Submitted Paper

Inserted: 9 jan 2023
Last Updated: 9 jan 2023

Year: 2022

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.

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• AreaStrict Final.pdf