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

Taheri, Ali (2003) On Artin's braid groups and polyconvexity in the calculus of variations. Journal of the London Mathematical Society, 67 (3). pp. 752-768. ISSN 00246107

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Abstract

Let Omega subset of R-2 be a bounded Lipschitz domain and let F : Omega X R-+(2x2) --> R be a Caratheodory integrand such that F (x, (.)) is polyconvex for L-2-a.e. x is an element of Omega. Moreover assume that F is bounded from below and satisfies the condition F(x,xi) SE arrow infinity as det xi SE arrow 0 for L-2-a.e. x is an element of Omega. The paper describes the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional F[u] := integral(Omega) F(x,delu(x))dx, where the map u lies in the Sobolev space W-id(1,p)(Omega,R-2) with p greater than or equal to 2 and satisfies the pointwise condition det delu(x) > 0 for L-2-a.e. x is an element of Omega. The question is settled by establishing that F[(.)] admits a set of strong local minimizers on W-id(1,p)(Omega,R-2) that can be indexed by the group P-n circle plus Z(n), the direct sum of Artin's pure braid group on it strings and n copies of the infinite cyclic group. The dependence on the domain topology is through the number of holes n in Omega and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Ali Taheri
Date Deposited: 06 Feb 2012 19:30
Last Modified: 09 Jul 2012 14:11
URI: http://sro.sussex.ac.uk/id/eprint/20893
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