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.
Reviewer: L.Rodino (Torino)


35S05 Pseudodifferential operators as generalizations of partial differential operators
47F05 General theory of partial differential operators
47G30 Pseudodifferential operators