Lusin's condition and the distributional determinant for deformations with finite energy

Abstract

Based on a previous work by the authors on the modelling of cavitation and fracture in nonlinear elasticity, we give an alternative proof of a recent result by Csornyei, Hencl and Malý on the regularity of the inverse of homeomorphisms in the Sobolev space W 1;n 1 . With this aim, we show that the notion of fracture surface introduced by the authors in their model corresponds precisely to the original notion of cavity surface in the cavitation models of Muller and Spector (1995) and Conti and De Lellis (2003). We also find that the surface energy introduced in the model for cavitation and fracture is related to Lusin's condition (N) on the non-creation of matter. A fundamentalquestionunderlyingthis paperis whetherDetDu D detDunecessarily implies that the deformation u opens no cavities. We show that this is not true unless Muller and Spector's condition INV for the non-interpenetration of matter is satisfied. Having thus provided an additionaljustification of its importance, we provethe stability of this condition with respect to weak convergence in the critical spaceW 1;n 1 . Combining this with the work by Conti and De Lellis, we obtain an existence theory for cavitation in this critical case.