Polyconvexity, generalised twists and self-maps of annuli in the multi-dimensional calculus of variations

Let be a bounded Lipschitz domain and consider the energy functional over the space of admissible maps In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler¿Lagrange equations associated with over . The main result is a complete characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions. Indeed we show that in even dimensions the latter system of equations admit infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler¿Lagrange equations.