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• W1606495360 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 $| {operatorname {dist}} (Dv_j, K)|_infty to 0$ 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 $|Dv_j |_infty le C$. 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." @default.
• W1606495360 created "2016-06-24" @default.
• W1606495360 creator A5017451822 @default.
• W1606495360 date "1999-07-21" @default.
• W1606495360 modified "2023-09-26" @default.
• W1606495360 title "A sharp version of Zhang’s theorem on truncating sequences of gradients" @default.
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• W1606495360 doi "https://doi.org/10.1090/s0002-9947-99-02520-9" @default.
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