M\"uller-Zhang truncation for general linear constraints with first or second order potential

Abstract

Let $\mathcal{B}$ be a homogeneous differential operator of order $l=1$ or $l=2$. We show that a sequence of functions of the form $(\mathcal{B}u_j)_j$ converging in the $L^1$-sense to a compact, convex set $K$ can be modified into a sequence converging uniformly to this set provided that the derivatives of order $l$ are uniformly bounded. We prove versions of our result on the whole space, an open domain, and for $K$ varying uniformly continuously on an open, bounded domain. This is a conditional generalization of a theorem proved by S. M\"uller for sequences of gradients, cf. [6]. Moreover, a potential of order two for the linearized isentropic Euler system is constructed.