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Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

Author: Ali Taheri
Journal: Proc. Amer. Math. Soc. 131 (2003), 3101-3107
MSC (2000): Primary 49J10, 49J45
Published electronically: January 28, 2003
MathSciNet review: 1993219
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Abstract: Let $\Omega \subset {\mathbb R}^n$ be a bounded starshaped domain. In this note we consider critical points $\bar{u} \in \bar{\xi} y + W_0 ^{1,p} (\Omega ; {\mathbb R}^m)$ of the functional

\begin{displaymath}{\mathcal F}(u, \Omega) := \int_{\Omega} f( \nabla u(y)) \, dy, \end{displaymath}

where $f: {\mathbb R}^{m \times n} \to {\mathbb R}$ of class ${C}^1$ satisfies the natural growth

\begin{displaymath}\vert f (\xi)\vert \le c (1 + \vert \xi\vert^p) \end{displaymath}

for some $1 \le p < \infty$ and $c>0$, is suitably rank-one convex and in addition is strictly quasiconvex at $\bar{\xi} \in {\mathbb R}^{m \times n}$. We establish uniqueness results under the extra assumption that ${\mathcal F}$ is stationary at $\bar{u}$ with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).

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Additional Information

Ali Taheri
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany

Received by editor(s): July 31, 2001
Received by editor(s) in revised form: April 24, 2002
Published electronically: January 28, 2003
Communicated by: Bennett Chow
Article copyright: © Copyright 2003 American Mathematical Society