Compensated compactness: continuity in optimal weak topologies

Abstract

For $l$-homogeneous linear differential operators $\mathcal{A}$ of constant rank, we study the implication $v_j\rightharpoonup v$ in $X$ and $\mathcal{A} v_j\rightarrow \mathcal{A} v$ in $W^{-l}Y$ implies $F(v_j)\rightsquigarrow F(v)$ in $Z$, where $F$ is an $\mathcal{A}$-quasiaffine function and $\rightsquigarrow$ denotes an appropriate type of weak convergence. Here $Z$ is a local $L^1$-type space, either the space $\mathscr{M}$ of measures, or $L^1$, or the Hardy space $\mathscr{H}^1$; $X,\, Y$ are $L^p$-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of $X,\,Y,\,Z$ are 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 $\mathscr{H}^p$-bounds for the sequence $(F(v_j))_j$, for appropriate $p<1$, and new convergence results in the dual of H\"older spaces when $(v_j)$ is $\mathcal{A}$-free and lies in a suitable negative order Sobolev space $W^{-\beta,s}$. The choice of these H\"older spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.