New models for deformations: Linear Distortion and the failure of rank-one convexity

Abstract: In this article, we discuss new models for static nonlinear deformations via scale-invariant conformal energy functionals based on the linear distortion. In particular, we give examples to show that, despite equicontinuity estimates giving compactness, minimising sequences will have strictly lower energy than their limit, and that this energy gap can be quite large. We do this by showing that Iwaniec's theorem on the failure of rank-one convexity for the linear distortion of a specific family of linear mappings, is actually generic and we subsequently identify the optimal rank-one direction to deform a linear map to maximally decrease its distortion.