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
Bibtex
Keywords and phrases: quasiconvexity, homotopy theory, cohomology groups

Abstract:
The aim of this paper is to discuss the question of existence and multiplicity of local minimizers for a relatively large class of functionals from a purely topological point of view. The basic assumptions on are sequential lower semicontinuity with respect to -weak convergence and -weak coercivity and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds and . In the first part of the paper, we focus on the case where 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 into . Naturally enough, our results in this direction are of a cohomological nature.

We devote the second part to the case where is the Euclidean space and , with being a bounded smooth domain. In particular we consider integral functionals of the form

where the above assumptions on 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 .