Generalised twists, , and the p-energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the energy functional

with p]1,∞ over the space of measure preserving 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 surprising discrepancy between even and odd dimensions. Here 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 where a key ingredient is a necessary and sufficient condition for an associated vector field to be a gradient.