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MiS Preprint
18/1997
A sharp version of Zhang's theorem on truncating sequences of gradients
Stefan Müller
Abstract
Let $K \subset R^{mn}$ be a compact and convex set of $m \times n$ matrices and let $\{u_j\}$ be a sequence in $W^{1,1}_{loc} (R^n:R^m)$ that converges to K in the mean, i.e. $\int_{R^n} dist(Du_j ,K) \to 0$. I show that there exists a sequence $v_j$ of Lipschitz functions such that $||dist(Dv_j,K)||_\infty \to 0$ and $L^n(\{u_j \ne v_j \}) \to 0$. This refines a result of Kewei Zhang (Ann. S.N.S. Pisa 19 (1992), 313-326) who showed that one may assume $||Dv_j||_\infty \leq C $. 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 $R \cup \{+ \infty\}$ valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of K can be replaced by quasiconvexity.
