Compensated compactness: Continuity in optimal weak topologies

摘要

For l-homogeneous linear differential operators A of constant rank, we study the implication v(j) -> v in X Av(j) -> Av in W-l Y } double right arrow F(v(j)) (sic) F(v) in Z, where F is an A-quasiaffine function and (sic) denotes an appropriate type of weak convergence. Here Z is a local L-1-type space, either the space M of measures, or L-1, or the Hardy space H-1; X, Y are L-p-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of X, Y, Zare sharp. Analogous statements are also given in the case when F(v) is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove H-p-bounds for the sequence (F(v(j)))(j), for appropriate p < 1, and new convergence results in the dual of Holder spaces when (v(j)) is A-free and lies in a suitable negative order Sobolev space W--beta,W-s. The choice of these Holder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians. (C) 2022 Elsevier Inc. All rights reserved.