Remarks on the Levi conditions for differential systems

Abstract

In this paper we prove two results on the Levi conditions for weakly hyperbolic systems with characteristics of constant multiplicities. A first result concerns scalar operators: we prove that Levi conditions defined by the second author in [29] are equivalent to the usual Levi conditions for scalar operator. A second result concerns systems whose principal symbol has a Jordan form made of a large number of $2 \times 2$ blocks. For these systems we express the first Levi condition via an invariant constructed from the sub-characteristic matrix. Moreover we show that this condition is necessary for the $C^\infty$ well-posedness.