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

Shahrokhi-Dehkordi, M S and Taheri, A (2010) Quasiconvexity and uniqueness of stationary points in a space of measure preserving maps. Journal of Convex Analysis, 17 (1). 069-079. ISSN 0944-6532.

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## Abstract

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}(\nabla 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 \nabla 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].

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Mohammad Shahrokhi-Dehkordi 06 Feb 2012 20:38 11 Apr 2012 07:50 http://sro.sussex.ac.uk/id/eprint/27102