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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A sharp version of Zhang's theorem
on truncating sequences of gradients


Author: Stefan Müller
Journal: Trans. Amer. Math. Soc. 351 (1999), 4585-4597
MSC (1991): Primary 49J45
Published electronically: July 21, 1999
MathSciNet review: 1675222
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Abstract: Let $K \subset \mathbf{R}^{mn}$ be a compact and convex set of $m \times n$ matrices and let $\{u_j\}$ be a sequence in $W_{\operatorname{loc}} ^{1,1}(\mathbf{R}^n;\mathbf{R}^m)$ that converges to $K$ in the mean, i.e. $\int _{\mathbf{R}^n} {\operatorname{dist}} (Du_j, K) \to 0$. I show that there exists a sequence $v_j$ of Lipschitz functions such that $\parallel\!\! {\operatorname{dist}} (Dv_j, K)\!\!\parallel _\infty \ \to 0$ and $\mathcal{L}^n (\{u_j \not= v_j\}) \to 0$. This refines a result of Kewei Zhang (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 313-326), who showed that one may assume $\parallel\!\!Dv_j\!\! \parallel _\infty \ \le C$. Applications to gradient Young measures and to a question of Kinderlehrer and Pedregal (Arch. Rational Mech. Anal. 115 (1991), 329-365) regarding the approximation of $\mathbf{R} \cup \{+\infty\}$ valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of $K$ can be replaced by quasiconvexity.


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Additional Information

Stefan Müller
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany
Email: sm@mis.mpg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02520-9
PII: S 0002-9947(99)02520-9
Keywords: Young measures, quasiconvexity, truncation
Received by editor(s): June 23, 1997
Published electronically: July 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society