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 $\mathcal{A}{v}_j\to \mathcal{A}v$ in $W^{-l}Y$ implies $F(v_j)⇝F(v)$ in $Z$, where $F$ is an $\mathcal{A}$-quasiaffine function and $⇝$ denotes an appropriate type of weak convergence. Here $Z$ is a local $L^1$-type space, either the space $\mathcal{M}$ of measures, or $L^1$, or the Hardy space ${\mathcal{H}}^1$; $X,Y$ 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 ${\mathcal{H}}^p$-bounds for the sequence $(F(v_j){)}_j$, for appropriate $p<1$, and new convergence results in the dual of Hölder spaces when $(v_j)$ is $\mathcal{A}$-free and lies in a suitable negative order Sobolev space $W^{-\beta ,s}$. The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.

arXiv comment39 pages. New examples added to show optimality of Theorem D

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Cited by in corpus (4)

• Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints
• On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_3$ Structure
• On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators
• A variational view on constitutive laws in parabolic problems