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

Abstract

In this paper, we prove that for each closed differential form $$\displaystyle u \in L^1(\mathbb {R}^N;(\mathbb {R}^N)^{*} \wedge ... \wedge (\mathbb {R}^N)^{*})$$ u ∈ L 1 ( R N ; ( R N ) ∗ ∧ . . . ∧ ( R N ) ∗ ) , which is almost in $$\displaystyle L^{\infty }$$ L ∞ in the sense that $$\begin{aligned} \int _{\{y \in \mathbb {R}^N :\vert u(y) \vert \ge L \}} \vert u(y) \vert dy< \varepsilon \end{aligned}$$ ∫ { y ∈ R N : | u ( y ) | ≥ L } | u ( y ) | d y < ε for some $$\displaystyle L>0$$ L > 0 and a small $$\displaystyle \varepsilon >0$$ ε > 0 , we may find a closed differential form v, such that $$\displaystyle \Vert u - v \Vert _{L^1}$$ ‖ u - v ‖ L 1 is again small, and v is, in addition, in $$\displaystyle L^{\infty }$$ L ∞ with a bound on its $$\displaystyle L^{\infty }$$ L ∞ norm depending only on N and L. In particular, the set $$\displaystyle \{ v \ne u\}$$ { v ≠ u } has measure at most $$\displaystyle CL^{-1} \varepsilon .$$ C L - 1 ε . As an application of this theorem, we are able to prove that the $$\displaystyle \mathcal {A}$$ A -p-quasiconvex hull of a set does not depend on p. Furthermore, we can prove a classification theorem for $$\displaystyle \mathcal {A}$$ A -$$\displaystyle \infty $$ ∞ -Young measures.

An alternative approach to solenoidal Lipschitz truncation

“…Consequently, Lipschitz truncation of a function u under a constraint A u = 0 is closely connected to the L ∞ -truncation of w (= Du) under some constraint Bw = 0. As mentioned before, the latter has been examined in [5,32] in greater detail; in particular (as above) the authors in [5] have conjectured that a low-regularity truncation is possible whenever B obeys the complex constant rank condition.…”

“…L ∞ -instead of W 1,∞ ), cf. [5,32] for a discussion of (T2b') and [21,22] for a discussion of (T2c') in that setting.…”

“…Corresponding to the W 1,p -case one might ask to truncate divergence-free L p functions to be in L ∞ (cf. [32]). While these questions turn out to be quite different, they both fit in a larger common framework.…”