On Artin's braid group and polyconvexity in the calculus of variations
UNSPECIFIED. (2003) On Artin's braid group and polyconvexity in the calculus of variations. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 67 (Part 3). pp. 752-768. ISSN 0024-6107Full text not available from this repository.
Official URL: http://dx.doi.org/10.1112/S0024610703004253
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:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Journal or Publication Title:||JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES|
|Publisher:||LONDON MATH SOC|
|Number of Pages:||17|
|Page Range:||pp. 752-768|
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