Archive for Rational Mechanics and Analysis

, Volume 176, Issue 3, pp 363–414

Local Minimizers and Quasiconvexity – the Impact of Topology

Authors

    • Mathematical InstituteUniversity of Oxford
Article

DOI: 10.1007/s00205-005-0356-7

Cite this article as:
Taheri, A. Arch. Rational Mech. Anal. (2005) 176: 363. doi:10.1007/s00205-005-0356-7

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 https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb1.gif : https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb2.gif from a purely topological point of view. The basic assumptions on https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb1.gif 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 https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb3.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb4.gif.

In the first part of the paper, we focus on the case where https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb4.gif 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 https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb3.gif into https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb4.gif. 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 https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb4.gif=ℝN and https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb5.gif is a bounded smooth domain. In particular we consider integrals

https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb6.gif

where the above assumptions on https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb1.gif 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 https://static-content.springer.com/image/art%3A10.1007%2Fs00205-005-0356-7/MediaObjects/s00205-005-0356-7flb1.gif 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