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. 752-768. ISSN 0024-6107

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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 L2-a.e. x ∈ Ω. Moreover assume that F is bounded from below and satisfies the condition F (x, ξ) ∞ as det ξ 0 for L2-a.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 L2-a.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:	0024-6107
Date:	June 2003
Volume:	Vol.67
Number:	No.3
Page Range:	pp. 752-768
Identification Number:	10.1112/S0024610703004253
Status:	Peer Reviewed
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/761

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