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Let Ω ⊂ 2 be a bounded Lipschitz domain and let\ud F : Ω × 2×2\ud +\ud −→ \ud be a Carathèodory integrand such that F (x, ·) is polyconvex for L2-a.e. x ∈ Ω. Moreover assume that\ud F is bounded from below and satisfies the condition F (x, ξ) ∞ as det ξ 0 for L2-a.e. x ∈ Ω. The paper describes the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional\ud [u] :=\ud \ud Ω\ud F (x,∇u (x)) dx,\ud where the map u lies in the Sobolev space W1,p\ud id (Ω,2) with p 2 and satisfies the pointwise condition\ud det ∇u (x) > 0 for L2-a.e. x ∈ Ω. The question is settled by establishing that [·] admits a set of strong\ud local minimizers on W1,p id (Ω,2) that can be indexed by the group n ⊕ n, 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 Ω and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation

Topics:
QA

Publisher: Cambridge University Press

Year: 2003

OAI identifier:
oai:wrap.warwick.ac.uk:761

Provided by:
Warwick Research Archives Portal Repository

- (1992). Boundary behaviour of conformal maps,G r a duate Texts in
- (1975). Braids, links and mapping class groups,A nnals
- (1948). Chow,‘ O nt h ea l g ebraic braid group’,
- (1938). Die Gruppe der Abbildungsklassen’,
- (1995). Functions of one complex variable II,
- (1977). Geometric topology in dimensions 2 and 3,
- (1977). Gol’dshtein,‘ Q u asiconformal mappings and spaces of functions with generalized ﬁrst derivatives’,
- (2002). L o c a lm i n imizers and quasiconvexity – the impact of topology’,
- (1962). Lickorish,‘ Ar e p r e sentation of orientable combinatorial 3-manifolds’,
- (1966). Multiple integrals in the calculus of variations,G r a duate Texts in
- (2002). O nc r i t ical points of functionals with polyconvex integrands’,
- (1997). On homotopy conditions and the existence of multiple equilibria in ﬁnite elasticity’,
- Q u asiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations’,
- (1947). T h e o r yo fbraids’,
- (1972). Uniqueness of non-linear elastic equilibrium for prescribed boundary displacement and suﬃciently small strains’,
- (1984). W1,p-quasiconvexity and variational problems for multiple integrals’,