A deformation in Nonlinear elasticity is represented by a Sobolev map. In order to preserve the orientation, the gradient determinant is required to be positive. The integrability exponents p of the gradient and q of the cofactor gradient give a measure of the regularity of the map. When p ≥ n-1 and q ≥ n/(n-1), maps are regular and locally invertible. When p ≥ n-1 and 1 < q < n/(n-1), cavitation can appear and is detected by the distributional determinant. Moreover, cavitation is the only possible singularity for this range of exponents, so in the absence of cavitation, the maps enjoy the same regularity as their counterparts for q ≥ n/(n-1). Finally, if p = n-1 and q = 1, a pathology of the sort of a harmonic dipole can emerge: two cavities appear, one of which reverses the orientation and creates a hole which is filled by material from elsewhere. The distributional determinant detects this singularity but its description needs to be complemented by the singular part of the inverse.
Joint work with Marco Barchiesi, Duvan Henao and Rémy Rodiac.