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

Ali Taheri

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Submission date: 05. Oct. 2001
Pages: 21
published in: The journal of the London Mathematical Society, 67 (2003) 3, p. 752-768 
DOI number (of the published article): 10.1112/S0024610703004253
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Abstract:
Let be a bounded Lipschitz domain and let be a Carathèodory integrand such that is polyconvex for - a.e. . Moreover assume that F is bounded from below and satisfies the condition as for - a.e. . In this article we study the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional

where the map u lies in the Sobolev space with and satisfies the pointwise condition for -a.e. . We settle the question by establishing that admits a set of strong local minimizers on that can be indexed by the group , 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 and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.