On Critical Points of Functionals with Polyconvex Integrands

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 vector-valued 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 $A$