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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact and convex set of matrices and let be a sequence in that converges to in the mean, i.e. . I show that there exists a sequence of Lipschitz functions such that and . 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 . 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 valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of 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:
https://doi.org/10.1090/S0002-9947-99-02520-9

Keywords:
Young measures,
quasiconvexity,
truncation

Received by editor(s):
June 23, 1997

Published electronically:
July 21, 1999

Article copyright:
© Copyright 1999
American Mathematical Society