Abstract

Let Ω⊂Rn\Omega \subset {\mathbb R}^n be a bounded starshaped domain and consider the energy functional F[u;Ω]:=∫ΩF(∇u(x)) dx,{\mathbb F}[u; \Omega]:= \int_\Omega {\bf F}(\nabla u(x)) \, dx, over the space of measure preserving maps Ap(Ω)={u∈ξˉx+W01,p(Ω,Rn):det⁡∇u=1 a.e. in Ω},{\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\}, with p∈[1,∞[p \in [1, \infty[, ξˉ∈Mn×n\bar \xi \in {\mathbb M}_{n \times n} and det⁡ξˉ=1\det \bar \xi =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]

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Mohammad Sadegh Shahrokhi-Dehkordi

Department of Mathematics, University of Sussex, Falmer BN1 9RF, England

References

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Suggested citation

M. S. Shahrokhi-Dehkordi, A. Taheri. “Quasiconvexity and Uniqueness of Stationary Points on a Space of Measure Preserving Maps.” Journal of Convex Analysis 17 (2010), No. 1, 69–79. https://doi.org/10.68381/jca17006

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.