Archive for Rational Mechanics and Analysis

, Volume 176, Issue 3, pp 363–414

Local Minimizers and Quasiconvexity – the Impact of Topology


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


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 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 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 whereℝN 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].

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

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