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## On Artin's braid group and polyconvexity in the calculus of variations

### Abstract

Let Ω ⊂ 2 be a bounded Lipschitz domain and let\ud F : Ω × 2×2\ud +\ud −→ \ud be a Carathèodory integrand such that F (x, ·) is polyconvex for L2-a.e. x ∈ Ω. Moreover assume that\ud 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\ud [u] :=\ud \ud Ω\ud F (x,∇u (x)) dx,\ud where the map u lies in the Sobolev space W1,p\ud id (Ω,2) with p 2 and satisfies the pointwise condition\ud det ∇u (x) > 0 for L2-a.e. x ∈ Ω. The question is settled by establishing that [·] admits a set of strong\ud 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

Topics: QA
Publisher: Cambridge University Press
Year: 2003
OAI identifier: oai:wrap.warwick.ac.uk:761

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