Twist maps as energy minimisers in homotopy classes: Symmetrisation and the coarea formula

Let X = X[a,b] = {x : a < |x| < b} ? Rn with 0 < a < b < 8 fixed be an open annulus and consider the energy functional, F[u;X] = 1/2 X |?u|2 |u|2 dx, over the space of admissible incompressible Sobolev maps Af(X) ={u ? W1,2(X,Rn) : det?u = 1 a.e. in X and u|?X = f}, where f is the identity map of X. Motivated by the earlier works (Taheri (2005), (2009)) in this paper we examine the twist maps as extremisers of F over Af(X) and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case n = 2 where Af(X) is a union of infinitely many disjoint homotopy classes we establish the minimality of these extremising twists in their respective homotopy classes a result that then leads to the latter twists being L1-local minimisers of F in Af(X). We discuss variants and extensions to higher dimensions as well as to related energy functionals.