Reflection of ${\mathrm{C}}^{\infty }$ Singularities for a Class of Operators With Multiple Characteristics

Abstract

Let $P,$ a partial differential operator with real principal part and constant multiplicity characteristics. We have shown in some previous works [1], [2] that several results about operators with simple characteristics can be generalized to these operators with multiple characterics if we add an hypothesis, named Levi's condition, on the lower order terms. For instance, there is still propagation along bicharacteristics for the singularities of distributions $u$ such that $Pu\in {\mathrm{C}}^{\infty }(\Omega )$; if moreover the principal part of $P$ is hyperbolic, the Cauchy problem is well posed in the ${\mathrm{C}}^{\infty }$ setting.

In this paper, we shall give a generalization to such operators of the theorem of Lax–Nirenberg [6] concerning the reflection at the boundary $\partial \Omega$ of the singularities of distributions $u$ satisfying $Pu\in {\mathrm{C}}^{\infty }(\bar{\Omega })$. In order to get a more precise result of regularity up to the boundary, we shall differ from Nirenberg's proof in the details, but the principle will be the same. That is, we factorise micro locally $P$ in an elliptic and two hyperbolic factors satisfying the Levi condition; then the proof of the theorem will be reduce to the study of micro local regularity results for elliptic and hyperbolic boundary problems.