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

Morris, Charles and Taheri, Ali (2017) Annular rearrangements, incompressible axi-symmetric whirls and L1-local minimisers of the distortion energy. Nonlinear Differential Equations and Applications, 25 (2). ISSN 1021-9722

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Abstract

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}:

Aφ(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 Aφ(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 Aφ(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 Aφ(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 Aφ(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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Billy Wichaidit
Date Deposited: 06 Nov 2017 09:55
Last Modified: 19 Jan 2018 10:54
URI: http://sro.sussex.ac.uk/id/eprint/70935

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