On the complex constant rank condition and inequalities for differential operators

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.