Classification of strict limits of planar BV homeomorphisms

Abstract

We present a classification of m-strict limits (i.e. ▪ and |D1fk|(Ω)+|D2fk|(Ω)→|D1f|(Ω)+|D2f|(Ω)) of planar BV homeomorphisms; a class previously studied by the authors and S. Hencl in [6]. There it was shown that such mappings allow for cavitations and fractures singularities but fulfil a suitable generalization of the INV condition. As pointed out by J. Ball [3], these features are physically expected by limit configurations of elastic deformations. In the present work we develop a suitable generalization of the no-crossing condition introduced by De Philippis and Pratelli in [8] to describe weak limits of planar Sobolev homeomorphisms that we call the no-crossing BV condition, and we show that a planar mapping satisfies this property if and only if it can be approximated m-strictly by homeomorphisms of bounded variations.