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

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.