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

Abstract

Let X=X[a,b]={x:a<|x|<b}⊂Rn with 0<a<b<∞ fixed be an open annulus and consider the energy functional, F[u;X]=12∫X|∇u|2|u|2dx, over the space of admissible incompressible Sobolev maps Aϕ(X)={u∈W1,2(X,Rn):det∇u=1 a.e. in X and u|∂X=ϕ}, where ϕ 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 Aϕ(X) and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case n=2 where Aϕ(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 Aϕ(X). We discuss variants and extensions to higher dimensions as well as to related energy functionals.