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

Leichtnam, Eric

Ann. Sci. Éc. Norm. Supér. (4) 23, No. 3, 369-443 (1990).

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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