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On Artin's braid group and polyconvexity in the calculus of variations
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Taheri, Ali. (2003) On Artin's braid group and polyconvexity in the calculus of variations. Journal of the London Mathematical Society, Vol.67 (No.3). pp. 752768. ISSN 00246107

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Official URL: http://dx.doi.org/10.1112/S0024610703004253
Abstract
Let Ω ⊂ 2 be a bounded Lipschitz domain and let
F : Ω × 2×2
+
−→
be a Carathèodory integrand such that F (x, ·) is polyconvex for L2a.e. x ∈ Ω. Moreover assume that
F is bounded from below and satisfies the condition F (x, ξ) ∞ as det ξ 0 for L2a.e. x ∈ Ω. The paper describes the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional
[u] :=
Ω
F (x,∇u (x)) dx,
where the map u lies in the Sobolev space W1,p
id (Ω,2) with p 2 and satisfies the pointwise condition
det ∇u (x) > 0 for L2a.e. x ∈ Ω. The question is settled by establishing that [·] admits a set of strong
local minimizers on W1,p id (Ω,2) that can be indexed by the group n ⊕ n, the direct sum of Artin’s pure braid group on n strings and n copies of the infinite cyclic group. The dependence on the domain topology is through the number of holes n in Ω and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Braid theory, Mathematical optimization, Coxeter groups, Calculus of variations, Lipschitz spaces, Sobolev spaces 
Journal or Publication Title:  Journal of the London Mathematical Society 
Publisher:  Cambridge University Press 
ISSN:  00246107 
Official Date:  June 2003 
Volume:  Vol.67 
Number:  No.3 
Page Range:  pp. 752768 
Identification Number:  10.1112/S0024610703004253 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. E. Artin, ‘Theory of braids’, Ann. of Math. 48 (1947) 101–126. 
URI:  http://wrap.warwick.ac.uk/id/eprint/761 
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