University of Sussex
Browse
Morris-Taheri-Uniqueness2018.pdf (413.96 kB)

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

Download (413.96 kB)
journal contribution
posted on 2023-06-09, 16:13 authored by Ali TaheriAli Taheri, Charles Morris
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.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Journal of Mathematical Analysis and Applications

ISSN

0022-247X

Publisher

Elsevier

Issue

18

Volume

473

Page range

1-26

Department affiliated with

  • Physics and Astronomy Publications

Research groups affiliated with

  • Analysis and Partial Differential Equations Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-12-13

First Open Access (FOA) Date

2019-10-31

First Compliant Deposit (FCD) Date

2018-12-13

Usage metrics

    University of Sussex (Publications)

    Categories

    No categories selected

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC