A sharp version of Zhang's theorem on truncating sequences of gradients
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Submission date: 31. Jul. 1997
published in: Transactions of the American Mathematical Society, 351 (1999) 11, p. 4585-4597
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Let be a compact and convex set of matrices and let be a sequence in that converges to K 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. S.N.S. Pisa 19 (1992), 313-326) who showed that one may assume . Applications to gradient Young measures and to a question of Kinderlehrer and Pedregal (Arch. Rat. 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 K can be replaced by quasiconvexity.