Annular rearrangements, incompressible axi-symmetric whirls and L1-local minimisers of the distortion energy

In this paper we consider a variational problem consisting of an energy functional defined by the integral, F[u, X] = 1/2?x |?u|²/|u|² dx, and an associated mapping space, here, the space of incompressible Sobolev mappings of the symmetric annular domain X = {x ? R n : a < |x| < b}: Af(X) = { u ? W1,2 (X, R n ) : det ?u = 1 a.e. and u|?X = x } . The goal is then two fold. Firstly to establish and highlight an unexpected difference in the symmetries of the extremiser and local minimisers of F over Af(X) in the two special cases n = 2 and n = 3. More specifically, that when n = 3, despite the inherent rotational symmetry in the problem, there are NO non-trivial rotationally symmetric critical points of F over Af(X), whereas in sharp contrast, when n = 2, not only that there is an infinitude of rotationally symmetric critical points of the energy but also there is an infinitude of local minimisers of F over Af(X) with respect to the L¹ -metric. At the heart of this analysis is an investigation into the rich homotopy structure of the space of self-mappings of annuli. The second aim is to introduce and implement a novel symmetrisation technique in the planar case n = 2 for Sobolev mappings u in Af(X) that lowers the energy whilst keeping the homotopy class of u invariant. We finally generalise and extend some of these results to higher dimensions, in particular, we show that only in even dimensions do we have an infinitude of non-trivial rotationally symmetric critical points.