Local minimizers and quasiconvexity - the impact of topology

Taheri, Ali (2005) Local minimizers and quasiconvexity - the impact of topology. Archive for Rational Mechanics and Analysis, 176 (3). pp. 363-414. ISSN 0003-9527

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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 F[.] : W-1,W-p(X, Y) -> R from a purely topological point of view. The basic assumptions of F[.] are sequential lower semicontinuity with respect to W-1,W-p-weak convergence and W-1,W-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 X and Y. In the first part of the paper, we focus on the case where Y 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 X into Y. 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 Y = R-N and X subset of R-n is a bounded smooth domain. In particular we consider integrals F[u] := integral(X)F(x,u(x), del u(x))dx, where the above assumptions on F[.] 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 F[.] that in turn settles a longstanding open problem in the multi-dimensional calculus of variations as described in [6].

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Ali Taheri
Date Deposited: 06 Feb 2012 19:38
Last Modified: 09 Jul 2012 14:45
URI: http://sro.sussex.ac.uk/id/eprint/21587
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