MiS Preprint Repository

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a starshaped domain. In this note we consider critical points $\bar{u}\in \xi x=W_0^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m)$ of the functional $$F(u,\mathrm{\Omega }):={\int }_{\mathrm{\Omega }}f(\mathrm{\nabla }u(x))dx$$ where $f:{\mathbb{R}}^{m\times n}\to \mathbb{R}$ of class $C^1$, is suitably rank-one convex, and in addition strictly quasiconvex at $\xi \in R^{m\times n}$. We establish uniqueness results under the extra assumption that $F$ is stationary at $\bar{u}$ with respect to variations of the domain. These results should be compared to the ones by Knops and Stuart and recent counterexamples produced by Müller and Sverak.