Remarks on quasiconvexity and stability of equilibria for variational integrals

Let $F:{{\mathbf{R}}^{nN}} \to {\mathbf{R}}$ be a uniformly strictly quasiconvex function (see [3, 4]) of class ${C^{2 + \alpha }},(0 < \alpha < 1)$, and be of polynomial growth. Then every smooth solution of the Euler-Lagrangian equation of the multiple integral $I\left( {u;\Omega } \right) = {\smallint _\Omega }F(Du(x))dx$ is a minimum of $I$ for variations of sufficiently small supports contained in $\Omega$.