Classification of strict limits of planar BV homeomorphisms

TL;DR

This paper classifies the strict limits of planar BV homeomorphisms, showing they can be approximated by homeomorphisms and exhibit cavitations and fractures, extending previous conditions to include these singularities.

Contribution

It introduces the BV no-crossing condition as a generalization of the no-crossing condition, characterizing limits of planar Sobolev homeomorphisms with singularities.

Findings

Strict limits can be approximated by homeomorphisms.

Limits exhibit cavitations and fractures.

BV no-crossing condition characterizes these limits.

Abstract

We present a classification of strict limits of planar BV homeomorphisms. The authors and S. Hencl showed in a previous work \cite{CHKR} that such mappings allow for cavitations and fractures singularities but fulfill a suitable generalization of the INV condition. As pointed out by J. Ball \cite{B}, these features are physically expected by limit configurations of elastic deformations. In the present work we develop a suitable generalization of the \emph{no-crossing} condition introduced by De Philippis and Pratelli in \cite{PP} to describe weak limits of planar Sobolev homeomorphisms that we call \emph{BV no-crossing} condition, and we show that a planar mapping satisfies this property if and only if it can be approximated strictly by homeomorphisms of bounded variations.