Archive for Rational Mechanics and Analysis

, Volume 176, Issue 3, pp 363–414

Local Minimizers and Quasiconvexity – the Impact of Topology

Article

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 Open image in new window : Open image in new window from a purely topological point of view. The basic assumptions on Open image in new window are sequential lower semicontinuity with respect to W1,p-weak convergence and W1,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 Open image in new window and Open image in new window.

In the first part of the paper, we focus on the case where Open image in new window 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 Open image in new window into Open image in new window. 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 Open image in new window=ℝN and Open image in new window is a bounded smooth domain. In particular we consider integrals

Open image in new window

where the above assumptions on Open image in new window 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 Open image in new window that in turn settles a longstanding open problem in the multi-dimensional calculus of variations as described in [6].

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordEngland, U.K

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