Quasiconvexity and Uniqueness of Stationary Points on a Space of Measure Preserving Maps

Let $\Omega \subset {\mathbb R}^n$ be a bounded starshaped domain and consider the energy functional $${\mathbb F}[u; \Omega]:= \int_\Omega {\bf F}(\nabla u(x)) \, dx,$$ over the space of measure preserving maps {\mathcal A}_p(\Omega)=\bigg\{u \in \bar \xi x + W_0^{1,p}(\Omega, {\mathbb R}^n): \det \nabla u = 1 \mbox{ $a.e.$ in $\Omega$} \bigg\}, 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]