On the uniqueness and monotonicity of energy minimisers in the homotopy classes of incompressible mappings and related problems

Taheri, Ali and Morris, Charles (2019) On the uniqueness and monotonicity of energy minimisers in the homotopy classes of incompressible mappings and related problems. Journal of Mathematical Analysis and Applications, 473 (18). pp. 1-26. ISSN 0022-247X

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Abstract

The goal of this paper is to prove the existence and uniqueness of the so-called energy minimisers in homotopy classes for the variational energy integral

F[u; X] = Z X F(|x| 2 , |u| 2 )|∇u| 2 /2 dx,

with F ≥ c > 0 of class C 2 and satisfying suitable conditions and u lying in the Sobolev space of weakly differentiable incompressible mappings of a finite open symmetric plane annulus X onto itself, specifically, lying in A(X) = {u ∈ W 1,2 (X, R 2 ) : det ∇u = 1 a.e. in X, and u ≡ x on ∂X}. It is well known that the space A(X) admits a countably infinite homotopy class decomposition A(X) = S Ak (with k ∈ Z). We prove that the energy integral F has a unique minimiser in each of these homotopy classes Ak. Furthermore we show that each minimiser is a homeomorphic, monotone, radially symmetric twist mapping of class C 3 (X, X) or as smooth as F allows thereafter whilst also being a local minimiser of F over A(X) with respect to the L 1 -metric. To our best knowledge this is the first uniqueness result for minimisers in homotopy classes in the context of incompressible mappings.

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
Related URLs:
Depositing User: Ali Taheri
Date Deposited: 13 Dec 2018 09:49
Last Modified: 02 Jul 2019 13:19
URI: http://sro.sussex.ac.uk/id/eprint/80749

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