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**Invariants of systems of differential operators and asymptotic formal sums.
(Invariants des systèmes d’opérateurs différentiels et sommes formelles asymptotiques.)**
*(French)*
Zbl 0941.35138

This paper is part of a general research program of the author, concerning well-posedness of the Cauchy problem in \(C^\infty\) and Gevrey classes for systems of linear partial differential equations, under Levi and Levi-Gevrey conditions, cf. the preceding contributions of the author [C. R. Acad. Sci., Paris, Sér. I 313, 227-230 (1991; Zbl 0755.35064), C. R. Acad. Sci., Paris, Sér. I 320, 1469-1474 (1995; Zbl 0836.35033), Pitman Res. Notes Math. Ser. 349, 209-229 (1996; Zbl 0868.35068)].

The present paper is devoted to the construction of asymptotic solutions, giving immediately necessary conditions for the \(C^\infty\) well-posedness. Necessary conditions in the Gevrey frame are announced to be studied in a future paper, whereas the construction of parametrices, proving the sufficiency, is sketched here.

The class of operators under examination is, in short, the following. One considers a system of first order linear partial differential operators, and assumes that the determinant of the principal symbol \(A\) has constant multiplicity: \(\text{det} A=H^mK\), where \(HK\) is strictly hyperbolic. The author studies in detail Levi conditions for the case of multiplicity \(m=5\), arriving to different subcases denoted by (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1) according to the structure of the matrix \(A\). In each subcase asymptotic solutions are constructed explicitly.

The present paper is devoted to the construction of asymptotic solutions, giving immediately necessary conditions for the \(C^\infty\) well-posedness. Necessary conditions in the Gevrey frame are announced to be studied in a future paper, whereas the construction of parametrices, proving the sufficiency, is sketched here.

The class of operators under examination is, in short, the following. One considers a system of first order linear partial differential operators, and assumes that the determinant of the principal symbol \(A\) has constant multiplicity: \(\text{det} A=H^mK\), where \(HK\) is strictly hyperbolic. The author studies in detail Levi conditions for the case of multiplicity \(m=5\), arriving to different subcases denoted by (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1) according to the structure of the matrix \(A\). In each subcase asymptotic solutions are constructed explicitly.

Reviewer: L.Rodino (Torino)

### MSC:

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

47F05 | General theory of partial differential operators |

47G30 | Pseudodifferential operators |