Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

Let $\Omega \subset {\mathbb R}^n$ be a bounded starshaped domain. In this note we consider critical points $\bar{u} \in \bar{\xi} y + W_0 ^{1,p} (\Omega ; {\mathbb R}^m)$ of the functional

where $f: {\mathbb R}^{m \times n} \to {\mathbb R}$ of class ${C}^1$ satisfies the natural growth

for some $1 \le p < \infty$ and $c>0$, is suitably rank-one convex and in addition is strictly quasiconvex at $\bar{\xi} \in {\mathbb R}^{m \times n}$. We establish uniqueness results under the extra assumption that ${\mathcal F}$ is stationary at $\bar{u}$ with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).