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

Let X = X [ a , b ] = { x : a < | x | < b } ⊂ R n with 0 < a < b < ∞ fixed be an open annulus and consider the energy functional, F [ u ; X ] = 1 2 ∫ X | ∇ u | 2 | u | 2 d x , over the space of admissible incompressible Sobolev maps A ϕ ( X ) = { u ∈ W 1 , 2 ( X , R n ) : 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 A ϕ ( X ) and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case n = 2 where A ϕ ( 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 L 1 -local minimisers of F in A ϕ ( X ) . We discuss variants and extensions to higher dimensions as well as to related energy functionals. [ABSTRACT FROM AUTHOR]