Inner and Outer Versions of Hyper-Elasticity

Through suitable changes of variables for a typical problem in hyper-elasticity, either in the reference or deformed configurations, one can setup and analyze versions of the same problem in terms of inner or outer maps or variations. Though such kind of transformations are part of the classical background in the Calculus of Variations, we explore under what sets of hypotheses such versions can be shown to have minimizers and be equivalent to the standard form of the problem. Such sets of hypotheses lead naturally to some distinct poly-convex energy densities for hyper-elasticity. Likewise we explore optimality in either of the two forms through a special way to generate one-parameter families of feasible deformations, feasibility including injectivity and non-interpenetration of matter.