On the complex constant rank condition and inequalities for differential operators

TL;DR

This paper investigates the complex constant rank condition for differential operators and explores its role in establishing coercive inequalities that generalize classical Korn and Sobolev inequalities on bounded domains.

Contribution

It introduces the complex constant rank condition for differential operators and analyzes its implications for inequalities relating different differential operators in bounded domains.

Findings

The complex constant rank condition ensures coercivity of certain differential inequalities.

Generalization of Korn's and Sobolev's inequalities under the complex constant rank framework.

Applicability to differential operators with constant coefficients on bounded domains.

Abstract

In this note, we study the complex constant rank condition for differential operators and its implications for coercive differential inequalities. These are inequalities of the form \[ \Vert A u \Vert_{L^p} \leq \Vert \mathscr{A} u \Vert_{L^q}, \] for exponents $1\leq p,q <\infty$ and homogeneous constant-coefficient differential operators $A$ and $\mathscr{A}$. The functions $u \colon \Omega \to \mathbb{R}^d$ are defined on open and bounded sets $\Omega \subset \mathbb{R}^N$ satisfying certain regularity assumptions. Depending on the order of $A$ and $\mathscr{A}$, such an inequality might be viewed as a generalisation of either Korn's or Sobolev's inequality, respectively. In both cases, as we are on bounded domains, we assume that the Fourier symbol of $\mathscr{A}$ satisfies an algebraic condition, the complex constant rank property.