
Journal of Convex Analysis 09 (2002), No. 1, 055072 Copyright Heldermann Verlag 2002 On Critical Points of Functionals with Polyconvex Integrands Ali Taheri MaxPlanckInstitute for Mathematics, Inselstr. 2226, 04103 Leipzig, Germany [Abstractpdf] Let $\Omega \subset {\mathbb R}^n$ be a bounded domain with Lipschitz boundary, and assume that $f: \Omega \times {\mathbb R}^{m \times n} \to {\mathbb R}$ is a Carath\'eodory integrand such that $f(x, \cdot)$ is {\it polyconvex} for ${\mathcal L}^n$ a.e. $x \in \Omega$. In this paper we consider integral functionals of the form $$ {\mathcal F}(u, \Omega) := \int_{\Omega} f(x, Du(x)) \, dx, $$ where $f$ satisfies a growth condition of the type $$ f(x,A) \le c (1 + A^p), $$ for some $c>0$ and $1 \le p < \infty$, and $u$ lies in the Sobolev space of vectorvalued functions $W^{1,p}(\Omega, {\mathbb R}^m)$. We study the implications of a function $u_0$ being a critical point of ${\mathcal F}$. In this regard we show among other things that if $f$ does not depend on the spatial variable $x$, then every piecewise affine critical point of ${\mathcal F}$ is a global minimizer subject to its own boundary condition. Moreover for the general case, we construct an example exhibiting that the uniform positivity of the second variation at a critical point is {\it not} sufficient for it to be a strong local minimizer. In this example $f$ is discontinuous in $x$ but smooth in $A$. [ Fulltextpdf (505 KB)] 