$L^{\infty }$-truncation of closed differential forms

Abstract

In this paper, we prove that for each closed differential form $u\in L^1({\mathbb{R}}^N,({\mathbb{R}}^N{)}^{\ast }\wedge ...\wedge ({\mathbb{R}}^N{)}^{\ast })$, which is almost in $L^{\infty }$ in the sense that \[ \int_{\{y \in \mathbb{R}^N \colon \vert u(y) \vert \geq L \}} \vert u(y) \vert dy < \varepsilon \] for some $L>0$ and a small $\epsilon >0$, we may find a closed differential form $v$, such that $\Vert u-v{\Vert }_{L^1}$ is again small, and $v$ is, in addition, in $L^{\infty }$ with a bound on its $L^{\infty }$ norm depending only on $N$ and $L$. In particular, the set $\{v\ne u\}$ has measure at most $C\epsilon$. We then look at applications of this theorem. We are able to prove that the $\mathcal{A}$-$p$-quasiconvex hull of a set does not depend on $p$. Furthermore, we can prove a classification theorem for $\mathcal{A}$-$\infty$-Young measures.

37 pages, 2 figures