Comparison of polynomial matrix differential operators

Abstract

We characterize matrix polynomials $P,Q$ such that the inequality $${\Vert Q(D)u\Vert }_{L^2}\le C{\Vert P(D)u\Vert }_{L^2}\text{for all }u\in C_c^{\infty }(\hat{I}\text{c◯}),$$ holds on bounded open sets $\hat{I}\text{c◯}$. We also characterize the operators $P,Q$ for which the linear continuous embedding above is compact, i.e., if $u_n\in C_c^{\infty }(\hat{I}\text{c◯})$ are such that $(P(D)u_n{)}_{n\ge 1}$ is bounded in $L^2$, then $(Q(D)u_n{)}_{n\ge 1}$ is strongly compact in $L^2$.

15 pages