, Volume 176, Issue 3, pp 363-414
Date: 20 Apr 2005

Local Minimizers and Quasiconvexity – the Impact of Topology

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Abstract.

The aim of this paper is to discuss the question of existence and multiplicity of strong 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 W 1, p -weak convergence and W 1, p -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 . As this in turn can be tackled by the so-called obstruction method, it is evident that our results in this direction are of a cohomological nature.

The second part is devoted to the case where =ℝ N and is a bounded smooth domain. In particular we consider integrals

where the above assumptions on can be verified when the integrand F is quasiconvex and pointwise p-coercive with respect to the gradient argument. We introduce and exploit the notion of a topologically non-trivial domain and under this establish the first existence and multiplicity result for strong local minimizers of that in turn settles a longstanding open problem in the multi-dimensional calculus of variations as described in [6].

Communicated by C.A. Stuart