Preprint 27/2002

Local minimizers and quasiconvexity - the impact of Topology

Ali Taheri

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Submission date: 14. Mar. 2002
Pages: 51
published in: Archive for rational mechanics and analysis, 176 (2005) 3, p. 363-414 
DOI number (of the published article): 10.1007/s00205-005-0356-7
Keywords and phrases: quasiconvexity, homotopy theory, cohomology groups

The aim of this paper is to discuss the question of existence and multiplicity of local minimizers for a relatively large class of functionals formula29 from a purely topological point of view. The basic assumptions on formula31 are sequential lower semicontinuity with respect to formula33-weak convergence and formula33-weak coercivity and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds formula37 and formula39. In the first part of the paper, we focus on the case where formula39 is non contractible and proceed by establishing a link between the latter problem and the question of enumeration of homotopy classes of continuous maps from various skeleta of formula37 into formula39. Naturally enough, our results in this direction are of a cohomological nature.

We devote the second part to the case where formula39 is the Euclidean space formula49 and formula51, with formula53 being a bounded smooth domain. In particular we consider integral functionals of the form
where the above assumptions on formula31 can be verified when the integrand F is appropriately quasiconvex and pointwise p-coercive with respect to the gradient argument. We introduce the notion of a topologically non trivial domain and under this hypothesis establish the required multiplicity result for strong local minimizers of formula31.

18.10.2019, 02:11