

Preprint 27/2002
Local minimizers and quasiconvexity - the impact of Topology
Ali Taheri
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Submission date: 14. Mar. 2002
Pages: 51
published in: Archive for rational mechanics and analysis, 176 (2005) 3, p. 363-414
DOI number (of the published article): 10.1007/s00205-005-0356-7
Bibtex
Keywords and phrases: quasiconvexity, homotopy theory, cohomology groups
Abstract:
The aim of this paper is to discuss the question of existence and
multiplicity of 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
-weak convergence and
-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
. Naturally enough, our results in this direction
are of a cohomological nature.
We devote the second part to the case where is the
Euclidean space
and
, with
being a bounded smooth domain. In particular we
consider integral functionals of the form
where the above assumptions on can be verified
when the integrand F is appropriately quasiconvex and pointwise
p-coercive with respect to the gradient argument. We introduce the
notion of a topologically non trivial domain and under this
hypothesis establish the required multiplicity result for strong
local minimizers of
.