Regularity of inverses of Sobolev deformations with finite surface energy

Let u be a Sobolev W1,pW1,p map from a bounded open set Ω⊂RnΩ⊂Rn to RnRn. We assume u to satisfy some invertibility properties that are natural in the context of nonlinear elasticity, namely, the topological condition INV and the orientation-preserving constraint det⁡Du>0det⁡Du>0. These deformations may present cavitation, which is the phenomenon of void formation. We also assume that the surface created by the cavitation process has finite area. If p>n−1p>n−1, we show that a suitable defined inverse of u is a Sobolev map. A partial result is also given for the critical case p=n−1p=n−1. The proof relies on the techniques used in the study of cavitation.