## Le problème de Cauchy ramifié. (The ramified Cauchy problem).(French)Zbl 0717.35018

The author considers the Cauchy problem $(1)\quad a(x,D)u(x)=v;\quad D^ h_{x_ 0}u(x)|_ S=u_ h(x'),\quad 0\leq h<m,\quad x=(x_ 0,x')\in C^{n+1},$ where a(x,D) is a differential operator of order m with multiple characteristics and S is the hypersurface of equation $$x_ 0=0.$$
The main result of the paper is the following
Theorem: Let U be a neighbourhood of $$0\in C^{n+1}$$ and v a holomorphic function defined on a universal covering of U without the union of the characteristic surfaces. Then there exists a neighbourhood $$\Omega\subseteq U$$ of 0 and a solution of problem (1), holomorphic on the universal covering of $$\Omega$$ without the characteristic surfaces. Furthermore, if a(x,D) has simple characteristics and the data belong to the Nilsson class then the solution belongs to the Nilsson class.
Reviewer: R.Salvi

### MSC:

 35G10 Initial value problems for linear higher-order PDEs 35A20 Analyticity in context of PDEs

### Keywords:

Cauchy problem; multiple characteristics; Nilsson class
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### References:

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