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

Morris, C and Taheri, A (2017) Twist maps as energy minimisers in homotopy classes: Symmetrisation and the coarea formula. Nonlinear Analysis: Theory, Methods and Applications, 152. pp. 250-275. ISSN 0362-546X

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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] = 1/2 X |∇u|2 |u|2 dx, 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Physics and Astronomy
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Subjects: Q Science > QA Mathematics
Depositing User: Billy Wichaidit
Date Deposited: 13 Jul 2017 09:22
Last Modified: 11 Sep 2017 14:34
URI: http://sro.sussex.ac.uk/id/eprint/69221

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