Preprint 71/2001

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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Let tex2html_wrap_inline27 be a bounded Lipschitz domain and let tex2html_wrap_inline29 be a Carathèodory integrand such that tex2html_wrap_inline31 is polyconvex for tex2html_wrap_inline33- a.e. tex2html_wrap_inline35. Moreover assume that F is bounded from below and satisfies the condition tex2html_wrap_inline39 as tex2html_wrap_inline41 for tex2html_wrap_inline33- a.e. tex2html_wrap_inline35. 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 tex2html_wrap_inline51 with tex2html_wrap_inline53 and satisfies the pointwise condition tex2html_wrap_inline55 for tex2html_wrap_inline33-a.e. tex2html_wrap_inline35. We settle the question by establishing that tex2html_wrap_inline61 admits a set of strong local minimizers on tex2html_wrap_inline63 that can be indexed by the group tex2html_wrap_inline65, 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 tex2html_wrap_inline73 and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.

24.11.2021, 02:11