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
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 ApplicationsISSN
0022-247XPublisher
ElsevierExternal DOI
Issue
18Volume
473Page range
1-26Department 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
2018-12-13First Open Access (FOA) Date
2019-10-31First Compliant Deposit (FCD) Date
2018-12-13Usage metrics
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