Heriot-Watt Mathematics Report Series
HWM02-5, 21 Feb 2002
Local minimizers and quasiconvexity - the impact of topology
A Taheri
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 $F[.]: W^{1,p}(X,Y) \to \R$ from a purely topological point of view.
The basic assumptions on $F[.]$ are sequential lower semicontinuity with respect to $W^{1,p}$-weak convergence and $W^{1,p}$-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 $Y$. Naturally enough our results in this direction are of a cohomological nature.
We devote the second part to the case where $Y=\R^N$ and $X=\Omega$ with $\Omega$ being a bounded smooth domain in $\R^n$. In particular we consider integral functionals of the form
$$
F[u]=\int_{\Omega} F(x,u,\nabla u)
$$
where the above assumptions on $F[.]$ 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 assumption establish the required multiplicity result for strong local minimizers of $F[.]$.
Google Scholar Search: links, citations and journal (if available)
Contact Details | 2002 Reports Index |
Full Index