossing BV} condition is equivalent with a seemingly stronger version of the same condition.