Quasiconvexity and uniqueness of stationary points in a space of measure preserving maps

Let $\Omega \subset {\mathbb R}^n$ be a bounded starshaped domain and consider the energy functional \begin{equation*} {\mathbb F}[u; \Omega] := \int_\Omega {\bf F}(
abla u(x)) \, dx, \end{equation*} over the space of measure preserving maps \begin{equation*} {\mathcal A}_p(\Omega)=\bigg\{u \in \bar \xi x + W_0^{1,p}(\Omega, {\mathbb R}^n) : \det
abla u = 1 \mbox{ $a.e.$ in $\Omega$} \bigg\}, \end{equation*} with $p \in [1, \infty[$, $\bar \xi \in {\mathbb M}_{n \times n}$ and $\det \bar \xi =1$. In this short note we address the question of {\it uniqueness} for solutions of the corresponding system of Euler-Lagrange equations. In particular we give a new proof of the celebrated result of R. J. Knops and C. A. Stuart [Arch. Rational Mech. Anal. 86, No. 3 (1984) 233--249] using a method based on {\it comparison} with homogeneous degree-one extensions as introduced by the second author in his recent paper "Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations" [Proc. Amer. Math. Soc. 131, (2003) 3101--3107].