Fracture surfaces and the regularity of inverses for BV deformations

Motivated by nonlinear elasticity theory, we study deformations that are weakly approximately differentiable, orientation-preserving and one-to-one almost everywhere, and in addition have finite surface energy. This surface energy is a functional E introduced by the authors in a previous paper, and has connections with the theory of currents. In the present paper we prove that E measures exactly the area of the surface created by the deformation. This is done through a proper definition of created surface, which is related to the set of discontinuity points of the inverse of the deformation. In doing that, we also obtain an SBV regularity result for the inverse.