On Artin's Braid Group And Polyconvexity In The Calculus Of Variations |
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Authors: | Taheri Ali |
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Institution: | Department of Mathematics, University of Warwick Coventry CV4 7AL taheri{at}maths.warwick.ac.uk |
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Abstract: | Let R2 be a bounded Lipschitz domain and let
be a Carathèodory integrand such that F(x,·) is polyconvex for L2-a.e. x . Moreover assume thatF is bounded from below and satisfies the condition as det for L2-a.e. x . The paper describes the effect of domain topologyon the existence and multiplicity of strong local minimizersof the functional
wherethe map u lies in the Sobolev space Wid1,p (, R2) with p 2and satisfies the pointwise condition u(x) >0 for L2-a.e.x . The question is settled by establishing that F·]admits a set of strong local minimizers on that can be indexed by the group Pn Zn, the directsum of Artin's pure braid group on n strings and n copies ofthe infinite cyclic group. The dependence on the domain topologyis through the number of holes n in and the different mechanismsthat give rise to such local minimizers are fully exploitedby this particular representation. |
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