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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Email: taheri@mis.mpg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06852-7
PII: S 0002-9939(03)06852-7
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