A Sharp Version of Zhang’s Theorem on Truncating Sequences of Gradients

Let K ⊂ Rmn be a compact and convex set of m×n matrices and let {uj} be a sequence in W 1,1 loc (Rn;Rm) that converges to K in the mean, i.e. ∫ Rn dist(Duj , K) → 0. I show that there exists a sequence vj of Lipschitz functions such that ‖dist(Dvj , K)‖∞ → 0 and L({uj 6= vj}) → 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 ‖Dvj ‖∞ ≤ 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 R ∪ {+∞} valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of K can be replaced by quasiconvexity. 1. Main results Let {uj} be a sequence of weakly differentiable functions uj : R → R whose gradients approach the ball B(0, R) in the mean, i.e. ∫ Rn dist(Duj , B(0, R))dx → 0. (1.1) Motivated by work of Acerbi and Fusco [1], [2], and Liu [13], Kewei Zhang showed that the sequence can be modified on a small set in such a way that the new sequence is uniformly Lipschitz. The following theorem is a slight variant of Lemma 3.1 in [21]. Theorem 1 (Zhang). There exists a constant c(n, m) with the following property. If (1.1) holds, then there exists a sequence of functions vj : R → R such that ||Dvj ||∞ ≤ c(n, m)R, L({uj 6= vj})→ 0. In fact one has the seemingly stronger conclusions L({uj 6= vj or Duj 6= Dvj}) → 0, ∫ Rn |Duj −Dvj |dx → 0. For the first conclusion it suffices to note that for weakly differentiable functions u and v the implication u = v a.e. in A =⇒ Du = Dv a.e. in A (1.2) Received by the editors June 23, 1997. 1991 Mathematics Subject Classification. Primary 49J45.