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

Let

be a bounded starshaped domain and consider the energy functional

{\mathbb F}[u; \Omega]:= \int_\Omega {\bf F}(\nabla u(x)) \, dx, F [ u ; Ω ] : = ∫ Ω F ( ∇ u ( x ) )   d x ,

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 \text{ $a.e.$ in $\Omega$} \bigg\}, A p ( Ω ) = { u ∈ ξ ˉ x + W 0 1 , p ( Ω , R n ) : det ⁡ ∇ u = 1   a . e .  in  Ω } ,

with

p \in [1, \infty[ p ∈ [ 1 , ∞ [

,

\bar \xi \in {\mathbb M}_{n \times n} ξ ˉ ∈ M n × n

and

\det \bar \xi =1 det ⁡ ξ ˉ = 1

. In this short note we address the question of 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 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]