A simple construction of potential operators for compensated compactness

TL;DR

This paper presents a concise proof that homogeneous linear differential operators of constant rank have associated potential operators, refining previous results and providing insights into their structure for applications in compensated compactness.

Contribution

It offers a simplified proof and refinements showing that such operators admit potential operators, advancing theoretical understanding in the field.

Findings

Homogeneous linear differential operators of constant rank have potential operators.

Refined the original proof with a shorter, more direct argument.

Clarified the structure of kernels and images for these operators.

Abstract

We give a short proof of the fact that each homogeneous linear differential operator $A$ of constant rank admits a homogeneous potential operator $B$, meaning that $\ker A(x)=\mathrm{im\,}B(x)$ for $x\neq 0$. We make some refinements of the original result and some related remarks.